**4. Generation of Analytic and** *C***∞-Semigroups**

In this section, under the assumption (2) on the order of the symbols, we will prove the main result of this paper (Theorem 1). For that we will need to estimate the norm *bλ*(*D*)*uW<sup>k</sup> <sup>p</sup>* (R*n*,C*q*) .

Let *<sup>A</sup>*(*D*) be <sup>Λ</sup>-elliptic with *rij* <sup>&</sup>gt; 0, *<sup>ρ</sup>* <sup>≥</sup> *<sup>ρ</sup>n*, and suppose that the assumption (2) holds. Then, note that *rij* + *rji* = *ri* + *rj* for *i*, *j* = 1, ... , *q*. Moreover, let *r*<sup>+</sup> := max 1≤*i*≤*q* {*ri*},

$$\tau r\_- := \min\_{1 \le i,j \le q} \{ r\_{ij} \}, \omega \ge 1 \text{ and}$$

$$\Lambda\_{\omega} := \Lambda(\theta)\_{\omega} := \{ \lambda \in \Lambda = \Lambda(\theta) : |\lambda| \ge \omega \}.$$

Note that for *<sup>b</sup>λ*, as in corollary 1, *<sup>u</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>∞</sup> *<sup>b</sup>* (R*n*, <sup>C</sup>*q*) <sup>∩</sup> *<sup>W</sup><sup>k</sup> <sup>p</sup>*(R*n*, <sup>C</sup>*q*) and *<sup>β</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>0</sup> , we have

$$\begin{split} \partial\_{\mathbf{x}}^{\frac{\delta}{2}} (b\_{\lambda}(D)\boldsymbol{u})(\mathbf{x}) &= \operatorname{Os} - \iint \operatorname{el}^{i\overline{\mathbf{x}} \cdot \boldsymbol{\eta}} b\_{\lambda}(\overline{\boldsymbol{\xi}}) (\partial\_{\mathbf{x}}^{\overline{\mathbf{c}}} \boldsymbol{u}) (\mathbf{x} - \boldsymbol{\eta}) \frac{d (\overline{\boldsymbol{\xi}}, \boldsymbol{\eta})}{(2\pi)^{n}} \\ &= \lim\_{\boldsymbol{\mathfrak{x}} \searrow \boldsymbol{0}} \int \operatorname{K}\_{\boldsymbol{\mathfrak{t}}} (\boldsymbol{\eta}, \lambda) (\partial\_{\mathbf{x}}^{\overline{\mathbf{c}}} \boldsymbol{u}) (\mathbf{x} - \boldsymbol{\eta}) d \boldsymbol{\eta} \end{split} \tag{8}$$

with

$$K\_{\varepsilon}(\eta,\lambda) := \int\_{\mathbb{R}^n} e^{i\frac{\pi}{\varepsilon}\cdot\eta} \chi\_{\varepsilon}(\xi,\eta;\lambda) b\_{\lambda}(\xi) \frac{d\xi}{(2\pi)^n} \,, \tag{9}$$

and

$$\chi\_{\varepsilon}(\tilde{\mathfrak{s}}, \eta; \lambda) := \chi\_{\varepsilon}(\tilde{\mathfrak{s}}; \lambda)\psi\_{\varepsilon}(\eta)$$

for *<sup>ξ</sup>*, *<sup>η</sup>* <sup>∈</sup> <sup>R</sup>*n*, 0 <sup>&</sup>lt; *<sup>ε</sup>* <sup>&</sup>lt; 1, where *<sup>ψ</sup>* is a function in <sup>S</sup>(R*n*) with *<sup>ψ</sup>*(0) <sup>=</sup> 1, *ψε*(*η*) :<sup>=</sup> *<sup>ψ</sup>*(*εη*), *χε*(*ξ*; *λ*) := *ϕε* (|*ξ*| <sup>2</sup> <sup>+</sup> <sup>|</sup>*λ*<sup>|</sup> 2/*r*<sup>+</sup> )1/2 with *ϕε*(*x*) :<sup>=</sup> *<sup>ϕ</sup>*(*εx*) for *<sup>x</sup>* <sup>∈</sup> <sup>R</sup> and *<sup>ϕ</sup>* ∈ S(R) satisfies *ϕ*(0) = 1.

It was proven in [20] (p. 845) that for *<sup>α</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>0</sup> , there exists a constant *<sup>C</sup><sup>α</sup>* <sup>&</sup>gt; 0 such that for all *<sup>ξ</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* and *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>*ω*,

$$\left|\partial\_{\xi}^{a}\chi\_{\varepsilon}(\xi;\lambda)\right| \leq \mathbb{C}\_{\mathfrak{a}}\left(|\xi|^{2} + |\lambda|^{2/r^{+}}\right)^{-|a|/2} \qquad (0 < \varepsilon < 1).$$

Now, due to

$$\begin{aligned} \frac{\omega^{2/r^+} + 1}{\omega^{2/r^+}} (|\xi|^2 + |\lambda|^{2/r^+}) &\geq \frac{1}{\omega^{2/r^+}} (\omega^{2/r^+} |\xi|^2 + \omega^{2/r^+} |\lambda|^{2/r^+} + |\lambda|^{2/r^+})\\ &\geq |\xi|^2 + |\lambda|^{2/r^+} + 1, \end{aligned}$$

we have

$$\left|\partial\_{\xi}^{a}\chi\_{\varepsilon}(\xi;\lambda)\right| \leq \overline{\mathfrak{C}}\_{a}\left\langle\xi,\left|\lambda\right|^{1/r^{+}}\right\rangle^{-|a|} \qquad (0 < \varepsilon < 1). \tag{10}$$

We will obtain some estimate for *K<sup>ε</sup>* with help of (10) and the following lemma and remark.

**Lemma 3** ([15], Lemma 6.3)**.** *Let <sup>χ</sup>* ∈ S(R*n*) *with <sup>χ</sup>*(0) <sup>=</sup> <sup>1</sup>*. Then:*

(a) *<sup>χ</sup>*(*εx*) −→*ε*<sup>0</sup> <sup>1</sup> *uniformly on all compact subset of* <sup>R</sup>*n.*

(b) *∂<sup>α</sup> <sup>x</sup>χ*(*εx*) −→*ε*<sup>0</sup> <sup>0</sup> *uniformly on* <sup>R</sup>*n, if <sup>α</sup>* <sup>=</sup> <sup>0</sup>*.*

(c) *For all <sup>α</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>0</sup> *, there exists some C<sup>α</sup>* <sup>&</sup>gt; <sup>0</sup>*, independent on* <sup>0</sup> <sup>&</sup>lt; *<sup>ε</sup>* <sup>&</sup>lt; <sup>1</sup>*, such that*

$$|\partial\_x^{\mathfrak{a}}\chi(\varepsilon x)| \le C\_{\mathfrak{a}}\langle x\rangle^{-(|\mathfrak{a}|-\sigma)} \text{ for all } \mathfrak{x} \in \mathbb{R}^n \text{ and } 0 \le \sigma \le |\mathfrak{a}|.$$

**Remark 3.** *Note that, if* <sup>1</sup> <sup>&</sup>lt; *<sup>r</sup>*−*, then for all* <sup>1</sup> <sup>≤</sup> *<sup>δ</sup>* <sup>&</sup>lt; *<sup>r</sup>*−*, we obtain* <sup>&</sup>lt; *<sup>δ</sup>* <sup>+</sup> *rij* <sup>≤</sup> *rji* <sup>+</sup> *rij* <sup>=</sup> *ri* <sup>+</sup> *rj (for all i*, *j),*

*and*

$$\begin{split} \left\langle |\lambda|^{1/r^+} \mathfrak{F}\_{\prime} |\lambda|^{1/r\_i} \right\rangle^{-r\_i} &\leq \frac{1}{\left( |\lambda|^{2/r^+} |\xi|^2 + |\lambda|^{2/r\_i} \right)^{r\_i/2}} \\ &\leq \frac{1}{\left( |\lambda|^{2/r^+} |\xi|^2 + |\lambda|^{2/r^+} \right)^{r\_i/2}} \\ &= |\lambda|^{-r\_i/r^+} \langle \xi \rangle^{-r\_i} \end{split} \tag{11}$$

*for all <sup>ξ</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup> and <sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>*ω. Moreover, <sup>σ</sup>* :<sup>=</sup> *<sup>r</sup>*<sup>−</sup> *<sup>r</sup>*<sup>+</sup> <sup>∈</sup> (1/2*r*+, 1] *and <sup>μ</sup>* :<sup>=</sup> <sup>|</sup>*λ*<sup>|</sup> 1/*r*<sup>+</sup> *, with λ* ∈ Λ*ω, satisfies*

$$\mu^{-r\_{ij}} = \frac{1}{|\lambda|^{r\_{ij}/r^+}} \le \frac{1}{|\lambda|^{\sigma}} \quad \text{for all } i, j.$$

Now, we will establish a key lemma for the generation of analytic semigroup. In the lemma, *σ* and *μ* are as in Remark 3.

**Lemma 4.** *Let* <sup>1</sup> <sup>2</sup> <sup>≤</sup> *<sup>δ</sup>* <sup>&</sup>lt; min{1,*r*−} *and K<sup>ε</sup> as in* (9)*. Then:*

(a) *There exists a constant C* <sup>&</sup>gt; <sup>0</sup> *such that for all <sup>ε</sup>* <sup>∈</sup> (0, 1)*, <sup>η</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup> and <sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>*<sup>ω</sup> it holds*

$$\|\left(1+|\mu\eta|\right)|\mu\eta|^{n}\|\mathcal{K}\_{\varepsilon}(\eta,\lambda)\|\_{\mathcal{L}(\mathbb{C}^{q})} \leq \frac{\mathsf{C}}{|\lambda|^{\sigma}}\mu^{n}|\mu\eta|^{\delta}.\tag{12}$$

(b) *There exists a strongly measurable function <sup>K</sup>* : <sup>R</sup>*<sup>n</sup>* <sup>×</sup> <sup>Λ</sup>*<sup>ω</sup>* → L(C*q*) *with <sup>K</sup>ε*(*η*, *<sup>λ</sup>*) <sup>→</sup> *K*(*η*, *λ*) (*ε* 0) *pointwise, and the estimate* (12) *holds with K<sup>ε</sup> being replaced by K. In consequence there exists a constant M* > <sup>0</sup>*, independent on <sup>λ</sup>, such that*

$$\|K(\cdot,\lambda)\|\_{L^{1}(\mathbb{R}^{n},\mathcal{L}(\mathbb{C}^{\mathbb{J}}))} \leq \frac{M}{|\lambda|^{\sigma}} \quad \forall \lambda \in \Lambda\_{\omega}.\tag{13}$$

**Proof.** (a) First, note that with the change *ξ* → *μξ* we obtain

$$K\_{\varepsilon}(\eta,\lambda) = \mu^n \int\_{\mathbb{R}^n} e^{i\mu\_{\mathbb{S}}^\pi \cdot \eta} \chi\_{\varepsilon}(\mu\_{\mathbb{S}}^\pi,\eta;\lambda) b\_{\lambda}(\mu\_{\mathbb{S}}^\pi) \frac{d\xi}{(2\pi)^n}.$$

Note also that, for *<sup>α</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>0</sup> with 0 <sup>&</sup>lt; <sup>|</sup>*α*<sup>|</sup> <sup>≤</sup> *<sup>ρ</sup>*, it holds

$$\int\_{\mathbb{R}^n} D\_{\xi}^{\alpha} \left( \chi\_{\varepsilon} (\mu \xi, \eta; \lambda) b\_{\lambda} (\mu \xi) \right) \frac{d\xi}{\left( 2\pi \right)^n} = 0.$$

With this, *eiμξ*·*<sup>η</sup>* <sup>−</sup> <sup>1</sup> ≤ 2|*μξ*| *δ* |*η*| *<sup>δ</sup>* for all *<sup>ξ</sup>*, *<sup>η</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* and *<sup>δ</sup>* <sup>∈</sup> (0, 1), partial integration, Leibniz rule, (10), corollary 1, Lemma 3, and Remark 3, we obtain for all *<sup>α</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>0</sup> with <sup>|</sup>*α*<sup>|</sup> <sup>=</sup> *<sup>n</sup>* <sup>+</sup> *<sup>l</sup>*, *<sup>l</sup>* <sup>=</sup> 0, 1, and <sup>1</sup> <sup>2</sup> <sup>≤</sup> *<sup>δ</sup>* <sup>&</sup>lt; min{1,*r*−}, that

 (*μη*) *α Kε*(*η*, *λ*) <sup>L</sup>(C*q*) = *μn* R*n e <sup>i</sup>μξ*·*ηD<sup>α</sup> <sup>ξ</sup>* (*χε*(*μξ*, *<sup>η</sup>*; *<sup>λ</sup>*)*bλ*(*μξ*)) *<sup>d</sup><sup>ξ</sup>* (2*π*) *n* <sup>L</sup>(C*q*) = *μn* R*n* (*e <sup>i</sup>μξ*·*<sup>η</sup>* <sup>−</sup> <sup>1</sup>)*ψ*(*εη*)*D<sup>α</sup> <sup>ξ</sup>* (*χε*(*μξ*; *<sup>λ</sup>*)*bλ*(*μξ*)) *<sup>d</sup><sup>ξ</sup>* (2*π*) *n* <sup>L</sup>(C*q*) <sup>≤</sup> *<sup>μ</sup><sup>n</sup>* R*n* 2|*μξ*| *δ* |*η*| *δ* |*ψ*(*εη*)| ∑ *γ*≤*α Cγα Dα*−*<sup>γ</sup> <sup>ξ</sup>* (*χε*(*μξ*; *λ*)) *∂ γ <sup>ξ</sup>* (*bλ*(*μξ*)) <sup>L</sup>(C*q*) *dξ* <sup>≤</sup> <sup>2</sup>*μn*|*μη*<sup>|</sup> *δ* R*n* |*ξ*| *<sup>δ</sup>* ∑ *γ*≤*α Cγαμ*|*α*<sup>|</sup> *Cα*−*<sup>γ</sup>μξ*, *μ* |*γ*|−|*α*| (*∂ γ <sup>ξ</sup> bλ*)(*μξ*) <sup>L</sup>(C*q*) *dξ* <sup>≤</sup> <sup>2</sup>*μn*|*μη*<sup>|</sup> *δ* R*n* |*ξ*| *δ* ∑ *i*,*j* ∑ *γ*≤*α Cγαμ*|*α*<sup>|</sup> *Cα*−*<sup>γ</sup>μξ*, *μ* |*γ*|−|*α*| (*∂ γ <sup>ξ</sup> gij*(*ξ*; *λ*))(*μξ*) *dξ* <sup>≤</sup> *<sup>C</sup>μn*|*μη*<sup>|</sup> *δ* R*n* |*ξ*| *δ* \* *q* ∑ *i*=1 *<sup>C</sup>*0*αμ*|*α*<sup>|</sup> *μξ*, *μ* −|*α*| + *μξ*, |*λ*| 1/*ri* ,−*ri* + *q* ∑ *j*=1 ∑ Z1 *<sup>C</sup>αμ*|*α*<sup>|</sup> *μξ*, *μ* |*γ*|−|*α*| *μξ rj*−|*γ*| + *μξ*, |*λ*| 1/*ri* ,−*ri* + *μξ*, |*λ*| 1/*rj* ,−*rj* + ∑ Z2 *<sup>C</sup>γαμ*|*α*<sup>|</sup> *μξ*, *μ* |*γ*|−|*α*| *μξ rij*−|*γ*| + *μξ*, |*λ*| 1/*ri* ,−*ri* + *μξ*, |*λ*| 1/*rj* ,−*rj dξ* (11) <sup>≤</sup> *<sup>C</sup>μn*|*μη*<sup>|</sup> *δ* R*n* |*ξ*| *δ* \* *q* ∑ *i*=1 *<sup>C</sup>*0*αμ*|*α*<sup>|</sup> |*μξ*, *μ*| −|*α*| *<sup>μ</sup>*−*ri<sup>ξ</sup>* −*ri* + *q* ∑ *j*=1 ∑ Z1 *<sup>C</sup>αμ*|*α*<sup>|</sup> *μξ rj*−|*α*| *<sup>μ</sup>*−*ri<sup>ξ</sup>* −*ri μ*−*rj ξ* −*rj* + ∑ Z2 *<sup>C</sup>γαμ*|*α*<sup>|</sup> *μξ rij*−|*α*| *<sup>μ</sup>*−*ri<sup>ξ</sup>* −*ri μ*−*rj ξ* −*rj dξ* <sup>≤</sup> *<sup>C</sup>μn*|*μη*<sup>|</sup> *δ* \* 1 |*λ*| *σ q* ∑ *i*=1 R*n ξ* −(*ri*+*n*+*l*−*δ*) *dξ* \$ %& ' <sup>=</sup>:*Cil*<<sup>∞</sup> + *q* ∑ *i*,*j*=1 R*n μn*+*l*−*ri*−*rj* |*ξ*| *δ μξ rij*−(*n*+*l*) *ξ* −*ri*−*rj dξ* \$ %& ' <sup>=</sup>:*Iijl* - .

Let <sup>Ω</sup><sup>1</sup> :<sup>=</sup> {*<sup>ξ</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* : <sup>|</sup>*ξ*<sup>|</sup> <sup>&</sup>lt; <sup>1</sup>}, <sup>Ω</sup><sup>2</sup> :<sup>=</sup> <sup>R</sup>*n*\Ω<sup>1</sup> and

$$I\_{ijl}^{(k)} := \int\_{\Omega\_k} \mu^{n+l-r\_i-r\_j} |\mathfrak{J}|^{\delta} \langle \mu \mathfrak{J} \rangle^{r\_{ij}-(n+l)} \langle \mathfrak{J} \rangle^{-r\_i-r\_j} d\mathfrak{J}^{\delta}$$

for *k* = 1, 2. Since *Iijl* = *I* (1) *ijl* + *I* (2) *ijl* , we will estimate *I* (*k*) *ijl* . We consider two cases: Case 1. If *rij* ≤ *n* + *l* for some *i*, *j*, it holds

$$\begin{aligned} I\_{ijl}^{(1)} &\leq \int\_{\Omega\_1} \mu^{n+l-r\_i-r\_j} |\xi|^\delta \mu^{r\_{ij}-(n+l)} |\xi|^{r\_{ij}-(n+l)} d\xi \\ &= \mu^{-r\_{\vec{\mu}}} \int\_{\Omega\_1} |\xi|^{\delta+r\_{ij}-(n+l)} d\xi \leq \frac{\mathsf{C}}{|\lambda|^{\mathsf{C}}} \end{aligned}$$

since <sup>1</sup> <sup>2</sup> <sup>≤</sup> *<sup>δ</sup>* <sup>&</sup>lt; *rij* (thus *<sup>δ</sup>* <sup>+</sup> *rij* <sup>&</sup>gt; 1). Furthermore,

$$\begin{split} I\_{ijl}^{(2)} &\leq \int\_{\Omega} \mu^{n+l-r\_i-r\_j} |\mathfrak{F}|^\delta \mu^{r\_{ij}-(n+l)} |\mathfrak{F}|^{r\_{ij}-(n+l)} |\mathfrak{F}|^{-r\_i-r\_j} d\mathfrak{F} \\ &= \mu^{-r\_{ji}} \int\_{\Omega} |\mathfrak{F}|^{\delta-r\_{ji}-n-l} d\mathfrak{F} \leq \frac{\mathsf{C}}{|\lambda|^{\sigma}}, \end{split}$$

due to *<sup>δ</sup>* <sup>&</sup>lt; *rji* <sup>+</sup> *<sup>l</sup>* for *<sup>l</sup>* <sup>=</sup> 0, 1. Therefore,

$$I\_{ijl} \le \frac{\widetilde{\mathbb{C}}}{|\lambda|^{\sigma}} \qquad (l = 0, 1). \tag{14}$$

Case 2. Suppose *rij* <sup>&</sup>gt; *<sup>n</sup>* <sup>+</sup> *<sup>l</sup>* for some *<sup>i</sup>*, *<sup>j</sup>*. Since *<sup>μ</sup>* <sup>≥</sup> 1, then we get

$$\begin{split} I\_{ijl}^{(1)} &\leq \int\_{\Omega\_{1}} \mu^{n+l-r\_{i}-r\_{j}} |\mathfrak{f}|^{\delta} \left( 1+\mu^{2} |\mathfrak{f}|^{2} \right)^{\frac{r\_{ij}-(n+l)}{2}} d\mathfrak{f} \\ &\leq \int\_{\Omega\_{1}} \mu^{n+l-r\_{i}-r\_{j}} |\mathfrak{f}|^{\delta} 2^{\frac{r\_{ij}-n-l}{2}} \mu^{r\_{ij}-n-l} d\mathfrak{f} \\ &= 2^{\frac{r\_{ij}-n-l}{2}} \mu^{-r\_{ji}} \int\_{\Omega\_{1}} |\mathfrak{f}|^{\delta} d\mathfrak{f} \leq \frac{\mathbb{C}}{|\lambda|^{\sigma}}. \end{split}$$

Moreover,

$$\begin{split} I\_{ijl}^{(2)} &= \int\_{\Omega} \mu^{n+l-r\_i-r\_j} |\mathfrak{f}|^{\delta} \left( 1 + \mu^2 |\mathfrak{f}|^2 \right)^{\frac{r\_{ij}^{\*}-(n+l)}{2}} \langle \mathfrak{f} \rangle^{-r\_i-r\_j} d\mathfrak{f} \\ &\leq \int\_{\Omega} \mu^{n+l-r\_i-r\_j} |\mathfrak{f}|^{\delta} |\mathfrak{f}|^{r\_{ij}-(n+l)} \left( 1 + \mu^2 \right)^{\frac{r\_{ij}^{\*}-(n+l)}{2}} |\mathfrak{f}|^{-r\_i-r\_j} d\mathfrak{f} \\ &\leq \int\_{\Omega} \mu^{n+l-r\_i-r\_j} |\mathfrak{f}|^{\delta-r\_{ji}-(n+l)} 2^{\frac{r\_{ij}^{\*}-n-l}{2}} \mu^{r\_{ij}-n-l} d\mathfrak{f} \\ &= 2^{\frac{r\_{ij}-n-l}{2}} \mu^{-r\_{ji}} \int |\mathfrak{f}|^{\delta-r\_{ji}-n-l} d\mathfrak{f} \leq \frac{C}{|\lambda|^{\sigma}}. \end{split}$$

Thus, (14) holds too. In consequence

$$\left\| \left( (\mu\eta)^{a} K\_{\varepsilon}(\eta,\lambda) \right) \right\|\_{\mathcal{L}(\mathbb{C}^{4})} \leq \frac{\mathsf{C}}{|\lambda|^{\varepsilon}} \mu^{n} |\mu\eta|^{\delta}$$

for all *<sup>η</sup>* <sup>∈</sup> <sup>R</sup>*n*, *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>*ω*, *<sup>α</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>0</sup> with <sup>|</sup>*α*<sup>|</sup> <sup>=</sup> *<sup>n</sup>* <sup>+</sup> *<sup>l</sup>*, *<sup>l</sup>* <sup>=</sup> 0, 1, *<sup>ε</sup>* <sup>∈</sup> (0, 1) and <sup>1</sup> <sup>2</sup> <sup>≤</sup> *<sup>δ</sup>* <sup>&</sup>lt; min{1,*r*−}. Therefore, we have

$$\|\mu\eta|^{n+l} \|\mathcal{K}\_{\varepsilon}(\eta,\lambda)\|\_{\mathcal{L}(\mathbb{C}^{\mathfrak{q}})} \le n^{\frac{n+l}{2}} \sum\_{|a|=n+l} \left\|(\mu\eta)^{a} \mathcal{K}\_{\varepsilon}(\eta,\lambda)\right\|\_{\mathcal{L}(\mathbb{C}^{\mathfrak{q}})} \le \frac{\mathcal{C}}{|\lambda|^{\sigma}} \mu^{n} |\mu\eta|^{\delta}.$$

Adding these inequalities for *l* = 0 and *l* = 1, we obtain the assertion (a). (b) Let *<sup>ε</sup>*,*ε* <sup>∈</sup> (0, 1), *<sup>η</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* and *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>*ω*. From the proof of (a) we see that

$$\begin{split} \left(\mu\eta\right)^{a} \left(K\_{\varepsilon}(\eta,\lambda) - K\_{\varepsilon'}(\eta,\lambda)\right) \\ = & \mu^{n} \int\_{\mathbb{R}^{n}} (e^{i\mu\overline{\xi}\cdot\eta} - 1) D\_{\xi}^{a} \left[ \left(\chi\_{\varepsilon}(\mu\overline{\xi},\eta;\lambda) - \chi\_{\varepsilon'}(\mu\overline{\xi},\eta;\lambda)\right) b\_{\lambda}(\mu\overline{\xi}) \right] \frac{d\overline{\xi}}{\left(2\pi\right)^{n}}.\end{split} \tag{15}$$

From Lemma 3 we know that *D<sup>γ</sup> <sup>ξ</sup>* (*χε*(*μξ*, *η*; *λ*) − *χε*(*μξ*, *η*; *λ*)) −→ 0 (*ε*,*ε* 0) for all *<sup>γ</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>0</sup> and all *ξ*, *η*. Therefore the integrand in (15) converges pointwise to zero for *ε*,*ε* 0. Furthermore, in the same way of the proof of part (a) we have that

$$\begin{split} & \left\| \left( \epsilon^{j\mu\_{5}^{\pi}\eta} - 1 \right) D\_{\xi}^{\pi} [ (\chi\_{\ell}(\mu\_{5}^{\pi}\eta;\lambda) - \chi\_{\ell^{\prime}}(\mu\_{5}^{\pi}\eta;\lambda)) b\_{\lambda}(\mu\_{5}^{\pi}) ] \right\|\_{\mathcal{L}(\mathbb{C}^{q})} \\ & \leq C |\mu\eta|^{\delta} \Big[ \frac{1}{|\lambda|^{\sigma}} \sum\_{i=1}^{q} \langle \xi \rangle^{-(r\_{i} + n + l - \delta)} \\ & \qquad + \sum\_{i,j=1}^{q} \mu^{n+l-r\_{i}-r\_{j}} |\xi|^{\delta} \langle \mu\xi \rangle^{r\_{ij}-(n+l)} \langle \xi \rangle^{-r\_{i}-r\_{j}} \Big] \in L^{1} \Big( \mathbb{R}\_{\delta}^{n} \Big). \end{split}$$

Hence, by dominated convergence we get for fixed (*η*, *<sup>λ</sup>*) <sup>∈</sup> (R*<sup>n</sup>* - {0}) × Λ*<sup>ω</sup>* that *Kε*(*η*, *<sup>λ</sup>*) <sup>−</sup> *<sup>K</sup>ε*(*η*, *<sup>λ</sup>*)L(C*q*) −→ <sup>0</sup> (*ε*,*ε* <sup>0</sup>). Therefore there exists a strongly measurable function *<sup>K</sup>* : <sup>R</sup>*<sup>n</sup>* <sup>×</sup> <sup>Λ</sup>*<sup>ω</sup>* → L(C*q*) with *<sup>K</sup>ε*(*η*, *<sup>λ</sup>*) <sup>→</sup> *<sup>K</sup>*(*η*, *<sup>λ</sup>*) (*<sup>ε</sup>* <sup>0</sup>) pointwise a.e. Then, inequality (12) holds for *K*(*η*, *λ*) instead of *Kε*(*η*, *λ*) and in consequence (13) is true due to

$$\int\_{\mathbb{R}^n} \frac{\mu^n |\mu\eta|^{\delta - n}}{1 + |\mu\eta|} d\eta < \infty$$

**Proposition 1.** *Let <sup>A</sup>*(*D*) *be* <sup>Λ</sup>*-elliptic with <sup>ρ</sup>* <sup>≥</sup> *<sup>ρ</sup>n,* <sup>1</sup> <sup>2</sup> <sup>&</sup>lt; *<sup>r</sup>*<sup>−</sup> *and let <sup>b</sup>λ*(·) :<sup>=</sup> (*Aα*<sup>0</sup> (·) <sup>−</sup> *<sup>λ</sup>*) −1 *for all <sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>*ω. If k* <sup>∈</sup> <sup>N</sup><sup>0</sup> *and* <sup>1</sup> <sup>≤</sup> *<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup>*, then bλ*(*D*) ∈ L *W<sup>k</sup> <sup>p</sup>*(R*n*, C*q*) *with*

$$\left\| \left\| b\_{\lambda}(D) \right\| \right\|\_{\mathcal{L}\left(\mathcal{W}^{k}\_{p}(\mathbb{R}^{n},\mathbb{C}^{q})\right)} \leq \frac{M}{|\lambda|^{\sigma}} \quad \forall \lambda \in \Lambda\_{\omega\prime} $$

*where the constant M* > <sup>0</sup> *is independent on <sup>λ</sup> and <sup>σ</sup>.*

**Proof.** Let *<sup>β</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>0</sup> with <sup>|</sup>*β*<sup>|</sup> <sup>≤</sup> *<sup>k</sup>*, *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>*ω*, *<sup>u</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>∞</sup> *<sup>b</sup>* (R*n*, <sup>C</sup>*q*) <sup>∩</sup> *<sup>W</sup><sup>k</sup> <sup>p</sup>*(R*n*, <sup>C</sup>*q*) and *<sup>x</sup>* <sup>∈</sup> <sup>R</sup>*n*. Then (see (8))

$$\partial\_x^{\mathcal{S}}(b\_\lambda(D)\mu)(\mathbf{x}) = \lim\_{\varepsilon \searrow 0} \int\_{\mathbb{R}^n} K\_\varepsilon(\eta, \lambda)(\partial\_x^{\mathcal{S}}\mu)(\mathbf{x} - \eta)d\eta \tag{16}$$

with *Kε* as in (9). From (16), Lemma 4 and dominated convergence, we get

$$\partial\_x^{\mathcal{S}}(b\_\lambda(D)u)(\mathbf{x}) = \int\_{\mathbb{R}^n} \mathcal{K}(\eta, \lambda)(\partial\_x^{\mathcal{S}}u)(\mathbf{x} - \eta)d\eta = (\mathcal{K}(\cdot, \lambda) \* (\partial\_x^{\mathcal{S}}u))(\mathbf{x}),$$

where ∗ stands for the standard convolution. Since *∂ β xu* <sup>∈</sup> *<sup>L</sup>p*(R*n*, <sup>C</sup>*q*), we have *<sup>K</sup>*(·, *<sup>λ</sup>*) <sup>∗</sup> (*∂ β xu*) <sup>∈</sup> *<sup>L</sup>*1(R*n*, <sup>C</sup>*q*) and

$$\begin{aligned} \left\| \partial\_{\mathbf{x}}^{\mathcal{G}} (\mathsf{b}\_{\lambda}(D)\boldsymbol{\mu}) \right\|\_{L^{p}(\mathbb{R}^{n}, \mathbb{C}^{q})} &\leq \left\| K(\cdot, \lambda) \right\|\_{L^{1}(\mathbb{R}^{n}, L(\mathbb{C}^{q}))} \left\| \partial\_{\mathbf{x}}^{\mathcal{G}} \boldsymbol{\mu} \right\|\_{L^{p}(\mathbb{R}^{n}, \mathbb{C}^{q})} \\ &\leq \frac{M}{|\lambda|^{\sigma}} \left\| \boldsymbol{\mu} \right\|\_{W^{k}\_{p}(\mathbb{R}^{n}, \mathbb{C}^{q})} \end{aligned}$$

due to Lemma 4 (b). This implies that

$$\left\|\left\|b\_{\lambda}(D)u\right\|\right\|\_{\mathcal{W}^{k}\_{p}(\mathbb{R}^{n},\mathbb{C}^{q})} \leq \frac{\dot{M}}{\left\|\lambda\right\|^{\mathcal{C}}} \left\|u\right\|\_{\mathcal{W}^{k}\_{p}(\mathbb{R}^{n},\mathbb{C}^{q})}\tag{17}$$

for all *<sup>u</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>∞</sup> *<sup>b</sup>* (R*n*, <sup>C</sup>*q*) <sup>∩</sup> *<sup>W</sup><sup>k</sup> <sup>p</sup>*(R*n*, <sup>C</sup>*q*) and *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>*ω*. Because of 1 <sup>≤</sup> *<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup>, <sup>S</sup>(R*n*, <sup>C</sup>*q*) is dense in *W<sup>k</sup> <sup>p</sup>*(R*n*, <sup>C</sup>*q*) which gives *<sup>b</sup>λ*(*D*) ∈ L *W<sup>k</sup> <sup>p</sup>*(R*n*, C*q*) and the estimate on its norm.

For *<sup>k</sup>* <sup>∈</sup> <sup>N</sup><sup>0</sup> and 1 <sup>≤</sup> *<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup>, we define the *<sup>W</sup><sup>k</sup> <sup>p</sup>*(R*n*, <sup>C</sup>*q*)-realization *<sup>A</sup>α*0,*<sup>k</sup>* of the system *Aα*<sup>0</sup> (*D*) as the unbounded operator given by

$$\begin{aligned} D(A\_{\mathfrak{a}\_0,k}) &:= \left\{ \mu \in \mathcal{W}\_p^k(\mathbb{R}^n, \mathbb{C}^q) : A\_{\mathfrak{a}\_0}(D)\mu \in \mathcal{W}\_p^k(\mathbb{R}^n, \mathbb{C}^q) \right\}, \\ A\_{\mathfrak{a}\_0,k}\mu &:= A\_{\mathfrak{a}\_0}(D)\mu \text{ for } \mu \in D(A\_{\mathfrak{a}\_0,k}). \end{aligned}$$

Now we are able to show the main result of this paper. We recall that *<sup>ρ</sup>* <sup>≥</sup> *<sup>ρ</sup>n*, <sup>1</sup> <sup>2</sup> <sup>&</sup>lt; *<sup>r</sup>*<sup>−</sup> and *σ* = *<sup>r</sup>*<sup>−</sup> *<sup>r</sup>*<sup>+</sup> <sup>∈</sup> ( <sup>1</sup> <sup>2</sup>*r*<sup>+</sup> , 1].

**Theorem 1.** *Let <sup>A</sup>*(*D*) *be* <sup>Λ</sup>(*θ*)*-elliptic with* <sup>0</sup> < *<sup>θ</sup>* < *<sup>π</sup>*/2 *and <sup>ϑ</sup>* :<sup>=</sup> *<sup>π</sup>* <sup>−</sup> *<sup>θ</sup>. Let <sup>k</sup>* <sup>∈</sup> <sup>N</sup>0*,* <sup>1</sup> <sup>≤</sup> *<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup> *and <sup>A</sup>α*0,*<sup>k</sup> be the <sup>W</sup><sup>k</sup> <sup>p</sup>*(R*n*, C*q*)*-realization of <sup>A</sup>α*<sup>0</sup> (*D*)*. Then, for the resolvent set <sup>ρ</sup>*(−*Aα*0,*k*) *of* <sup>−</sup>*Aα*0,*<sup>k</sup> we have <sup>ρ</sup>*(−*Aα*0,*k*) <sup>⊃</sup> <sup>Σ</sup>*ϑ*,*<sup>ω</sup>* :<sup>=</sup> " *<sup>λ</sup>* <sup>∈</sup> <sup>C</sup> : <sup>|</sup>*λ*<sup>|</sup> <sup>≥</sup> *<sup>ω</sup> and* <sup>|</sup>arg *<sup>λ</sup>*<sup>|</sup> <sup>≤</sup> *<sup>ϑ</sup>* # *and*

$$\left\|\left(\lambda + A\_{\mathfrak{a}\_0,k}\right)^{-1}\right\|\_{\mathcal{L}\left(\mathcal{W}\_p^k(\mathbb{R}^n, \mathbb{C}^q)\right)} \le \frac{M}{|\lambda|^{\sigma}} \qquad \left(\lambda \in \Sigma\_{\mathfrak{d},\omega}\right). \tag{18}$$

*for some constant <sup>M</sup>* <sup>&</sup>gt; <sup>0</sup>*. Therefore,* <sup>−</sup>*Aα*0,*<sup>k</sup>* : *<sup>W</sup><sup>k</sup> <sup>p</sup>*(R*n*, <sup>C</sup>*q*) <sup>⊃</sup> *<sup>D</sup>*(*Aα*0,*k*) <sup>→</sup> *<sup>W</sup><sup>k</sup> <sup>p</sup>*(R*n*, C*q*) *generates an infinitely differentiable semigroup on W<sup>k</sup> <sup>p</sup>*(R*n*, C*q*)*, which is analytic and strongly continuous if σ* = 1 *(i.e., r*<sup>1</sup> = ··· = *rq* = *r*−*).*

**Remark 4.** *The semigroup is given by* (*e* <sup>−</sup>*τAα*0,*<sup>k</sup>* )*τ*≥<sup>0</sup> *with e*−0*Aα*0,*<sup>k</sup>* :<sup>=</sup> *I and*

$$e^{-\tau A\_{\mathfrak{a}\_0k}} := \frac{1}{2\pi i} \int\_{\Gamma} e^{-\tau \lambda} (\lambda I - A\_{\mathfrak{a}\_0k})^{-1} d\lambda \quad (\tau > 0),$$

*where* <sup>Γ</sup> : *<sup>λ</sup>* <sup>=</sup> *<sup>ω</sup>* <sup>+</sup> *iy,* <sup>−</sup><sup>∞</sup> < *<sup>y</sup>* < <sup>∞</sup> *stands for a lying in <sup>ρ</sup>*(−*Aα*0,*k*) *path, and* [*<sup>t</sup>* <sup>→</sup> *<sup>e</sup>* <sup>−</sup>*τAα*0,*<sup>k</sup>* ] <sup>∈</sup> *<sup>C</sup>*∞((0, <sup>∞</sup>);L(*W<sup>k</sup> <sup>p</sup>*(R*n*, C*q*)))*. See [21, Theorem 3.4, Ch. 1] for a reference. Further results about differential and analytical properties of semigroups of operators can be found also in [22] and in the references therein.*

**Proof of Theorem 1.** Because of the density of <sup>S</sup>(R*n*, <sup>C</sup>*q*) in *<sup>W</sup><sup>k</sup> <sup>p</sup>*(R*n*, C*q*) and

$$(A\_{\mathfrak{a}\_0,k} - \overline{\lambda})b\_{\overline{\lambda}}(D)\mathfrak{u} = b\_{\overline{\lambda}}(D)\left(A\_{\mathfrak{a}\_0,k} - \overline{\lambda}\right)\mathfrak{u} = \mathfrak{u}$$

for all *<sup>u</sup>* ∈ S(R*n*, <sup>C</sup>*q*) and <sup>0</sup> *λ* ∈ Λ(*θ*)*ω*, it follows from (17) that Λ(*θ*)*<sup>ω</sup>* ⊂ *ρ Aα*0,*<sup>k</sup>* and *b*0 *<sup>λ</sup>*(*D*) <sup>=</sup> *Aα*0,*<sup>k</sup>* − 0 *λ* <sup>−</sup><sup>1</sup> in *W<sup>k</sup> <sup>p</sup>*(R*n*, <sup>C</sup>*q*). Now, if *<sup>λ</sup>* <sup>∈</sup> <sup>Σ</sup>*ϑ*,*ω*, then <sup>0</sup> *λ* := −*λ* ∈ Λ(*θ*)*ω*. Therefore we have

$$(\lambda + A\_{\mathfrak{a}\_0,k})b\_{-\lambda}(D)\mathfrak{u} = b\_{-\lambda}(D)(\lambda + A\_{\mathfrak{a}\_0,k})\mathfrak{u} = \mathfrak{u}$$

for all *<sup>u</sup>* ∈ S(R*n*, <sup>C</sup>*q*) and *<sup>λ</sup>* <sup>∈</sup> <sup>Σ</sup>*ϑ*,*ω*. It follows that <sup>Σ</sup>*ϑ*,*<sup>ω</sup>* <sup>⊂</sup> *<sup>ρ</sup>* −*Aα*0,*<sup>k</sup>* and *<sup>b</sup>*−*λ*(*D*) = *λ* + *Aα*0,*<sup>k</sup>* <sup>−</sup><sup>1</sup> for *<sup>λ</sup>* <sup>∈</sup> <sup>Σ</sup>*ϑ*,*ω*. Then (18) follows from Proposition 1.

The above result on the generation of semigroup in *W<sup>k</sup> <sup>p</sup>*(R*n*, C*q*) allow us to solve non-autonomous Cauchy problems, based on an abstract result in [23], Chapter IV. For this, let *<sup>T</sup>* <sup>&</sup>gt; 0 and assume <sup>A</sup> <sup>=</sup> {*A*(*t*, *<sup>D</sup>*) : *<sup>t</sup>* <sup>∈</sup> [0, *<sup>T</sup>*]} to be a uniformly bounded family of <sup>Λ</sup>elliptic systems. For *<sup>k</sup>* <sup>∈</sup> <sup>N</sup><sup>0</sup> and 1 <sup>≤</sup> *<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup>, we denote by *Ak*(*t*) the *<sup>W</sup><sup>k</sup> <sup>p</sup>*(R*n*, C*q*)-realization of *A*(*t*, *D*). Then, we study the Cauchy problem

$$\begin{cases} \partial\_t u(t) + A\_k(t)u(t) = f(t), \quad t \in (0, T],\\ u(0) = u\_0. \end{cases} \tag{19}$$

A function *<sup>u</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>1</sup> (0, *T*], *W<sup>k</sup> <sup>p</sup>*(R*n*, C*q*) ∩ *C* [0, *T*], *W<sup>k</sup> <sup>p</sup>*(R*n*, C*q*) is called a classical solution of (19), if *u*(*t*) ∈ *D*(*Ak*(*t*)) for all *t* ∈ (0, *T*], *∂tu*(*t*) + *Ak*(*t*)*u*(*t*) = *f*(*t*) for all *t* ∈ (0, *T*] and *u*(0) = *u*0.

Using Theorem 1 and the abstract result on Cauchy problems given in Theorem 2.5.1 of Chapter IV in [23], we obtain, in the same way to the proof of Theorem 4.3 in [6], the following result.

**Theorem 2.** *Let* A = {*A*(*t*, *D*) : *t* ∈ [0, *T*]} *be a uniformly bounded family of* Λ(*θ*)*-elliptic systems,* <sup>0</sup> < *<sup>θ</sup>* < *<sup>π</sup>*/2*, with symbols aij*(*t*, *ξ*) <sup>1</sup>≤*i*,*j*≤*<sup>q</sup> for all <sup>t</sup>* <sup>∈</sup> [0, *<sup>T</sup>*]*, such that* . *t* → *aij*(*t*, ·) / ∈ *<sup>C</sup>α*([0, *<sup>T</sup>*], *<sup>S</sup>rij*,*ρ*(R*n*)) *for all <sup>i</sup>*, *<sup>j</sup>* <sup>=</sup> 1, ..., *<sup>q</sup> and some <sup>α</sup>* <sup>∈</sup> (0, 1)*, with <sup>r</sup>*<sup>1</sup> <sup>=</sup> ··· <sup>=</sup> *rq* <sup>=</sup> *<sup>r</sup>*<sup>−</sup> <sup>&</sup>gt; 1/2*. Furthermore, suppose that there exists <sup>α</sup>*<sup>0</sup> <sup>∈</sup> <sup>R</sup> *such that <sup>A</sup>α*<sup>0</sup> (*t*, *<sup>D</sup>*) :<sup>=</sup> *<sup>A</sup>*(*t*, *<sup>D</sup>*) + *<sup>α</sup>*<sup>0</sup> *is* <sup>Λ</sup>(*θ*) *elliptic,* <sup>0</sup> < *<sup>θ</sup>* < *<sup>π</sup>*/2*, with the same constant <sup>C</sup> and <sup>R</sup>* <sup>=</sup> <sup>0</sup>*, for all <sup>t</sup>* <sup>∈</sup> [0, *<sup>T</sup>*] *(see Definition 3 and Remark 2). Moreover, let <sup>k</sup>* <sup>∈</sup> <sup>N</sup>0*,* <sup>1</sup> <sup>≤</sup> *<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup> *and <sup>ε</sup>* <sup>∈</sup> (0, 1)*. Then, for every <sup>u</sup>*<sup>0</sup> <sup>∈</sup> *<sup>W</sup><sup>k</sup> <sup>p</sup>*(R*n*, C*q*) *and f* <sup>∈</sup> *<sup>C</sup><sup>ε</sup>* [0, *T*], *W<sup>k</sup> <sup>p</sup>*(R*n*, C*q*) *, the Cauchy problem*

$$\begin{cases} \partial\_t v(t) + A\_{a\_0,k}(t)v(t) = e^{-a\_0 t} f(t), & t \in (0, T],\\ v(0) = u\_0. \end{cases} \tag{20}$$

*has a unique classical solution, where Aα*0,*k*(*t*) *is the W<sup>k</sup> <sup>p</sup>*(R*n*, C*q*)*-realization of Aα*<sup>0</sup> (*t*, *<sup>D</sup>*)*.*

**Corollary 2.** *Suppose that the same hypothesis from Theorem 2 hold. Then, there exists a unique classical solution of problem* (19)*.*

**Proof.** First note that *Aα*0,*k*(*t*) = *Ak*(*t*) + *α*0. Now, let *v*(*t*), *t* ∈ [0, *T*], be the classical solution of problem (20) and set *u*(*t*) := *eα*0*<sup>t</sup> v*(*t*) for *t* ∈ [0, *T*]. Then *u* is the unique classical solution of problem (19).

**Remark 5.** *If* −*Aα*0,*k*(*t*)*, t* ∈ [0, *T*]*, generates only an infinitely differentible semigroup on W<sup>k</sup> <sup>p</sup>*(R*n*, <sup>C</sup>*q*) *and, <sup>A</sup>α*0,*k*(·)−<sup>1</sup> *is strongly continuously differentiable on* [0, *<sup>T</sup>*] *and satisfies some additional conditions, Theorems 4.3, 4.4, and Remark 4.5 in [24] imply the existence and uniqueness of a strict solution of* (20)*, and therefore of* (19)*, for each <sup>u</sup>*<sup>0</sup> <sup>∈</sup> *<sup>W</sup><sup>k</sup> <sup>p</sup>*(R*n*, C*q*)*. Such strict solution is taken in sense of Definition 1.1 in [24].*

**Remark 6.** *(i) With the method used in this paper some better assertions could be obtained, for instance maximal Lp-regularity or the existence of a H*∞*-calculus as in [4].*

*(ii) Using some ideas from [4], one could change the basic space W<sup>k</sup> <sup>p</sup>*(R*n*, C*q*) *by q* ∏ *i*=1 *<sup>W</sup>k*−*li <sup>p</sup>* (R*n*)

*for some suitable integers li, i* = 1, ... , *q. Thus one could obtain similar result as in Theorem 1, but under weaker assumption on the structure of the system. This remark will be useful for the analysis, in a forthcoming paper, of the generalized thermoelastic plate equations with fractional damping.*

#### **5. Examples**

In this section, we will consider some examples where we could apply our results. Initially, as a naive example, we consider the Cauchy problem associated to the *n*-dimensional linear heat equation in the whole space. That is

$$\begin{cases} u\_t(\mathbf{x}, t) - a \Delta u(\mathbf{x}, t) = 0 & (\mathbf{x} \in \mathbb{R}^n, t > 0), \\ u(\mathbf{x}, 0) = u\_0(\mathbf{x}) & (\mathbf{x} \in \mathbb{R}^n), \end{cases} \tag{21}$$

where *<sup>α</sup>* > 0 is related to the thermal diffusivity and *<sup>u</sup>*(*x*, *<sup>t</sup>*) represents the temperature in point *x* at time *t*. The differential equation in (21) can be written in the form

$$u\_t - A(D)u = 0\_\prime$$

where

$$A(\xi) = -a|\xi|^2, \quad \xi \in \mathbb{R}^n.$$

Note that in this case *<sup>r</sup>*<sup>1</sup> <sup>=</sup> <sup>2</sup> <sup>&</sup>gt; 1/2 and therefore the condition (2) holds trivially. Let define

$$A\_-(\xi) := -A(\xi).$$

Now, for all *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>(*θ*) with 0 <sup>&</sup>lt; *<sup>θ</sup>* <sup>&</sup>lt; *<sup>π</sup>*/2, and all <sup>|</sup>*ξ*<sup>|</sup> <sup>≥</sup> <sup>√</sup><sup>1</sup> 3 , it can be shown that

$$|\det(\lambda - A\_{-}(\xi))| = |\lambda - a|\xi|^2| \ge M(\langle \xi \rangle^2 + |\lambda|).$$

Then *A*(*D*) is Λ(*θ*)-elliptic and we can apply corollary 2 to solve problem (21).

Consider now the thermoelastic plate equations on R*<sup>n</sup>* given by

$$\begin{cases} v\_{tt} + \Delta^2 v + \Delta \theta = 0, \\ \theta\_t - \Delta \theta - \Delta v\_t = 0 \end{cases} \tag{22}$$

together with the initial conditions

$$v(0, \cdot) = v\_{0\prime} \ v\_t(0, \cdot) = v\_{1\prime} \ \theta(0, \cdot) = \theta\_0.$$

The equations in (22) were derived in [25], where *v* denotes a mechanical variable representing the vertical displacement of the plate, while *θ* denotes a thermal variable describing the temperature relative to a constant reference temperature *θ*.

Using the substitution *u* = (*θ*, *vt*, −Δ*v*), the system (22) can be written as

$$
\mu\_t - A(D)\mu = 0,\tag{23}
$$

where

$$A(\mathfrak{f}) := \begin{pmatrix} -|\mathfrak{f}|^2 & -|\mathfrak{f}|^2 & 0 \\ |\mathfrak{f}|^2 & 0 & -|\mathfrak{f}|^2 \\ 0 & |\mathfrak{f}|^2 & 0 \end{pmatrix}. \tag{24}$$

Note that in this case, *rij* <sup>=</sup> <sup>2</sup> <sup>&</sup>gt; 1/2 for all *<sup>i</sup>*, *<sup>j</sup>* <sup>=</sup> 1, 2, 3, and assumption (2) holds. Now, we define

$$A\_{-}(\xi) := -A(\xi) \tag{25}$$

and consider the determinant of *λ* − *A*−(*ξ*), which is given by:

$$\det(\lambda - A\_{-}(\xi)) = \lambda^3 - |\xi|^2 \lambda^2 + 2|\xi|^4 \lambda - |\xi|^6.$$

It is easy to see that

$$\det(\lambda - A\_{-}(\xi)) = |\xi|^{6} p \left(\frac{\lambda}{|\xi|^{2}}\right),\tag{26}$$

where

$$p(t) = t^3 - t^2 + 2t - 1.$$

Since *<sup>p</sup>*(0) < 0, *<sup>p</sup>*(1) > 0 and *<sup>p</sup>* (*t*) > 0 for all *<sup>t</sup>* <sup>∈</sup> <sup>R</sup>, there exists a unique real number *α* ∈ (0, 1) such that *p*(*α*) = 0. Now, since *p* is a polynomial with real coefficients, there exist positive constants *β* and *γ*, such that

$$p(t) = (t - \lambda\_1)(t - \lambda\_2)(t - \lambda\_3) \tag{27}$$

with *λ*<sup>1</sup> = *α*, *λ*<sup>2</sup> = *β* + *γi* and *λ*<sup>3</sup> = *λ*2. In particular, we get *λ*<sup>1</sup> + *λ*<sup>2</sup> + *λ*<sup>3</sup> = 1, and therefore *β* = <sup>1</sup>−*<sup>α</sup>* <sup>2</sup> <sup>&</sup>gt; 0. Hence, according to (26) and (27), it follows that

$$\det(\lambda - A\_{-}(\vec{\xi})) = \left(\lambda - |\vec{\xi}|^{2}\lambda\_{1}\right)\left(\lambda - |\vec{\xi}|^{2}\lambda\_{2}\right)\left(\lambda - |\vec{\xi}|^{2}\lambda\_{3}\right). \tag{28}$$

By inequality (2.7) in [26], there exists *<sup>π</sup>* <sup>2</sup> <sup>&</sup>lt; *<sup>ϑ</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>π</sup>* such that

$$\left|\lambda\lambda\_j^{-1} + |\xi|^2\right| \ge \mathcal{C}(|\lambda| + |\xi|^2) \quad (j = 1, 2, 3) \quad \forall \lambda \in -\Lambda(\theta\_0) \text{ and } \xi \in \mathbb{R}^n,\tag{29}$$

where <sup>−</sup>Λ(*ϑ*0) :<sup>=</sup> {−*<sup>λ</sup>* : *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>(*ϑ*0)}. Hence, for all *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>(*ϑ*0) and <sup>|</sup>*ξ*| ≥ <sup>√</sup><sup>1</sup> 3 , we have

$$\begin{aligned} \left| \lambda - |\mathfrak{f}|^2 \lambda\_{\mathfrak{f}} \right| &= |\lambda\_{\mathfrak{f}}| \left| (-\lambda)\lambda\_{\mathfrak{f}}^{-1} + |\mathfrak{f}|^2 \right| \\ &\geq c|\lambda\_{\mathfrak{f}}| (|\lambda| + |\mathfrak{f}|^2) \\ &\geq \mathsf{C} (|\lambda| + |\mathfrak{f}|^2) \end{aligned} \tag{30}$$

for *<sup>j</sup>* <sup>=</sup> 1, 2, 3. Note that 2*r*|*ξ*<sup>|</sup> *<sup>r</sup>* <sup>≥</sup>*<sup>ξ</sup> <sup>r</sup>* if <sup>|</sup>*ξ*| ≥ <sup>√</sup><sup>1</sup> <sup>3</sup> and *<sup>r</sup>* ≥ 0.

**Proposition 2.** *Let <sup>A</sup>*−(*ξ*) *be defined as in* (25)*. Then <sup>A</sup>*−(*D*) *is* <sup>Λ</sup>(*ϑ*)*-elliptic with* <sup>0</sup> < *<sup>ϑ</sup>* < *<sup>π</sup>*/2*.*

**Proof.** This follows from (28)–(30).

**Theorem 3.** *Let <sup>T</sup>* > <sup>0</sup>*, <sup>ε</sup>* <sup>∈</sup> (0, 1), *<sup>k</sup>* <sup>∈</sup> <sup>N</sup>0*,* <sup>1</sup> <sup>≤</sup> *<sup>p</sup>* < <sup>∞</sup>*, <sup>A</sup>*(*D*) *be defined by* (23) *and* (24) *and let Ak be the W<sup>k</sup> p* R*n*, C<sup>3</sup> *-realization of <sup>A</sup>*(*D*)*. Then, for each <sup>u</sup>*<sup>0</sup> <sup>∈</sup> *<sup>W</sup><sup>k</sup> p* R*n*, C<sup>3</sup> *and <sup>f</sup>* <sup>∈</sup> *<sup>C</sup><sup>ε</sup>* [0, *T*], *W<sup>k</sup> p* R*n*, C<sup>3</sup> *the Cauchy problem*

$$\begin{cases} \partial\_t \mu(t) + A\_k \mu(t) = f(t), \quad t \in (0, T],\\ \mu(0) = \mu\_0. \end{cases}$$

*has a unique classical solution.*

**Proof.** It follows from Proposition 2 and corollary 2.

Now, as a third example, we consider the lineal structurally damped plate equation on R*<sup>n</sup>*

$$
\sigma\_{tt} + \Delta^2 v - \rho \Delta v\_t = f\_\prime \tag{31}
$$

together with initial conditions

$$v(0, \cdot) = v\_{0\prime} \ v\_t(0, \cdot) = v\_1 \cdot$$

Here, *<sup>ρ</sup>* > 0 is a fixed parameter. A description of this equation can be found in [27] and the references therein.

Using the substitution *u* = (*vt*, −Δ*v*) and *F* = (*f* , 0) *<sup>T</sup>*, the Equation (31) can be written as

$$
\mu\_t - \omega'(D)\mu = F\_\prime
$$

where

$$\varkappa(\xi) := \begin{pmatrix} -\rho|\xi|^2 & -|\xi|^2 \\ |\xi|^2 & 0 \end{pmatrix}.$$

Note again that *rij* <sup>=</sup> <sup>2</sup> <sup>&</sup>gt; 1/2 for all *<sup>i</sup>*, *<sup>j</sup>* <sup>∈</sup> {1, 2}, and assumption (2) holds. Now, we define A−(*ξ*) := −A (*ξ*) and consider the determinant of *λ* − A−(*ξ*), which is given by

$$\det(\lambda - \varkappa \angle(\xi)) = \lambda^2 - \rho |\xi|^2 \lambda + |\xi|^4.$$

Note that det(*λ*<sup>±</sup> − A−(*ξ*)) = 0 if only if *λ*<sup>±</sup> = |*ξ*| 2 *ρ* 2 ± <sup>1</sup>*ρ*<sup>2</sup> <sup>−</sup> <sup>4</sup> 2 =: |*ξ*| <sup>2</sup>*z*±. If *<sup>ξ</sup>* <sup>=</sup> 0, then *<sup>λ</sup>*<sup>±</sup> <sup>&</sup>gt; 0 for *<sup>ρ</sup>* <sup>≥</sup> 2 and *<sup>λ</sup>*<sup>±</sup> <sup>∈</sup> <sup>C</sup> (with Re *<sup>z</sup>*<sup>+</sup> <sup>&</sup>gt; 0 and *<sup>z</sup>*<sup>−</sup> <sup>=</sup> *<sup>z</sup>*+) for 0 <sup>&</sup>lt; *<sup>ρ</sup>* <sup>&</sup>lt; 2. Therefore, det(*<sup>λ</sup>* <sup>−</sup> <sup>A</sup>−(*ξ*)) = *λ* − |*ξ*| <sup>2</sup>*z*<sup>+</sup> *<sup>λ</sup>* − |*ξ*<sup>|</sup> <sup>2</sup>*z*<sup>−</sup> and in consequence

$$|\det(\lambda - \omega \mathbb{1}\_{-}(\xi))| \ge \mathbb{C}(\left<\xi\right>^2 + |\lambda|)(\left<\xi\right>^2 + |\lambda|) \text{ } \forall \lambda \in \Lambda(\pi/2) \text{ and } |\xi| \ge 1/\sqrt{3}.$$

In consequence, <sup>A</sup>−(*D*) is <sup>Λ</sup>(*θ*)-elliptic with 0 < *<sup>θ</sup>* < *<sup>π</sup>*/2. Using the same arguments as in the previous example, we have that the Cauchy problem associated with (31) has a unique classical solution.

As a last example we consider a generalized plate equation in R*<sup>n</sup>* with intermediated damping. Let *<sup>α</sup>*, *<sup>ρ</sup>* <sup>&</sup>gt; 0, *<sup>β</sup>* <sup>∈</sup> [0, 1] and *<sup>L</sup>* :<sup>=</sup> (−Δ) *α* . Then the associated symbol of *L* is *p*(*ξ*) = |*ξ*| 2*α* , *<sup>ξ</sup>* <sup>∈</sup> <sup>R</sup>*n*. The generalized plate equation in <sup>R</sup>*<sup>n</sup>* with intermediated damping is given by

$$
\mu\_{tt} + Lu + \rho L^{\beta} u\_t = 0 \tag{32}
$$

together with the initial conditions

$$
u(0, \cdot) = \boldsymbol{u}\_{0\prime} \,\,\,\boldsymbol{u}\_{t}(0, \cdot) = \boldsymbol{u}\_{1} \,. \tag{33}$$

The generalized thermoelastic plate equation has been introduced in [28], a plate equation with intermediate damping was studied in [29] and a plate equation with intermediate rotational force and damping in [30]. For the particular case *α* = 2, (32) models the equation of a plate with: (i) frictional damping if *β* = 0, (ii) structural damping if *β* = 1/2 and (iii) Kelvin-Voigt damping if *β* = 1.

If *U* := *ut*, *L*1/2*u* , the equation (32) can equivalently be written as

$$\mathcal{U}l\_t + \vec{A}(D)\mathcal{U} = 0, \qquad \vec{A}(\xi) := \begin{bmatrix} |\rho|\xi|^{2a\beta} & |\xi|^a \\ -|\xi|^a & 0 \end{bmatrix}.$$

Now, let *χ*(*ξ*) be an arbitrary 0-excision function and *A*(*ξ*) := *χ*(*ξ*)*A*0(*ξ*). In the following we will omit without loss of generality the factor *χ*(*ξ*) in the definition of *A*(*ξ*) and we will assume that *ρ* = 1.

Using the ideas of the proof of Lemma 6.1 in [4] we obtain the following lemma.

**Lemma 5.** *Assume that the parameters <sup>α</sup>* <sup>&</sup>gt; <sup>0</sup> *and <sup>β</sup>* <sup>∈</sup> (0, 1) *satisfy the conditions*

$$
\alpha > \frac{1}{2} \quad \land \quad \frac{1}{4\alpha} < \beta < 1 - \frac{1}{4\alpha}.
$$

*Then, for the the following choice of orders:*

$$r\_1 = 2\alpha\beta, r\_2 = 2\alpha(1-\beta) \text{ and } r\_{12} = r\_{21} = \alpha,$$

*<sup>A</sup>*(*D*) *is* <sup>Λ</sup>(*θ*)*-elliptic for any* <sup>0</sup> <sup>&</sup>lt; *<sup>θ</sup>* <sup>&</sup>lt; *<sup>π</sup>.*

Under the hypotheses of the previous lemma we have that

$$\begin{aligned} r\_+ &= \begin{cases} 2a(1-\beta), & 0 < \beta < \frac{1}{2}, \\ a, & \beta = \frac{1}{2}, \\ 2a\beta, & \frac{1}{2} < \beta < 1, \end{cases} & r\_- = \begin{cases} 2a\beta, & 0 < \beta < \frac{1}{2}, \\ a, & \beta = \frac{1}{2}, \\ 2a(1-\beta), & \frac{1}{2} < \beta < 1, \end{cases} \\\ \sigma = \frac{r\_-}{r\_+} &= \begin{cases} \frac{\beta}{1-\beta}, & 0 < \beta < \frac{1}{2}, \\ 1, & \beta = \frac{1}{2}, \\ \frac{1-\beta}{\beta}, & \frac{1}{2} < \beta < 1. \end{cases} \end{aligned}$$

Consequently, we can apply corollary 2 and Remark 5 to solve problem (32) and (33).

#### **6. Conclusions**

In this article, we have proved that the additive inverse of a suitable Sobolev space realization of a Λ-elliptic Fourier multipliers system (in the sense of the Definition 3) generates an infinitely differentiable semigroup on such Sobolev space, and that under certain additional conditions, it generates an analytic semigroup on the same Sobolev space (see Theorem 1). We emphasize again in these conclusions that the proof of the generation of semigroups was done directly using an approach based on oscillatory integrals and non trivial kernel estimates for them. With the results about generation of semigroups we addressed the analysis of some application problems in Section 5 using well-known statements for the existence and uniqueness of solutions for abstract evolution equations. Now, regarding the possible future scope of this work, we recall Remark 6: using techniques similar to those in this paper, questions about maximal *Lp*-regularity, the existence of a *H*∞-calculus, the improvement of the basic spaces, and the weakening of the assumptions for the structure of the system of Fourier multipliers, would be addressed in a forthcoming paper. In the other direction, it is interesting to study assumptions, under which Λ-elliptic Fourier multipliers systems generate Cosine families of operators in some appropriate functional or distributional spaces, to consider control problems for fractional evolution inclusions or equations following ideas from, for example [9,11–14].

**Author Contributions:** Conceptualization, B.B.M. and J.H.M.; Formal analysis, B.B.M., J.G.O., R.G.A. and J.H.M.; Funding acquisition, B.B.M. and J.H.M.; Investigation, B.B.M., J.G.O., R.G.A. and J.H.M.; Supervision, B.B.M., J.G.O., R.G.A. and J.H.M.; Writing—original draft, B.B.M., J.G.O. and R.G.A.; Writing—review and editing, J.H.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by MINCIENCIAS-COLOMBIA (formerly COLCIENCIAS) grant number 121571250194.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Effects of the Wiener Process on the Solutions of the Stochastic Fractional Zakharov System**

**Farah M. Al-Askar 1, Wael W. Mohammed 2,3,\*, Mohammad Alshammari <sup>2</sup> and M. El-Morshedy 4,5**


**Abstract:** We consider in this article the stochastic fractional Zakharov system derived by the multiplicative Wiener process in the Stratonovich sense. We utilize two distinct methods, the Riccati– Bernoulli sub-ODE method and Jacobi elliptic function method, to obtain new rational, trigonometric, hyperbolic, and elliptic stochastic solutions. The acquired solutions are helpful in explaining certain fascinating physical phenomena due to the importance of the Zakharov system in the theory of turbulence for plasma waves. In order to show the influence of the multiplicative Wiener process on the exact solutions of the Zakharov system, we employ the MATLAB tools to plot our figures to introduce a number of 2D and 3D graphs. We establish that the multiplicative Wiener process stabilizes the solutions of the Zakharov system around zero.

**Keywords:** fractional Zakharov system; stochastic Zakharov system; Riccati–Bernoulli sub-ODE method; Jacobi elliptic function method

**MSC:** 60H15; 60H10; 35A20; 83C15; 35Q51

#### **1. Introduction**

In 1972, Zakharov [1] developed the Zakharov system. It is a group of coupled nonlinear wave equations that explains the interaction of high-frequency Langmuir (dispersive) and low-frequency ion-acoustic (roughly nondispersive) waves. In one dimension, the Zakharov system can be authored as

$$\begin{aligned} \left. v\_{tt} - v\_{xx} + (\left| u \right|^2)\_{xx} \right| &= \begin{array}{rcl} 0, & & \text{(1)} \\ \left. u\_t + u\_{xx} + 2uv \right|\_{x} &=& 0, \end{array} \end{aligned} \tag{1}$$

where *<sup>v</sup>* : <sup>Ω</sup> <sup>×</sup> <sup>R</sup><sup>+</sup> <sup>→</sup> <sup>R</sup> denotes the plasma density as determined by its equilibrium value, and *<sup>u</sup>* : <sup>Ω</sup> <sup>×</sup> <sup>R</sup><sup>+</sup> <sup>→</sup> <sup>C</sup> denotes the high-frequency electric field's envelope. The Zakharov system is similar to nonlinear Schrödinger equations and significant in plasma turbulence theory. As a result, the Zakharov system has piqued the interest of many physicists and mathematicians, and has been extensively studied both theoretically and numerically [2–6]. To solve system problems (1), researchers have used a variety of methods. For example, Song et al. [7] introduced unbounded wave solutions, kink wave solutions, and periodic wave solutions by utilizing bifurcation theory method. Wang and Li [8] used the extended Fexpansion method to obtain periodic wave solutions. Javidi et al. [9] applied the variational iteration technique to obtain solitary wave solutions. Taghizadeh et al. [10] obtained some

**Citation:** Al-Askar, F.M.; Mohammed, W.W.; Alshammari, M.; El-Morshedy, M. Effects of the Wiener Process on the Solutions of the Stochastic Fractional Zakharov System. *Mathematics* **2022**, *10*, 1194. https://doi.org/10.3390/ math10071194

Academic Editor: Patricia J. Y. Wong

Received: 16 March 2022 Accepted: 2 April 2022 Published: 6 April 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

exact solutions using infinite series method. Hong et al. [11] obtained a few new doubly periodic solutions utilizing the Jacobian elliptic function expansion method.

In recent years, the fractional derivatives are utilized to describe numerous physical phenomena in engineering applications, signal processing, electromagnetic theory, finance, physics, mathematical biology, and various scientific studies, see for instance [12–17]. For instance, the fractional derivative is utilized in control theory, controller tuning, optics, seismic wave analysis, dynamical system, signal processing, and viscoelasticity.

On the other hand, the benefits of taking random effects into consideration in predicting, simulating, analyzing and modeling complex phenomena has been extensively distinguished in biology, engineering, physics, geophysical, chemistry, climate dynamics, and other fields [18–21]. Stochastic partial differential equations (SPDEs) are suitable mathematical equations for complicated systems subject to noise or random influences. Normally, random influences can be thought of as a simple estimate of turbulence in fluids. Therefore, we have to generalize the Zakharov system by taking into account more elements due to some important effects such as ion nonlinearities and transit-time damping.

To achieve a higher level of qualitative agreement, we consider here the following stochastic fractional-space Zakharov system (SFSZS) with multiplicative noise in the Stratonovich sense:

$$
\dot{\nu}u\_t + \mathbb{T}\_{xx}^{\alpha}u + 2\mu v + i\sigma u \circ \mathcal{W}\_t = 0,\tag{2}
$$

$$
\sigma\_{tt} - \mathbb{T}\_{xx}^{\alpha} \upsilon + \mathbb{T}\_{xx}^{\alpha} (\left| u \right|^2) \quad = \quad 0,\tag{3}
$$

where <sup>T</sup>*<sup>α</sup>* is the conformable fractional derivative (CFD) [22], <sup>W</sup>(*t*) is standard Wiener process (SWP).

In [23,24], the stochastic dissipative Zakharov system are obtained by utilizing the global-random attractors provided with normal topology, while in [25], the uniqueness and existence of solutions of the Zakharov system with stochastic term are obtained by applying the method of Galerkin approximation.

The novelty of this paper is to construct the exact fractional stochastic solutions of the SFSZS (2)–(3). This study is the first one to obtain the exact solutions of the SFSZS (2)–(3). We use two distinct methods including the Jacobi elliptic function and the Riccati–Bernoulli sub-ODE to achieve a wide range of solutions, including hyperbolic, trigonometric, rational, and elliptic functions. Besides that, we employ Matlab tools to plot 3D and 2D graphs for some of the analytical solutions developed in this study to check the effect of the Wiener process on the solutions of SFSZS (2)–(3).

The following is how the paper is arranged. In Section 2, we define the CFD and Wiener process and we state some features about them. To obtain the wave equation of SFSZS (2)–(3), we use a suitable wave transformation in Section 3. In Section 4, we apply two different methods to construct the exact solutions of SFSZS (2)–(3). In Section 5, we study the effect of the SWP on the obtained solutions. Finally, we present the paper's conclusion.

#### **2. Preliminaries**

In this section, we introduce some definitions and features for CFD, which are reported in [22] and SWP.

**Definition 1.** *Assume f* : (0, <sup>∞</sup>) <sup>→</sup> <sup>R</sup>*; hence, the CFD of f of order <sup>α</sup> is defined as*

$$\mathbb{T}\_x^{\kappa}f(\mathfrak{x}) = \lim\_{h \to 0} \frac{f(\mathfrak{x} + h\mathfrak{x}^{1-\alpha}) - f(\mathfrak{x})}{h}.$$

**Theorem 1.** *Let <sup>f</sup>* , *<sup>g</sup>* : (0, <sup>∞</sup>) <sup>→</sup> <sup>R</sup> *be differentiable, and also <sup>α</sup> differentiable functions; then, the next rule holds:*

$$T\_x^{\mathfrak{a}}(f\circ \mathfrak{g})(\mathfrak{x}) = \mathfrak{x}^{1-\mathfrak{a}}\mathfrak{g}'(\mathfrak{x})f'(\mathfrak{g}(\mathfrak{x})) .$$

Let us state some properties of the CFD:


$$\mathbf{4.} \quad \mathbb{T}^{\alpha}\_{x}\mathcal{G}(\mathbf{x}) = \mathbf{x}^{1-\alpha}\frac{d\mathbf{g}}{dx'} $$

In the next definition, we define standard Wiener process W(*t*):

**Definition 2.** *stochastic process* {W(*t*)}*t*≥<sup>0</sup> *is called a Wiener process if it satisfies*


We know the stochastic integral *<sup>t</sup>* <sup>0</sup> Θ*d*Wmay be interpreted in a variety of ways [26]. The Stratonovich and Itô interpretations of a stochastic integral are widely used. The stochastic integral is Itô (denoted by *<sup>t</sup>* <sup>0</sup> Θ*d*W) when it is evaluated at the left-end, while a Stratonovich stochastic integral (denoted by *<sup>t</sup>* <sup>0</sup> Θ ◦ *d*W) is one that is calculated at the midpoint. The next is the relationship between the Stratonovich and Itô integral:

$$\int\_{0}^{t} \Theta(\mathbf{r}, Z\_{\tau}) d\mathcal{W}(\mathbf{r}) = \int\_{0}^{t} \Theta(\mathbf{r}, Z\_{\tau}) \diamond d\mathcal{W}(\mathbf{r}) - \frac{1}{2} \int\_{0}^{t} \Theta(\mathbf{r}, Z\_{\tau}) \frac{\partial \Theta(\mathbf{r}, Z\_{\tau})}{\partial z} d\mathbf{r}, \tag{4}$$

where Θ is supposed to be sufficiently regular and {*Zt*, *t* ≥ 0} is a stochastic process.

#### **3. Wave Equation for SFSZS**

To acquire the wave equation for the SFSZS (2)–(3), the next wave transformation is applied:

$$\mu(\mathbf{x},t) = \varphi(\mu)\varepsilon^{(i\theta - \sigma \mathcal{W}(t) - \sigma^2 t)}, \quad \mu = k(\frac{1}{a}\mathbf{x}^a - \lambda t) \text{ and } \theta = \frac{\lambda}{2a}\mathbf{x}^a + \rho t,\tag{5}$$

where *ϕ* is a deterministic function and *k*, *λ*, *ρ* are nonzero constants. Plugging Equation (5) into Equation (2) and using

$$\begin{aligned} \frac{du}{dt} &= \begin{aligned} (-\lambda k \varrho' + i\rho \varrho - \sigma \varrho \mathcal{W} l\_t - \frac{1}{2} \sigma^2 \varrho) e^{(i\theta - \sigma \mathcal{W}(t) - \sigma^2 t)} \end{aligned} \\ &= \begin{aligned} (-\lambda k \varrho' + i\rho \varrho - \sigma \varrho \circ \mathcal{W}\_l) e^{(i\theta - \sigma \mathcal{W}(t) - \sigma^2 t)} \end{aligned} \\ \begin{aligned} \mathcal{T}\_{xx}^a &= (k^2 \varrho'' + i\lambda k \varrho' - \frac{1}{4} \lambda^2 \varrho) e^{(i\theta - \sigma \mathcal{W}(t) - \sigma^2 t)} \end{aligned} \end{aligned} \tag{6}$$

where we used (4). We obtain, for the real part,

$$k^2 \varrho^{\prime\prime} - (\frac{1}{4}\lambda^2 + \rho)\varrho + 2\varrho v = 0.\tag{7}$$

Now, we suppose

$$
v(\mathbf{x}, t) = \psi(\boldsymbol{\mu}),
$$

where *ψ* is real deterministic function, to obtain

$$
\omega v\_t = -\lambda k \psi', \ v\_{tt} = \lambda^2 k^2 \psi'', \ \mathbb{T}\_{xx}^u v = k^2 \psi''. \tag{8}
$$

Substituting Equation (8) into Equation (3), we attain

$$(\lambda^2 - 1)\psi'' + (\varphi^2)''e^{(-2r\mathcal{W}(t) - 2r^2t)} = 0. \tag{9}$$

Taking expectation <sup>E</sup>(·) on both sides, we have

$$(\lambda^2 - 1)\psi'' + (\varrho^2)''e^{-2\sigma^2 t} \mathbb{E}(e^{-2\sigma \mathcal{W}(t)}) = 0. \tag{10}$$

Since <sup>W</sup>(*t*) is standard Gaussian process; hence, <sup>E</sup>(*e*W(*t*)) = *<sup>e</sup>* 2 <sup>2</sup> *<sup>t</sup>* for any real constant . Now, Equation (10) has the form

$$(\lambda^2 - 1)\psi'' + (\varrho^2)'' = 0,\tag{11}$$

Integrating Equation (11) twice and putting the constants of integration equal zero yields

$$(\lambda^2 - 1)\psi + \varrho^2 = 0.\tag{12}$$

Hence, Equation (12) becomes

$$
\psi = \frac{-q^2}{(\lambda^2 - 1)}.\tag{13}
$$

Putting Equation (13) into Equation (7), we obtain the next wave equation

$$
\Psi'' - \gamma\_1 \varrho^3 - \gamma\_2 \varrho = 0,\tag{14}
$$

where

$$\gamma\_1 = \frac{2}{k^2(\lambda^2 - 1)} \quad \text{and} \quad \gamma\_2 = \frac{1}{4k^2}(\lambda^2 + 4\rho). \tag{15}$$

#### **4. The Analytical Solutions of the SFSZS**

To find the solutions of Equation (14), we utilize two different methods: Riccati– Bernoulli sub-ODE [27] and the Jacobi elliptic function method [28]. Therefore, we acquire the analytical solutions of the SFSZS (2)–(3).

#### *4.1. Riccati–Bernoulli Sub-ODE Method*

Assume the following Riccati–Bernoulli equation:

$$
\boldsymbol{\varrho}^{\prime} = \boldsymbol{\hbar}\_1 \boldsymbol{\varrho}^2 + \boldsymbol{\hbar}\_2 \boldsymbol{\varrho} + \boldsymbol{\hbar}\_3 \tag{16}
$$

where 1, 2, <sup>3</sup> are undefined constants and *ϕ* = *ϕ*(*μ*).

Differentiating Equation (16) with respect to *μ*, we obtain

$$
\varphi'' = 2\hbar\_1 \varphi \varphi' + \hbar\_2 \varphi',
$$

and using Equation (16) yields

$$
\varphi^{\prime\prime} = 2\hbar\_1^2 \varphi^3 + 3\hbar\_1 \hbar\_2 \varphi^2 + \left(2\hbar\_1 \hbar\_3 + \hbar\_2^2\right) \varphi + \hbar\_2 \hbar\_3. \tag{17}
$$

Substituting (17) into (14), we have

$$(2\hbar\_1^2 - \gamma\_1)\varrho^3 + 3\hbar\_1\hbar\_2\varrho^2 + \left(2\hbar\_1\hbar\_3 + \hbar\_2^2 - \gamma\_2\right)\varrho + \hbar\_2\hbar\_3 = 0.1$$

Equating each coefficient of *ϕ<sup>i</sup>* (*i* = 0, 1, 2, 3) to zero, we achieve the next algebraic equations

$$
\hbar\_2 \hbar\_3 = 0,
$$

$$
(2\hbar\_1 \hbar\_3 + \hbar\_2^2 - \gamma\_2) = 0,
$$

$$
3\hbar\_1 \hbar\_2 = 0,
$$

$$
2\hbar\_1^2 - \gamma\_1 = 0.
$$

When the above equations are solved, the result is

$$
\hbar\_1 = \pm \sqrt{\frac{1}{2}\gamma\_1}, \,\hbar\_2 = 0, \,\hbar\_3 = \frac{\gamma\_2}{2\hbar\_1} = \pm \frac{\gamma\_2}{\sqrt{2\gamma\_1}}.\tag{18}
$$

There are numerous solutions to the Riccati–Bernoulli Equation (16) depending on <sup>1</sup> and 3.

*First case:* If <sup>3</sup> <sup>1</sup> = 0, then Riccati–Bernoulli Equation (16) has the solution

$$\wp(\mu) = \frac{-1}{\hbar\_1 \mu + \mathcal{C}}.$$

Hence, the SFSZS (2)–(3) has the analytical solutions

$$u(\mathbf{x},t) = \varphi(\mu)e^{(i\theta - \sigma \mathcal{W}(t) - \sigma^2 t)} = \frac{-1}{\hbar\_1(\frac{k}{\mu}\mathbf{x}^a - k\lambda t) + \mathbf{C}}e^{(i\theta - \sigma \mathcal{W}(t) - \sigma^2 t)}\,,\tag{19}$$

$$v(x,t) = \frac{-\varrho^2}{(\lambda^2 - 1)} = \frac{-1}{(\lambda^2 - 1)\left(\hbar\_1(\frac{k}{\mathfrak{a}}x^\alpha - k\lambda t) + C\right)^2}.\tag{20}$$

*Second case:* If <sup>3</sup> <sup>1</sup> <sup>&</sup>gt; 0, then the Riccati–Bernoulli equation (16) has the solution

$$\varphi(\mu) = \sqrt{\frac{\hbar\_3}{\hbar\_1}} \tan \left( \sqrt{\frac{\hbar\_3}{\hbar\_1}} (\hbar\_1 \mu + \mathbb{C}) \right).$$

or

$$\varphi(\mu) = -\sqrt{\frac{\hbar\_3}{\hbar\_1}} \cot \left( \sqrt{\frac{\hbar\_3}{\hbar\_1}} (\hbar\_1 \mu + \mathcal{C}) \right).$$

Therefore, SFSZSs (2)–(3) have the following solutions:

$$u(x,t) = e^{(i\theta - \sigma\mathcal{W}(t) - \sigma^2 t)} \sqrt{\frac{\hbar\_3}{\hbar\_1}} \tan\left(\sqrt{\frac{\hbar\_3}{\hbar\_1}} (\hbar\_1 (\frac{k}{a} x^a - k\lambda t) + \mathbb{C})\right),\tag{21}$$

$$w(\mathbf{x},t) = \frac{-\hbar\_3}{(\lambda^2 - 1)\hbar\_1} \tan^2\left(\sqrt{\frac{\hbar\_3}{\hbar\_1}}(\hbar\_1(\frac{k}{\alpha}\mathbf{x}^a - k\lambda t) + \mathbf{C})\right),\tag{22}$$

or

$$u(\mathbf{x},t) = -e^{\left(i\theta - \sigma\mathcal{W}(t) - \sigma^2 t\right)} \sqrt{\frac{\hbar\_3}{\hbar\_1}} \cot\left(\sqrt{\frac{\hbar\_3}{\hbar\_1}} \left(\hbar\_1 \left(\frac{k}{\alpha}\mathbf{x}^a - k\lambda t\right) + \mathbb{C}\right)\right),\tag{23}$$

$$v(\mathbf{x},t) = \frac{-\hbar\_3}{(\lambda^2 - 1)\hbar\_1} \cot^2\left(\sqrt{\frac{\hbar\_3}{\hbar\_1}} (\hbar\_1(\frac{k}{a}\mathbf{x}^\kappa - k\lambda t) + \mathbb{C})\right),\tag{24}$$

respectively.

*Third case:* If <sup>3</sup> <sup>1</sup> <sup>&</sup>lt; 0 and <sup>|</sup>*ϕ*<sup>|</sup> <sup>&</sup>lt; −3 1 , then Riccati–Bernoulli Equation (16) has the solution 3 3 .

$$\varphi(\mu) = -\sqrt{\frac{-\hbar\_3}{\hbar\_1}} \tanh\left(\sqrt{\frac{-\hbar\_3}{\hbar\_1}}(\hbar\_1\mu + \mathbb{C})\right)$$

Thus, the SFSZS (2)–(3) have the following analytical solutions:

$$u(\mathbf{x},t) = -e^{\left(i\boldsymbol{\theta} - \boldsymbol{\sigma}\mathcal{W}(t) - \boldsymbol{\sigma}^2 t\right)} \sqrt{\frac{-\hbar\_3}{\hbar\_1}} \tanh\left(\sqrt{\frac{-\hbar\_3}{\hbar\_1}} (\hbar\_1 (\frac{k}{a} \mathbf{x}^a - k\lambda t) + \mathbf{C})\right),\tag{25}$$

$$w(\mathbf{x},t) = \frac{-\hbar\_3}{(\lambda^2 - 1)\hbar\_1} \tanh^2\left(\sqrt{\frac{-\hbar\_3}{\hbar\_1}} (\hbar\_1(\frac{k}{\mathfrak{a}}\mathbf{x}^\mathfrak{a} - k\lambda t) + \mathbb{C})\right). \tag{26}$$

*Fourth case:* If <sup>3</sup> <sup>1</sup> <sup>&</sup>lt; 0 and *<sup>ϕ</sup>*<sup>2</sup> <sup>&</sup>gt; <sup>−</sup><sup>3</sup> <sup>1</sup> , then Riccati–Bernoulli Equation (16) has the solution

$$\varphi(\mu) = -\sqrt{\frac{-\hbar\_3}{\hbar\_1}} \coth\left(\sqrt{\frac{-\hbar\_3}{\hbar\_1}} (\hbar\_1 \mu + \mathbb{C})\right).$$

Consequently, the analytical solutions of the SFSZS (2)–(3) are

$$u(\mathbf{x},t) = -\epsilon^{(i\theta - \sigma \mathcal{W}(t) - \sigma^2 t)} \sqrt{\frac{-\hbar\_3}{\hbar\_1}} \coth\left(\sqrt{\frac{\hbar\_3}{\hbar\_1}} (\hbar\_1 (\frac{k}{a} \mathbf{x}^a - k\lambda t) + \mathbb{C})\right),\tag{27}$$

$$v(\mathbf{x},t) = \frac{-\hbar\_3}{(\lambda^2 - 1)\hbar\_1} \coth^2\left(\sqrt{\frac{\hbar\_3}{\hbar\_1}}(\hbar\_1(\frac{k}{a}\mathbf{x}^a - k\lambda t) + \mathbf{C})\right),\tag{28}$$

where <sup>1</sup> and <sup>2</sup> are defined in Equation (18).

#### *4.2. The Jacobi Elliptic Function Method*

Assuming that the solutions to Equation (14) are of the form

$$
\varphi(\mu) = a + b \text{sn}(\delta \mu),
\tag{29}
$$

where *sn*(*δμ*) = *sn*(*δμ*, *<sup>m</sup>*), for 0 < *<sup>m</sup>* < 1, is the Jacobi elliptic sine function and *<sup>a</sup>*, *<sup>b</sup>*, *<sup>δ</sup>* are unknown constants. Differentiate Equation (29) two times and we have

$$
\varphi''(\mu) = -(m^2 + 1)b\delta^2 sn(\delta\mu) + 2m^2 b\delta^2 sn^3(\delta\mu). \tag{30}
$$

Substituting Equations (29) and (30) into Equation (14), we attain

$$(2m^2b\delta^2 - \gamma\_1b^3)sn^3(\delta\mu) - 3\gamma\_1ab^2sn^2(\delta\mu)$$

$$-[(m^2+1)b\delta^2 + 3\gamma\_1a^2b + \gamma\_2b]sn(\delta\mu) - (\gamma\_1a^3 + a\gamma\_2) = 0.$$

Setting each coefficient of [*sn*(*δμ*)]*n*(*n* = 0, 1, 2, 3) equal to zero, we attain

$$
\gamma\_1 a^3 + a \gamma\_2 = 0,
$$

$$
(m^2 + 1)b\delta^2 + 3\gamma\_1 a^2 b + \gamma\_2 b = 0,
$$

$$
3\gamma\_1 ab^2 sn^2 = 0,
$$

and

$$2m^2b\delta^2 - \gamma\_1b^3 = 0.$$

Solving the above equations, we have

$$a = 0, \ b = \pm \sqrt{\frac{-2m^2 \gamma\_2}{(m^2 + 1)\gamma\_1}} \\ \delta^2 = \frac{-\gamma\_2}{(m^2 + 1)}.$$

Hence, the solution of Equation (14), by using (29), has the form

$$\varphi(\mu) = \pm \sqrt{\frac{-2m^2 \gamma\_2}{(m^2+1)\gamma\_1}} \mathfrak{sn}(\frac{-\gamma\_2}{(m^2+1)}\mu).$$

Therefore, the analytical solutions of the SFSZS (2)–(3) are

$$u(\mathbf{x},t) = \pm \sqrt{\frac{-2m^2\gamma\_2}{(m^2+1)\gamma\_1}} \text{sn}\left(\sqrt{\frac{-\gamma\_2}{(m^2+1)}} (\frac{k}{\alpha}\mathbf{x}^\alpha - k\lambda t)\right) e^{(i\theta - \sigma\mathcal{W}(t) - \sigma^2 t)}\,,\tag{31}$$

$$v(\mathbf{x},t) = \frac{k^2m^2\gamma\_2}{(m^2+1)}sn^2\left(\sqrt{\frac{-\gamma\_2}{(m^2+1)}}\left(\frac{k}{n}\mathbf{x}^a - k\lambda t\right)\right),\tag{32}$$

for *<sup>γ</sup>*<sup>2</sup> <sup>&</sup>lt; 0 and *<sup>γ</sup>*<sup>1</sup> <sup>&</sup>gt; 0. When *<sup>m</sup>* <sup>→</sup> 1, the solutions (31)–(32) transfer into

$$u(\mathbf{x},t) = \pm \sqrt{\frac{-\gamma\_2}{\gamma\_1}} \tanh\left(\sqrt{\frac{-\gamma\_2}{2}} (\frac{k}{\alpha}\mathbf{x}^{\alpha} - k\lambda t)\right) e^{(i\boldsymbol{\theta} - \boldsymbol{\sigma})\mathcal{W}(t) - \boldsymbol{\sigma}^2 t}\,,\tag{33}$$

$$v(x,t) = -\frac{k^2\gamma\_2}{2}\tanh^2\left(\sqrt{\frac{-\gamma\_2}{2}}(\frac{k}{a}x^a - k\lambda t)\right). \tag{34}$$

Analogously, we can replace *sn* in (29) by *cn* and *dn* in order to obtain the solutions of Equation (14), respectively, as follows:

$$\wp(\mu) = \pm \sqrt{\frac{-2m^2 \gamma\_2}{(2m^2 - 1)\gamma\_1}} cn(\frac{-\gamma\_2}{(2m^2 - 1)}\mu)\_r$$

and

$$\varphi(\mu) = \pm \sqrt{\frac{2m^2 \gamma\_2}{(2 - m^2)\gamma\_1}} dn(\frac{-\gamma\_2}{(2 - m^2)}\mu).$$

Consequently, the solutions of the SFSZS (2)–(3) have the following forms:

$$u(\mathbf{x},t) = \pm \sqrt{\frac{-2m^2\gamma\_2}{(2m^2-1)\gamma\_1}} cn \left(\sqrt{\frac{-\gamma\_2}{(2m^2-1)}} \left(\frac{k}{a}\mathbf{x}^a - k\lambda t\right)\right) e^{\left(i\boldsymbol{\theta} - \boldsymbol{\sigma}\boldsymbol{\mathcal{W}}(t) - \boldsymbol{\sigma}^2 t\right)},\tag{35}$$

$$v(x,t) = \frac{k^2m^2\gamma\_2}{(2m^2-1)}cn^2\left(\sqrt{\frac{-\gamma\_2}{(2m^2-1)}}\left(\frac{k}{a}x^a - k\lambda t\right)\right),\tag{36}$$

for *<sup>γ</sup>*<sup>2</sup> (2*m*2−1) <sup>&</sup>lt; 0, *<sup>γ</sup>*<sup>1</sup> <sup>&</sup>gt; 0, and

$$u(\mathbf{x},t) = \pm \sqrt{\frac{-2m^2\gamma\_2}{(2m^2-1)\gamma\_1}}dn\left(\sqrt{\frac{-\gamma\_2}{(2m^2-1)}}\left(\frac{k}{a}\mathbf{x}^a - k\lambda t\right)\right)e^{(i\boldsymbol{\theta}-\boldsymbol{\sigma}\cdot\boldsymbol{\mathcal{W}}(t)-\boldsymbol{\sigma}^2t)}\,,\tag{37}$$

$$v(\mathbf{x},t) = \frac{k^2m^2\gamma\_2}{(2-m^2)}dn^2\left(\sqrt{\frac{-\gamma\_2}{(2-m^2)}}(\frac{k}{a}\mathbf{x}^a - k\lambda t)\right),\tag{38}$$

for *<sup>γ</sup>*<sup>2</sup> <sup>&</sup>lt; 0, *<sup>γ</sup>*<sup>1</sup> <sup>&</sup>gt; 0, respectively. When *<sup>m</sup>* <sup>→</sup> 1, the solutions (35)–(36) and (37)–(38) transfer into

$$u(\mathbf{x},t) = \pm \sqrt{\frac{-2\gamma\_2}{\gamma\_1}} \text{sech}\left(\sqrt{-\gamma\_2} (\frac{k}{a}\mathbf{x}^a - k\lambda t)\right) e^{\left(i\theta - \sigma \mathcal{W}(t) - \sigma^2 t\right)}\,,\tag{39}$$

$$v(\mathbf{x},t) = k^2 m^2 \gamma\_2 \text{sech}^2 \left(\sqrt{-\gamma\_2} (\frac{k}{a} \mathbf{x}^a - k\lambda t)\right),\tag{40}$$

for *<sup>γ</sup>*<sup>2</sup> <sup>&</sup>lt; 0, *<sup>γ</sup>*<sup>1</sup> <sup>&</sup>gt; 0.

#### **5. The Influence of Noise on SFSZS Solutions**

The influence of the noise on the analytical solution of the SFSZS (2)–(3) is addressed here. Fix the parameters *k* = 1, *ρ* = 1, *m* = 0.5, and *λ* = 3. We introduce a number of simulations for various values of *σ* (noise intensity) and *α* (fractional derivative order). We employ the MATLAB tools to plot our figures. In Figures 1 and 2, if *σ* = 0, we see that the surface fluctuates for different values of *α*:

**Figure 1.** 3D graphs of the solution (31).

**Figure 2.** 3D graphs of the solution (32).

In the following Figures 3–5, we can see that after minor transit patterns, the surface becomes considerably flattered when noise is included and its strength is increased *σ* = 1, 2.

**Figure 3.** 3D graphs of the solution (31) with *α* = 1.

**Figure 4.** 3D graphs of the equation (31) with *α* = 0.5.

**Figure 5.** 3D graphs of the equation (21) with *α* = 1.

In Figure 6, we introduce 2D plots of the *u* in (31) with *σ* = 0, 0.5, 1, 2 and *α* = 1, which emphasize the results above.

**Figure 6.** 2D graphs of the *u* in (31).

From Figures 1–6, we deduce the following:


#### **6. Conclusions**

In this article, we provided a wide range of exact solutions of the stochastic fractional Zakharov system (2)–(3). We applied two different methods such as the Riccati–Bernoulli sub-ODE method and Jacobi elliptic function method to attain rational, trigonometric, hyperbolic, and elliptic stochastic fractional solutions. Such solutions are critical for comprehending certain essential, fundamental, complex phenomena. The solutions obtained will be extremely useful for further studies such as fiber applications, spatial plasma, quasi particle theory, coastal water motion, and industrial research. Finally, the effect of multiplicative Wiener process on the exact solution of Zakharov system (2)–(3) is demonstrated. In future research, we can address the fractional-time Zakharov system (2)–(3) with multidimensional multiplicative noise.

**Author Contributions:** Conceptualization, F.M.A.-A., W.W.M., M.A. and M.E.-M.; methodology, F.M.A.-A. and W.W.M.; software, W.W.M. and M.E.-M.; formal analysis, F.M.A.-A., W.W.M., M.A. and M.E.-M.; investigation, F.M.A.-A. and W.W.M.; resources, F.M.A.-A., W.W.M., M.A. and M.E.-M.; data curation, F.M.A.-A. and W.W.M.; writing—original draft preparation, F.M.A.-A., W.W.M., M.A. and M.E.-M.; writing—review and editing, F.M.A.-A. and W.W.M.; visualization, F.M.A.-A. and W.W.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R273), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Xuhao Li <sup>1</sup> and Patricia J. Y. Wong 2,\***

	- Singapore 639798, Singapore

**Abstract:** In this paper, a numerical scheme based on a general temporal mesh is constructed for a generalized time-fractional diffusion problem of order *α*. The main idea involves the generalized linear interpolation and so we term the numerical scheme the *gL***1 scheme**. The stability and convergence of the numerical scheme are analyzed using the energy method. It is proven that the temporal convergence order is (2 − *α*) for a general temporal mesh. Simulation is carried out to verify the efficiency of the proposed numerical scheme.

**Keywords:** generalized fractional derivative; time-diffusion problem; generalized linear interpolation; numerical scheme

**MSC:** 65M12; 65N12

#### **1. Introduction**

The fractional model has been shown to be a powerful tool in modeling various memory process in applications [1,2], such as diffusion process in a porous media. The essence of this tool lies in fractional derivative (or integral) which is an extension of integer derivative (or integral). Many well known mathematicians such as Euler, Lagrange, Liouville, Riemann, Grüwald, Letnikov and Caputo have devoted their efforts to fractional calculus and have made great contributions to this topic. The extension from integer derivative (or integral) to fractional derivative (or integral) may not be unique due to the different techniques applied. To unify the different approaches, Agrawal [3] has proposed some *generalized* fractional operators which may unify some well known fractional operators such as Caputo, Riemann-Louville and Hadamard. The generalized fractional operators incorporate a scale function *z*(*t*) and a weight function *w*(*t*). With these two functions, many equations can be written in a general form and thus can be solved in an elegant way as shown in [3]. Moreover, by choosing different *z*(*t*) and *w*(*t*), one can readily obtain the well known fractional operators.

The generalized fractional operators naturally lead to *generalized* fractional problems. As in the fractional situation, the analytical solutions may not be easily derived and hence the corresponding numerical solutions are both necessary and useful in applications. In fact, some pioneer works have been done on the numerical treatment of certain generalized fractional problems [4–10]. In the earlier papers [8–10], the convergence of the numerical scheme is established using the Lax–Richtmyer theorem but the order of convergence is not explicitly given. Inspired by [11,12], some generalized weighted shifted Grünwald-Letnikov (gWSGL) type approximations [4–6] and generalized Alikhanov's approximation [7] were recently proposed, which improve the accuracy of previous work. In fact, the convergence order of these methods are shown to be *O*(*τ*<sup>2</sup> *<sup>z</sup>* ) (or higher) based on a particular choice of the temporal mesh that closely depends on the scale function *z*(*t*). It is noteworthy that

**Citation:** Li, X.; Wong, P.J.Y. *gL*1 Scheme for Solving a Class of Generalized Time-Fractional Diffusion Equations. *Mathematics* **2022**, *10*, 1219. https://doi.org/ 10.3390/math10081219

Academic Editor: Christopher Goodrich

Received: 16 March 2022 Accepted: 5 April 2022 Published: 8 April 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

the higher convergence order comes at the expense of more computations and restrictions in reality.

In this paper, we shall consider a generalized time-fractional diffusion problem that is more general than that in [9]. As one of the pioneer numerical treatments for generalized time-fractional diffusion problems, the authors in [9] first present analytical solution of a generalized time-fractional problem involving second order spatial derivative. Then, based on a uniform temporal mesh, they construct a numerical scheme by finite difference method. The stability of the numerical scheme is investigated via an estimate of the inverse of the coefficient matrix, and the convergence then follows from the Lax–Richtmyer theorem without giving explicit convergence order. Different from the work of [9], we aim to derive a numerical scheme based on a more general temporal mesh than [9] and give the convergence order of the proposed scheme explicitly using energy method. Our major contributions in this paper are as follows:


We consider the following generalized time-fractional diffusion equation with weight function *w*(*t*) ≡ 1

$$\begin{cases} \ ^{\mathbb{C}}D\_{t;[z(t),1]}^{\mathfrak{a}}u(\mathbf{x},t) = \mathcal{L}u(\mathbf{x},t) + f(\mathbf{x},t), & (\mathbf{x},t) \in (0,1) \times (0,1) \\\ u(\mathbf{x},0) = \mathfrak{p}(\mathbf{x}), & \mathbf{x} \in [0,1] \\\ u(0,t) = \phi\_1(t), & u(1,t) = \phi\_2(t), & t \in (0,1] \end{cases} \tag{1}$$

where L is a linear operator defined by

$$
\mathcal{L}u(\mathbf{x},t) = (p(\mathbf{x},t)u\_{\mathbf{x}}(\mathbf{x},t))\_{\mathbf{x}} - q(\mathbf{x},t)u(\mathbf{x},t)\_{\mathbf{x}},
$$

with *<sup>p</sup>*(*x*, *<sup>t</sup>*) <sup>≥</sup> *<sup>p</sup>*<sup>0</sup> <sup>&</sup>gt; 0 and *<sup>q</sup>*(*x*, *<sup>t</sup>*) <sup>≥</sup> 0, *<sup>C</sup>* <sup>0</sup> *<sup>D</sup><sup>α</sup> t*;[*z*(*t*),1] *<sup>u</sup>*(*x*, *<sup>t</sup>*)(<sup>0</sup> < *<sup>α</sup>* < <sup>1</sup>) is the generalized Caputo fractional derivative given by

$$\, \_0^C D\_{t; [z(t), 1]}^a u(x, t) = \frac{1}{\Gamma(1 - a)} \int\_0^t \frac{1}{[z(t) - z(s)]^a} \frac{\partial u(x, s)}{\partial s} ds,\tag{2}$$

and *φ*1(*t*), *φ*2(*t*), *ψ*(*x*), *f*(*x*, *t*) are given functions that are sufficiently smooth. We remark that (i) a generalized fractional problem of type (1) with any weight function *w*(*t*) (not necessarily 1) can be converted to (1) by a simple formula *u*(*x*, *t*) = *w*(*t*)*v*(*x*, *t*) (see [4–7]); and (ii) the generalized fractional equation considered in [9] is a special case of (1) when *p*(*x*, *t*) ≡ 1 and *q*(*x*, *t*) ≡ 0.

The plan of the paper is as follows. In Section 2, we shall develop a numerical scheme for the problem (1) based on a more general temporal mesh than [9]. Then, the stability as well as the convergence of the proposed numerical scheme will be established rigorously using energy method in Section 3. In Section 4, we carry out experiments to verify as well as to demonstrate the efficiency of the proposed numerical scheme. Finally, a brief conclusion is given in Section 5.

#### **2. Numerical Scheme**

In this section, we shall derive a numerical scheme for the problem (1) using the key idea of *generalized linear interpolation*. To begin, let

$$\Delta: 0 = t\_0 < t\_1 < \dots < t\_{N-1} < t\_N = 1 \text{ and } \Delta': 0 = \mathbf{x}\_0 < \mathbf{x}\_1 < \dots < \mathbf{x}\_{M-1} < \mathbf{x}\_M = 1$$

be *any* mesh in the temporal dimension and a uniform mesh in the spatial dimension with step size *h* = <sup>1</sup> *<sup>M</sup>* , respectively. Throughout, assume that the scale function *z*(*t*) is a strictly increasing function. Let *Z* : *t* → *z*(*t*) be a map from [0, 1] to [*z*(0), *z*(1)]. Denote *z*(*tn*) = *zn*. Moreover, denote by *U<sup>n</sup> <sup>j</sup>* the exact solution of (1) at (*xj*, *tn*), and by *<sup>u</sup><sup>n</sup> <sup>j</sup>* an approximation of *U<sup>n</sup> j* .

$$L\_{j,k-1}(z(t)) = \frac{z(t) - z\_{k-1}}{z\_k - z\_{k-1}} u\_j^k + \frac{z(t) - z\_k}{z\_{k-1} - z\_k} u\_j^{k-1}.\tag{3}$$

Note that *Lj*,*k*−1(*z*(*t*)) is a *generalized linear polynomial* of *<sup>z</sup>*(*t*) and it will be used to approximate *<sup>u</sup>*(*xj*, *<sup>t</sup>*) over the interval [*tk*−1, *tk*] in (2). Indeed, we derive the following approximation scheme at (*xj*, *tn*)

$$\begin{split} \, \_0^C D\_{t;[z(t),1]}^\mu \mu(x\_j, t\_n) &= \frac{1}{\Gamma(2-a)} \sum\_{k=1}^n \int\_{t\_{k-1}}^{t\_k} \frac{1}{[z(t\_n) - z(s)]^a} \frac{\partial u(x,s)}{\partial s} ds \\ &\approx \frac{1}{\Gamma(2-a)} \sum\_{k=1}^n \int\_{t\_{k-1}}^{t\_k} \frac{1}{[z(t\_n) - z(s)]^a} \left[ L\_{j,k-1}(z(s)) \right]^\prime ds \\ &= \frac{1}{\Gamma(2-a)} \sum\_{k=1}^n \int\_{t\_{k-1}}^{t\_k} \frac{1}{[z(t\_n) - z(s)]^a} \frac{u\_j^k - u\_j^{k-1}}{z\_k - z\_{k-1}} z'(s) ds \\ &= \mu \left[ \omega\_z^0 u\_j^n - \sum\_{k=1}^{n-1} \left( \omega\_z^{n-k-1} - \omega\_z^{n-k} \right) u\_j^k - \omega\_z^{n-1} u\_j^0 \right] := gL1[u(x\_j, t\_n)]. \end{split} \tag{4}$$

where *μ* = <sup>1</sup> <sup>Γ</sup>(2−*α*) and the coefficients

$$
\omega\_z^{\mathfrak{n}-k-1} = \frac{(z\_{\mathfrak{n}} - z\_k)^{1-\mathfrak{a}} - (z\_{\mathfrak{n}} - z\_{k+1})^{1-\mathfrak{a}}}{z\_{k+1} - z\_k}, \qquad 0 \le k \le n - 1. \tag{5}
$$

We shall call (4) the *gL***1 approximation** of the generalized Caputo fractional derivative.

#### **Remark 1.**


To proceed further, we shall investigate the accuracy of the *gL*1 approximation (4) and the properties of the coefficients *ω<sup>k</sup> <sup>z</sup>* which are both vital in subsequent analysis. For the former one, we introduce the following definition.

**Definition 1** ([14])**.** *Given the mesh* <sup>Δ</sup> : <sup>0</sup> <sup>=</sup> *<sup>t</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>t</sup>*<sup>1</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *tN*−<sup>1</sup> <sup>&</sup>lt; *tN* <sup>=</sup> 1, *denote <sup>z</sup>*(*tk*) = *zk*. *The mesh*

$$\Delta\_z: z(0) = z\_0 < z\_1 < \dots < z\_{N-1} < z\_N = z(1)$$

*is said to be quasi uniform if <sup>τ</sup>z*,max

$$\frac{\tau\_{z,\text{max}}}{\tau\_{z,\text{min}}} \le \rho\_{\text{\%}}$$

*where <sup>τ</sup>z*,max <sup>=</sup> max1≤*k*≤*<sup>N</sup>* <sup>|</sup>*zk* <sup>−</sup> *zk*−1|*, <sup>τ</sup>z*,min <sup>=</sup> min1≤*k*≤*<sup>N</sup>* <sup>|</sup>*zk* <sup>−</sup> *zk*−1<sup>|</sup> *and <sup>ρ</sup>* <sup>&</sup>gt; <sup>0</sup> *is a constant.*

**Theorem 1** (*gL***1 approximation**)**.** *Assume that for any fixed <sup>x</sup>* <sup>=</sup> *xj, <sup>u</sup>*(*xj*, *<sup>Z</sup>*−1(*z*)) = *<sup>g</sup>*(*z*) <sup>∈</sup> *<sup>C</sup>*2[*z*(0), *<sup>z</sup>*(1)]*. Suppose that the mesh* <sup>Δ</sup>*<sup>z</sup>* : *<sup>z</sup>*(0) = *<sup>z</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>z</sup>*<sup>1</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *zN*−<sup>1</sup> <sup>&</sup>lt; *zN* <sup>=</sup> *<sup>z</sup>*(1) *is quasi uniform. Then, we have for any fixed α* ∈ (0, 1)*,*

$$\,\_{0}^{C}D\_{t;[z(t),1]}^{a}\mu(x\_{j},t\_{n}) = gL1[\mu(x\_{j},t\_{n})] + O(r\_{z,\text{max}}^{2-a}).\tag{6}$$

**Proof.** From (4) we see that the error term *R<sup>n</sup> <sup>j</sup>* satisfies

$$\prescript{C}{0}{D}\_{t;[z(t),1]}^{\alpha}\mu(\boldsymbol{x}\_{j},t\_{n}) = \prescript{}{\mathcal{S}}{L}1[\mu(\boldsymbol{x}\_{j},t\_{n})] + \prescript{R}{R}{'}\_{j'} $$

and

$$R\_j^n = \frac{1}{\Gamma(1-a)} \sum\_{k=1}^n \int\_{t\_{k-1}}^{t\_k} [z(t\_n) - z(s)]^{-a} \left\{ \frac{\partial u(x\_j, s)}{\partial s} - [L\_{j, k-1}(z(s))]' \right\} ds.$$

Noting that *Lj*,*k*−1(*z*(*tk*)) = *<sup>u</sup><sup>k</sup> <sup>j</sup>* = *<sup>u</sup>*(*xj*, *tk*) and the relation *<sup>Z</sup>*−1(*z*) = *<sup>t</sup>*, after applying integration by parts, we get

$$R\_j^n = -\frac{a}{\Gamma(1-a)} \sum\_{k=1}^n \int\_{z\_{k-1}}^{z\_k} (z\_n - z)^{-a-1} \left[ u(x\_j, Z^{-1}(z)) - L\_{j,k-1}(z) \right] dz. \tag{7}$$

Since *<sup>u</sup>*(*xj*, *<sup>Z</sup>*−1(*z*)) = *<sup>g</sup>*(*z*) <sup>∈</sup> *<sup>C</sup>*2[*z*(0), *<sup>z</sup>*(1)], it is well known that

$$u(x\_j, Z^{-1}(z)) - L\_{j,k-1}(z) = (z - z\_{k-1})(z - z\_k) \frac{g\_{zz}(\tilde{\xi}\_k)}{2}, \qquad z\_{k-1} < \tilde{\xi}\_k < z\_k. \tag{8}$$

Denote *Mg* = max*z*(0)≤*z*≤*z*(1) |*gzz*(*z*)|. Upon substituting (8) into (7), we find

$$\begin{split} |R\_{j}^{n}| &\leq \frac{\alpha M\_{\mathcal{S}}}{2\Gamma(1-\alpha)} \sum\_{k=1}^{n} \int\_{z\_{k-1}}^{z\_{k}} (z\_{n}-z)^{-\alpha-1} (z-z\_{k-1})(z\_{k}-z) dz \\ &= \frac{M\_{\mathcal{S}}}{2\Gamma(2-\alpha)} \left[ \sum\_{k=1}^{n-1} \int\_{z\_{k-1}}^{z\_{k}} (z\_{n}-z)^{-\alpha-1} (z-z\_{k-1})(z\_{k}-z) dz + \int\_{z\_{n-1}}^{z\_{n}} \frac{z-z\_{n-1}}{(z\_{n}-z)^{\alpha}} dz \right] \\ &\leq \frac{M\_{\mathcal{S}}}{2\Gamma(2-\alpha)} \left[ \sum\_{k=1}^{n-1} \frac{(z\_{k}-z\_{k-1})^{2}}{4} \int\_{z\_{k-1}}^{z\_{k}} (z\_{n}-z)^{-\alpha-1} dz + \frac{(z\_{n}-z\_{n-1})^{2-\alpha}}{1-\alpha} \right]. \end{split}$$

This further gives

$$\begin{split} |R\_{\mathcal{I}}^{n}| &\leq \frac{M\_{\mathcal{S}}}{2\Gamma(2-a)} \left[ \frac{\tau\_{z,\text{max}}^{2}}{4} \int\_{z\_{0}}^{z\_{n-1}} (z\_{n}-z)^{-a-1} dz + \frac{\tau\_{z,\text{max}}^{2-a}}{1-a} \right] \\ &= \frac{M\_{\mathcal{S}}}{2\Gamma(2-a)} \left[ \frac{\tau\_{z,\text{max}}^{2}}{4a} \left( (z\_{n}-z\_{n-1})^{-a} - (z\_{n}-z\_{0})^{-a} \right) + \frac{\tau\_{z,\text{max}}^{2-a}}{1-a} \right] \\ &\leq \frac{M\_{\mathcal{S}}}{2\Gamma(2-a)} \left[ \frac{\tau\_{z,\text{max}}^{2}}{4a} (z\_{n}-z\_{n-1})^{-a} + \frac{\tau\_{z,\text{max}}^{2-a}}{1-a} \right] \\ &\leq \frac{M\_{\mathcal{S}}}{2\Gamma(2-a)} \left[ \frac{\rho^{\alpha}}{4\alpha} + \frac{1}{1-a} \right] \tau\_{z,\text{max}}^{2-a} . \end{split}$$

Hence, for any fixed *α* ∈ (0, 1), we get

$$|\mathcal{R}^n\_j| = O(r\_{z,\text{max}}^{2-a})\,.$$

This completes the proof.

**Remark 2.** *There is some relation between the commonly used uniform mesh of t and the quasi uniform mesh of z*(*t*)*.*

*(a) For finite intervals of t, say t* ∈ [0, 1], *in practice the partition of* [0, 1] *always results in finite number of subintervals, so we are able to find a constant ρ such that <sup>τ</sup>z*,max *<sup>τ</sup>z*,min ≤ *ρ*. *Hence, it is clear that any general mesh of t yields a quasi uniform mesh of z*(*t*)*. In particular, we can say that for finite intervals, the commonly used uniform mesh of t is a special case of the quasi uniform mesh of z*(*t*)*.*

*(b) For infinite intervals of t, in general the uniform mesh of t may not yield the quasi uniform mesh of <sup>z</sup>*(*t*)*. However, if* <sup>0</sup> < *<sup>c</sup>* < <sup>|</sup>*z* (*t*)<sup>|</sup> < *<sup>C</sup> for all <sup>t</sup> in the infinite interval, then the uniform mesh of t will give the quasi uniform mesh of z*(*t*)*, and so we can conclude here that the uniform mesh of t is a special case of the quasi uniform mesh of z*(*t*)*.*

Our next result gives the properties of the coefficients *ω<sup>k</sup> <sup>z</sup>* in (5) that is vital in subsequent analysis.

**Lemma 1.** *For fixed n, we have*

$$
\omega\_z^{n-k-1} \ge (1-a)(z\_n - z\_k)^{-n} > 0, \qquad 0 \le k \le n-1 \tag{9}
$$

$$
\omega\_z^{n-k-1} \ge \omega\_z^{n-k}, \qquad 0 \le k \le n-1. \tag{10}
$$

**Proof.** Let *<sup>F</sup>*(*z*)=(*zn* <sup>−</sup> *<sup>z</sup>*)1−*α*. Applying mean value theorem and noting the Definition (5), we get

$$F'(\xi) = -(1 - a)(z\_n - \xi)^{-n} = \frac{F(z\_{k+1}) - F(z\_k)}{z\_{k+1} - z\_k} = -\omega\_z^{n-k-1}, \qquad z\_k \le \xi \le z\_{k+1}$$

i.e.,

$$
\omega\_z^{n-k-1} = (1-a)(z\_n - \xi)^{-a}, \qquad z\_k \le \xi \le z\_{k+1}.\tag{11}
$$

Since (*zn* − ·)−*<sup>α</sup>* is an increasing function, (11) immediately leads to (9) and (10).

We are now ready to construct a numerical scheme for the generalized time-fractional diffusion Equation (1). Discretizing (1) at (*xj*, *tn*) and using the approximation (4) together with finite difference method in the spatial dimension yields

$$\begin{cases} \mu \left[ \omega\_z^0 u\_j^n - \sum\_{k=1}^{n-1} \left( \omega\_z^{n-k-1} - \omega\_z^{n-k} \right) u\_j^k - \omega\_z^{n-1} u\_j^0 \right] = \Lambda u\_j^n + f\_j^n, & 1 \le j \le M - 1 \\\ u\_j^0 = \psi(x\_j), \qquad 1 \le j \le M - 1 \\\ u\_0^n = \phi\_1(t\_n), \qquad u\_M^n = \phi\_1(t\_n), \qquad 0 \le n \le N \end{cases} \tag{12}$$

where *μ* = <sup>1</sup> <sup>Γ</sup>(2−*α*) and <sup>Λ</sup> is an operator given by

$$
\Lambda u\_j^n = \delta\_x (p \delta\_x u)\_j^n - q\_j^n u\_j^n \mu
$$

with

$$
\delta\_\mathbf{x} (p \delta\_\mathbf{x} \boldsymbol{\mu})^n\_j = \frac{1}{h} \left( p^n\_{j + \frac{1}{2}} \delta\_\mathbf{x} \boldsymbol{\mu}^n\_{j + \frac{1}{2}} - p^n\_{j - \frac{1}{2}} \delta\_\mathbf{x} \boldsymbol{\mu}^n\_{j - \frac{1}{2}} \right),
$$

*pn j*+ <sup>1</sup> 2 = *p*(*xj*<sup>+</sup> <sup>1</sup> 2 , *tn*), *q<sup>n</sup> <sup>j</sup>* = *<sup>q</sup>*(*xj*, *tn*) and *<sup>δ</sup>xu<sup>n</sup> j*+ <sup>1</sup> 2 = *un <sup>j</sup>*+<sup>1</sup> <sup>−</sup> *<sup>u</sup><sup>n</sup> j* /*h*. We shall call (12) the *gL***<sup>1</sup> scheme** of the generalized time-fractional diffusion Equation (1).

**Remark 3.** *It is easy to verify that the coefficient matrix of the system (12) is strictly diagonally dominated. Therefore, it is uniquely solvable.*

#### **3. Theoretical Results**

In this section, we shall analyze the stability as well as the convergence of the numerical scheme (12). To begin, let *Uh* = {*u* = (*u*0, *u*1, ··· , *uM*)|*u*<sup>0</sup> = *uM* = 0} and for any *u*, *v* ∈ *Uh*, define

$$\delta\_{\mathbf{x}} u\_{j+\frac{1}{2}} = \frac{1}{\hbar} (u\_{j+1} - u\_j), \qquad \delta\_{\mathbf{x}} u\_j = \frac{1}{\hbar} \left( u\_{j+\frac{1}{2}} - u\_{j-\frac{1}{2}} \right), \qquad \delta\_{\mathbf{x}}^2 u\_j = \frac{1}{\hbar} \left( \delta\_{\mathbf{x}} u\_{j+\frac{1}{2}} - \delta\_{\mathbf{x}} u\_{j-\frac{1}{2}} \right)$$

and

$$(u,v) = h \sum\_{j=1}^{M-1} u\_j v\_{j}, \qquad |u|\_1 = \sqrt{h \sum\_{j=0}^{M-1} \left(\delta\_{\ge} u\_{j + \frac{1}{2}}\right)^2}, \qquad ||u||\_{\infty} = \max\_{1 \le j \le M-1} |u\_j|. \tag{13}$$

Obviously, · <sup>=</sup> <sup>1</sup>(·, ·) is a norm defined over the space *Uh*.

Next, we shall present three lemmas that will be used later to establish the stability and convergence results. The first one gives a relation between |*u*|<sup>1</sup> and *u*∞.

**Lemma 2** ([15])**.** *For any u* <sup>∈</sup> *Uh, we have the inequality u*<sup>∞</sup> <sup>≤</sup> <sup>1</sup> <sup>2</sup> |*u*|1.

The next lemma is from [16] which reveals a relation between |*u*|<sup>1</sup> and *u*.

**Lemma 3** (Discrete Poincare inequality [16])**.** *Suppose that u* ∈ *Uh, then*

$$\frac{2}{h}\sin\left(\frac{\pi h}{2}\right)||u||\leq|u|\_1. \tag{14}$$

**Remark 4.** *If <sup>h</sup>* < <sup>1</sup> *which is the case in our method, from the fact* sin *<sup>π</sup>* 2 *h* ≥ *h, the inequality (14) yields* |*u*|<sup>1</sup> ≥ 2*u that will be used in subsequent analysis.*

The last lemma gives an explicit expression of −(Λ*u*, *u*).

**Lemma 4.** *For any u* ∈ *Uh, we have*

$$-(\Lambda \boldsymbol{u}, \boldsymbol{u}) = \boldsymbol{h} \sum\_{j=0}^{M-1} \boldsymbol{p}\_{j+\frac{1}{2}} \left(\delta\_{\boldsymbol{x}} \boldsymbol{u}\_{j+\frac{1}{2}}\right)^2 + \boldsymbol{h} \sum\_{j=1}^{M-1} \boldsymbol{q}\_j \boldsymbol{u}\_j^2.$$

**Proof.** Using the definition in (13), it is found that

$$\begin{split} -(\Lambda u, u) &= -h \sum\_{j=1}^{M-1} \left[ \delta\_{\mathbf{x}} (p \delta\_{\mathbf{x}} u)\_{j} - q\_{j} u\_{j} \right] u\_{j} = -h \sum\_{j=1}^{M-1} \left[ \delta\_{\mathbf{x}} (p \delta\_{\mathbf{x}} u)\_{j} \right] u\_{j} + h \sum\_{j=1}^{M-1} q\_{j} (u\_{j})^{2} \\ &= -\sum\_{j=1}^{M-1} \left( p\_{j + \frac{1}{2}} \delta\_{\mathbf{x}} u\_{j + \frac{1}{2}} - p\_{j - \frac{1}{2}} \delta\_{\mathbf{x}} u\_{j - \frac{1}{2}} \right) u\_{j} + h \sum\_{j=1}^{M-1} q\_{j} (u\_{j})^{2} \\ &= -\sum\_{j=1}^{M-1} p\_{j + \frac{1}{2}} u\_{j} \delta\_{\mathbf{x}} u\_{j + \frac{1}{2}} + \sum\_{j=1}^{M-1} p\_{j - \frac{1}{2}} u\_{j} \delta\_{\mathbf{x}} u\_{j - \frac{1}{2}} + h \sum\_{j=1}^{M-1} q\_{j} (u\_{j})^{2} \\ &= \sum\_{j=0}^{M-1} p\_{j + \frac{1}{2}} (u\_{j + 1} - u\_{j}) \delta\_{\mathbf{x}} u\_{j + \frac{1}{2}} + h \sum\_{j=1}^{M-1} q\_{j} (u\_{j})^{2}, \end{split}$$

where we have used *u*<sup>0</sup> = *uM* = 0 in the last equality. The result is immediate from the above equation.

**Remark 5.** *Since <sup>p</sup>*(*x*, *<sup>t</sup>*) <sup>≥</sup> *<sup>p</sup>*<sup>0</sup> <sup>&</sup>gt; <sup>0</sup> *and <sup>q</sup>*(*x*, *<sup>t</sup>*) <sup>≥</sup> <sup>0</sup>*, it is clear from Lemma 4 that* <sup>−</sup>(Λ*u*, *<sup>u</sup>*) <sup>≥</sup> *p*0|*u*| 2 <sup>1</sup> *for any u* ∈ *Uh*.

Now, let us present the stability and convergence of the numerical scheme (12). To this aim, we shall first consider (12) with zero boundary conditions.

**Theorem 2.** *Let* {*u<sup>n</sup> <sup>j</sup>* , 1 ≤ *j* ≤ *M* − 1, 1 ≤ *n* ≤ *N*} *be the solution of the system (12) with zero boundary conditions. Then, we have*

$$||\mu^n||^2 + \frac{p\_0}{\omega\_z^0 \mu} |\mu^n|\_1^2 \le E, \qquad 1 \le n \le N \tag{15}$$

*where μ* = <sup>1</sup> <sup>Γ</sup>(2−*α*), *<sup>ω</sup>*<sup>0</sup> *<sup>z</sup>* = (*zn* <sup>−</sup> *zn*−1)−*α*, *<sup>p</sup>*<sup>0</sup> <sup>&</sup>gt; <sup>0</sup> *and*

$$E = \|\mu^0\|^2 + \frac{\Gamma(1-a)(z(1)-z(0))^\kappa}{4p\_0} \max\_{1 \le n \le N} \|f^n\|^2.$$

**Proof.** Multiplying both sides of the first equation of (12) by <sup>−</sup>*u<sup>n</sup> <sup>j</sup>* and summing *j* from 1 to (*M* − 1) gives

$$-\mu \left[ \omega\_z^0 u\_j^n - \sum\_{k=1}^{n-1} \left( \omega\_z^{n-k-1} - \omega\_z^{n-k} \right) u\_j^k - \omega\_z^{n-1} u\_j^0 \right] = -(\Lambda u^n, u^n) - (f^n, u^n)\_+$$

which is rearranged to

$$
\mu \omega\_z^0(\boldsymbol{\mu}^n, \boldsymbol{\mu}^n) - (\boldsymbol{\Lambda}\boldsymbol{\mu}^n, \boldsymbol{\mu}^n) = \mu \sum\_{k=1}^{n-1} \left( \omega\_z^{n-k-1} - \omega\_z^{n-k} \right) (\boldsymbol{\mu}^k, \boldsymbol{\mu}^n) + \mu \omega\_z^{n-1}(\boldsymbol{\mu}^0, \boldsymbol{\mu}^n) + (f^n, \boldsymbol{\mu}^n). \tag{16}
$$

Since we consider zero boundary conditions here, it is obvious that *<sup>u</sup><sup>n</sup>* <sup>∈</sup> *Uh*. Therefore, using Remark 5, we get a lower bound for the left side of (16) below

$$
\mu \omega\_z^0(\mu^n, \mu^n) - (\Lambda \mu^n, \mu^n) \ge \mu \omega\_z^0 ||\mu^n||^2 + p \alpha |\mu^n|\_1^2.
$$

Noting (10) in Lemma 1 and using *xy* <sup>≤</sup> <sup>1</sup> <sup>2</sup> (*x*<sup>2</sup> + *<sup>y</sup>*2), an upper bound for the first two terms on the right side of (16) is found as follows

$$\begin{split} &\quad \mu \sum\_{k=1}^{n-1} \left( \omega\_z^{n-k-1} - \omega\_z^{n-k} \right) \left( u^k, u^n \right) + \mu \omega\_z^{n-1} \left( u^0, u^n \right) \\ &\leq \quad \frac{\mu}{2} \left[ \sum\_{k=1}^{n-1} \left( \omega\_z^{n-k-1} - \omega\_z^{n-k} \right) \left( u^k, u^k \right) + \omega\_z^{n-1} \left( u^0, u^0 \right) \right] + \frac{\mu}{2} \omega\_z^0 \left( u^n, u^n \right) \\ &= \quad \frac{\mu}{2} \left[ \sum\_{k=1}^{n-1} \left( \omega\_z^{n-k-1} - \omega\_z^{n-k} \right) \|u^k\|^2 + \omega\_z^{n-1} \|\|u^0\|^2 \right] + \frac{\mu}{2} \omega\_z^0 \|\|u^n\|^2. \end{split}$$

For the third term on the right side of (16), the Young's inequality gives the following upper bound

$$(f^n, \mu^n) \le \frac{||f^n||^2}{4\epsilon} + \epsilon ||\mu^n||^2, \qquad \forall \epsilon > 0.$$

Upon substituting the above upper and lower bounds into (16), we immediately get

$$\frac{\mu}{2}\omega\_z^0\|\boldsymbol{u}^n\|^2 + p\_0|\boldsymbol{u}^n|\_1^2 - \varepsilon\|\boldsymbol{u}^n\|^2 \le \frac{\mu}{2} \left[ \sum\_{k=1}^{n-1} \left( \omega\_z^{n-k-1} - \omega\_z^{n-k} \right) \|\boldsymbol{u}^k\|^2 + \omega\_z^{n-1} \|\boldsymbol{u}^0\|^2 \right] + \frac{1}{4\varepsilon} \|\boldsymbol{f}^n\|^2. \tag{17}$$

Next, using Lemma 3 and noting Remark 4, we have <sup>|</sup>*un*|<sup>1</sup> <sup>≥</sup> <sup>2</sup>*un*. Then, with = 2*p*0, the left side of (17) gives the following lower bound

$$\frac{\mu}{2}\omega\_z^0||u^n||^2 + p\_0|u^n|\_1^2 - \varepsilon||u^n||^2 \ge \frac{\mu}{2}\omega\_z^0||u^n||^2 + \left(p\_0 - \frac{\epsilon}{4}\right)|u^n|\_1^2 = \frac{\mu}{2}\omega\_z^0||u^n||^2 + \frac{p\_0}{2}|u^n|\_1^2.$$

Substituting the above into (17) leads to

$$\|\omega\_z^0\|\mu^n\|^2 + \frac{p\_0}{\mu} |\mu^n|\_1^2 \le \sum\_{k=1}^{n-1} \left(\omega\_z^{n-k-1} - \omega\_z^{n-k}\right) \|\mu^k\|^2 + \omega\_z^{n-1} \left(\|\mu^0\|^2 + \frac{1}{4p\_0\mu\omega\_z^{n-1}} \|f^n\|^2\right). \tag{18}$$

Further, from (9) we see that *<sup>ω</sup>n*−<sup>1</sup> *<sup>z</sup>* <sup>≥</sup> (<sup>1</sup> <sup>−</sup> *<sup>α</sup>*)(*zn* <sup>−</sup> *<sup>z</sup>*0)−*<sup>α</sup>* <sup>&</sup>gt; 0 and hence

$$\frac{1}{\omega\_z^{n-1}} \le \frac{(z\_\mathfrak{n} - z\_0)^\alpha}{1 - \alpha} \le \frac{(z(1) - z(0))^\alpha}{1 - \alpha}.$$

Using the above inequality and *μ* = <sup>1</sup> <sup>Γ</sup>(2−*α*) in (18), we find

$$\omega\_z^0 \left( \|\boldsymbol{u}^n\|\|^2 + \frac{p\_0}{\omega\_z^0 \mu} |\boldsymbol{u}^n|\_1^2 \right) \le \sum\_{k=1}^{n-1} \left( \omega\_z^{n-k-1} - \omega\_z^{n-k} \right) \|\boldsymbol{u}^k\|\|^2 + \omega\_z^{n-1} E\_\prime$$

where *E* is defined in the theorem. Noting (10), the above inequality readily leads to

$$\omega\_z^0 \left( \|u^n\|^2 + \frac{p\_0}{\omega\_z^0 \mu} |u^n|\_1^2 \right) \le \sum\_{k=1}^{n-1} \left( \omega\_z^{n-k-1} - \omega\_z^{n-k} \right) \left[ \|u^k\|^2 + \frac{p\_0(z\_k - z\_{k-1})^\alpha}{\mu} |u^k|\_1^2 \right] + \omega\_z^{n-1} E\_\epsilon$$

or equivalently

$$\begin{split} &\omega\_z^0 \left[ \|\boldsymbol{u}^n\|^2 + \frac{p\_0(\boldsymbol{z}\_n - \boldsymbol{z}\_{n-1})^\alpha}{\mu} |\boldsymbol{u}^n|\_1^2 \right] \\ &\leq \sum\_{k=1}^{n-1} \left( \omega\_z^{n-k-1} - \omega\_z^{n-k} \right) \left[ \|\boldsymbol{u}^k\|^2 + \frac{p\_0(\boldsymbol{z}\_k - \boldsymbol{z}\_{k-1})^\alpha}{\mu} |\boldsymbol{u}^k|\_1^2 \right] + \omega\_z^{n-1} \boldsymbol{E}. \end{split} \tag{19}$$

Now, we shall show by mathematical induction that

$$||u^n||^2 + \frac{p\_0(z\_n - z\_{n-1})^a}{\mu}|u^n|\_1^2 \le E\_\prime \qquad 1 \le n \le N. \tag{20}$$

In fact, let *n* = 1 in (19) and we get

$$
\omega\_z^0 \left[ ||u^1||^2 + \frac{p\_0(z\_1 - z\_0)^\alpha}{\mu} |u^1|\_1^2 \right] \le \omega\_z^0 E\_\prime \omega\_z^0
$$

which implies that (20) holds for *n* = 1. Suppose that (20) is true up to (*n* − 1). Then, from (19), we have

$$\begin{aligned} &\omega\_z^0 \left[ \|u^n\|^2 + \frac{p\_0(z\_n - z\_{n-1})^a}{\mu} |u^n|\_1^2 \right] \\ &\le \sum\_{k=1}^{n-1} \left( \omega\_z^{n-k-1} - \omega\_z^{n-k} \right) \left[ \|u^k\|^2 + \frac{p\_0(z\_k - z\_{k-1})^a}{\mu} |u^k|\_1^2 \right] + \omega\_z^{n-1} E\_\epsilon \\ &\le \sum\_{k=1}^{n-1} \left( \omega\_z^{n-k-1} - \omega\_z^{n-k} \right) E + \omega\_z^{n-1} E = \omega\_z^0 E\_\epsilon \end{aligned}$$

which immediately gives (20), or equivalently (15). This completes the proof.

**Remark 6** (**Stability**)**.** *Using a similar argument as in [4–7], it can readily be deduced from Theorem 2 that the numerical scheme (12) is robust (or stable) with respect to the initial data ψ*(*x*) *and the non-homogeneous data f*(*x*, *t*).

We are now ready to establish the convergence of the proposed scheme (12).

**Theorem 3** (**Convergence**)**.** *Assume that <sup>u</sup>*(*x*, *<sup>Z</sup>*−1(*z*)) = *<sup>u</sup>*¯(*x*, *<sup>z</sup>*) <sup>∈</sup> *<sup>C</sup>*4,2([0, 1] <sup>×</sup> [*z*(0), *<sup>z</sup>*(1)])*. Suppose that the mesh* <sup>Δ</sup>*<sup>z</sup>* : *<sup>z</sup>*(0) = *<sup>z</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>z</sup>*<sup>1</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *zN*−<sup>1</sup> <sup>&</sup>lt; *zN* <sup>=</sup> *<sup>z</sup>*(1) *is quasi uniform. Let* {*U<sup>n</sup> <sup>j</sup>* <sup>=</sup> *<sup>u</sup>*(*xj*, *tn*)} *be the exact solution of the problem (1),* {*u<sup>n</sup> <sup>j</sup>* } *be the numerical solution obtained from the scheme (12) and e<sup>n</sup> <sup>j</sup>* = *<sup>U</sup><sup>n</sup> <sup>j</sup>* <sup>−</sup> *<sup>u</sup><sup>n</sup> <sup>j</sup> be the error at* (*xj*, *tn*)*. Then, we have*

$$\left\| |e^{n}|\right\|^{2} + c(u, z) |e^{n}|\_{1}^{2} \le \left[ O\left(\tau\_{z, \text{max}}^{2-\mathfrak{a}} + h^{2}\right) \right]^{2}, \qquad 1 \le n \le N \tag{21}$$

*where c*(*α*, *z*) = *<sup>p</sup>*<sup>0</sup> *ω*0 *zμ .*

**Proof.** Since {*U<sup>n</sup> <sup>j</sup>* } is the exact solution of (1), it is clear that

$$\begin{cases} \begin{aligned} \mu \left[ \omega\_z^0 \mathcal{U}\_j^n - \sum\_{k=1}^{n-1} \left( \omega\_z^{n-k-1} - \omega\_z^{n-k} \right) \mathcal{U}\_j^k - \omega\_z^{n-1} \mathcal{U}\_j^0 \right] = \Lambda \mathcal{U}\_j^n + f\_j^n + T\_j^n, \qquad 1 \le j \le M - 1 \\\ \mathcal{U}\_j^0 = \mathfrak{p}(x\_j), \qquad 1 \le j \le M - 1 \\\ \mathcal{U}\_0^n = \mathfrak{\phi}\_1(t\_\mathbb{H}), \qquad \mathcal{U}\_M^n = \mathfrak{\phi}\_1(t\_\mathbb{H}), \qquad 0 \le n \le N \end{aligned} \tag{22}$$

where *T<sup>n</sup> <sup>j</sup>* is the local truncation error of the *j*-th equation.

Noting (6) in Theorem 1, and using Taylor expansion at *x* = *xj* in (22), we find that

$$T\_j^n = O(\tau\_{z,\text{max}}^{2-n} + h^2), \qquad 1 \le j \le M - 1, 1 \le n \le N. \tag{23}$$

Next, from (12) and (22) it is obvious that {*e<sup>n</sup> <sup>j</sup>* } is the solution of the system

$$\begin{cases} \begin{aligned} \mu \left[ \omega\_z^0 e\_j^n - \sum\_{k=1}^{n-1} \left( \omega\_z^{n-k-1} - \omega\_z^{n-k} \right) e\_j^k - \omega\_z^{n-1} e\_j^0 \right] = \Lambda e\_j^n + T\_j^n, & 1 \le j \le M - 1 \\\ e\_j^0 = 0, & 1 \le j \le M - 1 \\\ e\_0^n = e\_M^n = 0, & 0 \le n \le N. \end{aligned} \end{cases} \tag{24}$$

Hence, *<sup>e</sup><sup>n</sup>* <sup>∈</sup> *Uh*. Finally, applying Theorem 2 to (24) and noting *<sup>e</sup>*<sup>0</sup> *<sup>j</sup>* = 0, we obtain for 1 ≤ *n* ≤ *N*,

$$\|e^{n}\|^2 + c(a, z)|e^n|\_1^2 \le \frac{\Gamma(1 - a)(z(1) - z(0))^a}{4p\_0} \max\_{1 \le n \le N} \|T^n\|^2 = \left[O(\tau\_{z, \text{max}}^{2 - a} + h^2)\right]^2 \tag{25}$$

which completes the proof.

**Remark 7.** *From Theorem 3, it is easily seen that*

$$||e^{\mathfrak{n}}|| = O(\mathfrak{r}\_{z,\text{max}}^{2-\mathfrak{a}} + h^2).$$

*Hence, the temporal convergence order in the norm* · *is* (2 − *α*)*, which is optimal. On the other hand, the convergence order in* ·<sup>∞</sup> *is not optimal. In fact, from Lemma 2, (21) and the properties of quasi uniform mesh, we get*

$$\begin{split} \|\boldsymbol{\varepsilon}^{n}\|\_{\infty} &\leq \frac{1}{2} |\boldsymbol{\varepsilon}^{n}|\_{1} \quad \leq \sqrt{\frac{(\boldsymbol{z}\_{n} - \boldsymbol{z}\_{n-1})^{-a}}{4p\_{0}\Gamma(2-a)}} O(\boldsymbol{\tau}\_{\boldsymbol{z},\max}^{2-a} + h^{2}) \\ &\leq \sqrt{\frac{\boldsymbol{\tau}\_{\min,\boldsymbol{z}}^{-a}}{4p\_{0}\Gamma(2-a)}} O(\boldsymbol{\tau}\_{\boldsymbol{z},\max}^{2-a} + h^{2}) = O(\boldsymbol{\tau}\_{\boldsymbol{z},\max}^{2-\frac{3}{2}a} + h^{2}). \end{split}$$

**Remark 8.** *Theorem 3 is an extension of the work [9] in the sense that*


#### **4. Numerical Simulation**

In this section, we shall use two examples to demonstrate the efficacy of the proposed numerical scheme (12) and to verify the theoretical result in Remark 7. To be specific, we shall

• compute errors at *t* = 1 using

$$e(N, h) = \|e^N\| (\text{or} \|e^N\|\_{\infty})$$

as well as the corresponding temporal and spatial convergence orders by

$$\text{TCO} = \log\_2 \frac{\varepsilon(N, h)}{\varepsilon(2N, h)} \qquad \text{and} \qquad \text{SCO} = \log\_2 \frac{\varepsilon(N, h)}{\varepsilon(N, h/2)}$$

,

respectively;

	- **Uniform**: uniform mesh with respect to *t*;
	- **Graded**: graded mesh [17,18] with *tj* = *Z*−<sup>1</sup> (*z*(1) <sup>−</sup> *<sup>z</sup>*(0)) *<sup>j</sup> N r* + *z*(0) , <sup>0</sup> <sup>≤</sup> *<sup>j</sup>* <sup>≤</sup> *<sup>N</sup>* (let *<sup>r</sup>* <sup>=</sup> <sup>2</sup>−*<sup>α</sup> <sup>α</sup>* in the experiment to get optimal accuracy, refer to [17,18] for details);
	- **Uniform***z*: uniform mesh with respect to *<sup>z</sup>*(*t*).

In view of Remark 2, we note that all the above types of temporal meshes are particular cases of quasi uniform mesh of *z*(*t*).

Clearly, the exact solution is required to compute *e*(*N*, *h*). When the exact solution is not available (which is commonly encountered in applications), we shall use 'approximate' exact solution, which is obtained by the numerical scheme (12) with sufficiently small mesh sizes (e.g., *M* = *N* = 2000 in our experiments), as 'exact' solution to compute errors. This is reasonable as the numerical scheme (12) is convergent.

**Example 1** ([6,7,9])**.** *Consider the generalized time-fractional diffusion equation*

$$\begin{cases} \, \, ^C D\_{t; [z(t), 1]}^a u(\mathbf{x}, t) = u\_{\mathbf{x} \mathbf{x}}(\mathbf{x}, t) + f(\mathbf{x}, t), & (\mathbf{x}, t) \in (0, 1) \times (0, 1) \\\ u(\mathbf{x}, 0) = \sin(\pi \mathbf{x}), \qquad \mathbf{x} \in [0, 1] \\\ u(0, t) = 0, \qquad u(1, t) = 0, \qquad t \in (0, 1] \end{cases} \tag{26}$$

*where* <sup>0</sup> <sup>&</sup>lt; *<sup>α</sup>* <sup>&</sup>lt; <sup>1</sup>*, z*(*t*) *is a strictly increasing scale function and*

$$f(\mathbf{x}, t) = \frac{2}{\Gamma(2.15)} (\mathbf{x}^2 - \mathbf{x}) t^{1.15} + \pi^2 \sin(\pi \mathbf{x}) - 2t^2 \dots$$

*Note that when z*(*t*) = *t and α* = 0.85*, the exact solution of Equation (26) is*

$$
\mu(\mathfrak{x}, t) = \sin(\pi \mathfrak{x}) + \mathfrak{x}(\mathfrak{x} - 1)t^2.
$$

In this example, *p*(*x*, *t*) ≡ 1 and *q*(*x*, *t*) ≡ 0. Let us start with *α* = 0.85 and *z*(*t*) = *t*, *t* 0.5, *t* 2. Applying the numerical scheme (12) with fixed *h* = <sup>1</sup> <sup>512</sup> and varied *N*, we compute the errors and temporal convergence orders for three types of temporal meshes—Uniform, Graded, Uniform*z*. The results are displayed in Table 1. From the table, it is easily seen that the experimental temporal convergence orders (≈1.15) are consistent with the theoretical ones (=2 − *α*) for various scale functions *z*(*t*) and different types of temporal meshes. It is a pleasant surprise that the numerical performance in terms of maximum norm is better than the theoretical result in Remark 7.

Next, we investigate the performance of the proposed numerical scheme for different values of *α* and scale functions *z*(*t*). Here, we use the uniform mesh of *t* (Uniform) and compute *eN*.



**Table 1.** (Example 1) Temporal convergence order when *α* = 0.85, *h* = <sup>1</sup> <sup>512</sup> .

**Table 2.** (Example 1) Temporal convergence order for various *α* when *h* = <sup>1</sup> <sup>512</sup> .


**Table 3.** (Example 1) Spatial convergence order for various *α* when *N* = 2000.


Our next example is modified from an example in [19] and it involves a general operator L.

**Example 2** ([19])**.** *Consider the generalized time-fractional diffusion equation*

$$\begin{cases} \, \, ^C D\_{t; [z(t), 1]}^a u(\mathbf{x}, t) = \mathcal{L}u + f(\mathbf{x}, t), & (\mathbf{x}, t) \in (0, 1) \times (0, 1) \\\ u(\mathbf{x}, 0) = \sin(\pi \mathbf{x}), & \mathbf{x} \in [0, 1] \\\ u(0, t) = 0, & u(1, t) = 0, & t \in (0, 1] \end{cases} \tag{27}$$

*where* L*u* = *∂x*(*p*(*x*, *t*)*∂xu*) − *q*(*x*, *t*)*u,*

$$p(\mathbf{x},t) = 2 - \cos(\mathbf{x}t), \qquad q(\mathbf{x},t) = 1 - \sin(\mathbf{x}t)$$

*and*

$$\begin{aligned} f(\mathbf{x},t) &= \quad \left[ \frac{\Gamma(4+a)}{6}t^3 + \frac{2}{\Gamma(3-a)}t^{2-a} + \left(t^{3+a} + t^2 + 1\right) \left(\pi^2 p(\mathbf{x},t) + q(\mathbf{x},t)\right) \right] \sin(\pi \mathbf{x}) \\ &- \pi \left(t^{4+a} + t^3 + t\right) \sin(\pi t) \cos(\pi \mathbf{x}). \end{aligned}$$

*When z*(*t*) = *t, the exact solution of (27) is*

$$
\mu(\mathfrak{x}, t) = \sin(\pi \mathfrak{x}) (t^{3+\mathfrak{a}} + t^2 + 1).
$$

First, consider (27) when *α* = 0.5. With fixed *h* = <sup>1</sup> <sup>2000</sup> , we shall apply the numerical scheme (12) and compute the errors *eN*, *eN*<sup>∞</sup> and temporal convergence orders. The results are presented in Table 4, and it is clear that the numerical scheme (12) works well for different types of temporal meshes as well as for a wide range of problems (i.e., different *z*(*t*)). The experimental temporal convergence orders in · (≈1.5) agree with the theoretical ones (=2 − *α*), while once again it is a pleasant surprise that the numerical performance in terms of maximum norm is better than the theoretical result in Remark 7.

Next, we investigate the temporal convergence and spatial convergence of the numerical scheme (12) for different values of *α* and scale functions *z*(*t*). Here, we use the uniform mesh of *<sup>t</sup>* and compute *eN*. The results are presented in Tables 5 and 6. We observe that the experimental temporal/spatial convergence orders are consistent with the theoretical result.

**Table 4.** (Example 2) Temporal convergence order when *α* = 0.5, *h* = <sup>1</sup> <sup>2000</sup> .



**Table 5.** (Example 2) Temporal convergence order for various *α* when *h* = <sup>1</sup> <sup>2000</sup> .

**Table 6.** (Example 2) Spatial convergence order for various *α* when *N* = 2000.


#### **5. Conclusions**

In this paper, we derive a numerical scheme based on a general temporal mesh for the generalized time-fractional diffusion problem. The main idea involves the generalized linear interpolation. The stability and convergence of the proposed numerical scheme is established rigorously using energy method. More importantly, it is shown that the global convergence order is *O*(*τ*2−*<sup>α</sup> <sup>z</sup>*,max + *h*2) that extends the previous work [9]. For future work, we plan to investigate (i) the validity of the proposed scheme for nonsmooth data; (ii) high order methods based on general temporal mesh and spatial mesh for generalized fractional problems with smooth as well as nonsmooth data. We believe this will make the numerical scheme more applicable in reality.

**Author Contributions:** Formal analysis, X.L. and P.J.Y.W.; Investigation, X.L. and P.J.Y.W.; writing original draft, X.L. and P.J.Y.W.; writing—review and editing, X.L. and P.J.Y.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **To Solve Forward and Backward Nonlocal Wave Problems with Pascal Bases Automatically Satisfying the Specified Conditions**

**Chein-Shan Liu 1, Chih-Wen Chang 2, Yung-Wei Chen <sup>3</sup> and Jian-Hung Shen 3,\***


**Abstract:** In this paper, the numerical solutions of the backward and forward non-homogeneous wave problems are derived to address the nonlocal boundary conditions. When boundary conditions are not set on the boundaries, numerical instability occurs, and the solution may have a significant boundary error. For this reason, it is challenging to solve such nonlinear problems by conventional numerical methods. First, we derive a nonlocal boundary shape function (NLBSF) from incorporating the Pascal triangle as free functions; hence, the new, two-parameter Pascal bases are created to automatically satisfy the specified conditions for the solution. To satisfy the wave equation in the domain by the collocation method, the solution of the forward nonlocal wave problem can be quickly obtained with high precision. For the backward nonlocal wave problem, we construct the corresponding NLBSF and Pascal bases, which exactly implement two final time conditions, a leftboundary condition and a nonlocal boundary condition; in addition, the numerical method for the backward nonlocal wave problem under two-side, nonlocal boundary conditions is also developed. Nine numerical examples, including forward and backward problems, are tested, demonstrating that this scheme is more effective and stable. Even for boundary conditions with a large noise at final time, the solution recovered in the entire domain for the backward nonlocal wave problem is accurate and stable. The accuracy and efficiency of the method are validated by comparing the estimation results with the existing literature.

**Keywords:** backward nonlocal wave equation; Pascal bases automatically satisfying specified conditions; integral boundary condition; nonlocal boundary shape function

**MSC:** 35L70

#### **1. Introduction**

Integral-type, nonlocal boundary conditions (BCs) are an interesting area of a fastdeveloping differential equations theory. These problems arise in various fields of physics, mechanics, biology, biotechnology, etc. Nonlocal BCs may come up when the value of the solution on the boundary is connected with the values inside the domain. Theoretical and numerical investigation of this kind of problem is actually valuable, and much attention is given to it in the scientific literature [1–6]. Different, nonlocal BCs are also discussed in partial differential equations (PDEs), for example, Dehghan [7] proposed the numerical solution of several finite difference methods for the one-dimensional, non-classic boundary value problem. Saadatmandi and Dehghan [8] developed a numerical technique based on the shifted Legendre tau technique to demonstrate its validity and applicability for the hyperbolic partial differential equation with an integral condition. Dehghan and Saadatmandi [9] used the variational iteration method for solving the one-dimensional wave equation with classical and integral boundary conditions; this method changed the wave equation with nonlocal BCs into a direct problem. For forward problems, some

**Citation:** Liu, C.-S.; Chang, C.-W.; Chen, Y.-W.; Shen, J.-H. To Solve Forward and Backward Nonlocal Wave Problems with Pascal Bases Automatically Satisfying the Specified Conditions. *Mathematics* **2022**, *10*, 3112. https://doi.org/ 10.3390/math10173112

Academic Editor: Patricia J. Y. Wong

Received: 26 July 2022 Accepted: 24 August 2022 Published: 30 August 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

<sup>1</sup> Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan

solutions using theory and numerical methods for the nonlocal problems of the 1D wave equation were studied in [10–12].

As pointed out by Ames and Straughan [13], the backward wave problem has pivotal applications in optimal control theory and geophysics. The backward wave problem is an ill-posed problem, which has been studied in [14–20]. Especially when we consider the backward wave problem under nonlocal boundary conditions, the resulting problem is highly ill-posed, and the numerical method must be designed specifically to overcome the ill-posed property of the backward problem. The idea of a nonlocal boundary shape function (NLBSF) was first developed in [21] to solve the nonlocal, parabolic-type PDE, but has not yet been applied to hyperbolic-type PDE in the literature. We employ the NLBSF to resolve the nonlocal wave problem.

In this paper, we subject the wave equation to an unconventional right-boundary condition which includes an integral term over the spatial domain. In this situation, we encounter a nonlocal wave problem, the solution to which might suffer a large boundary error, since the BC is not given on a boundary. For this reason, it is hard to use the conventional numerical method to tackle this sort of problem. It is important in the field of numerical simulations of nonlocal wave problems to reduce the boundary error so that the error in the entire domain can be reduced. To guarantee the fulfilment of the nonlocal BC, a novel method with the Pascal bases automatically satisfying the specified conditions is pursued in the paper.

Sequentially, the forward wave problem of a one-dimensional wave equation under a nonlocal BC on the right-end is described in Section 2, wherein we construct the so-called NLBSF with the help of the nonlocal shape functions derived. The NLBSF satisfies all the conditions specified for the forward nonlocal wave problem with a free function involved. In Section 3, we develop a numerical method with two-parameter Pascal bases to solve the forward nonlocal wave problem. The bases satisfying all conditions are obtained by taking the Pascal polynomials for the free function. Four testing examples of the forward nonlocal wave problem are presented in Section 4, the high accuracy of which can be appreciated. In Section 5, we develop a numerical method with two-parameter Pascal bases relying on the Pascal polynomials to solve the backward nonlocal wave problem. The bases satisfying all the conditions are specified for the backward nonlocal wave problem. Three testing examples with a large noise being imposed on the final time data of the backward nonlocal wave problem are exhibited in Section 6, the robustness and high accuracy of which can be observed. In Section 7, the method of NLBSF is extended to solve the backward nonlocal wave problem under two-side nonlocal BCs. The conclusions are drawn in Section 8.

#### **2. A Nonlocal Wave Problem**

The one-dimensional wave equation is equipped with a nonlocal condition:

$$u\_{tt}(\mathbf{x},t) - u\_{xx}(\mathbf{x},t) = F(\mathbf{x},t), (\mathbf{x},t) \in \Omega := \left\{ (\mathbf{x},t) \Big| \begin{matrix} 0 < \mathbf{x} < l, \ 0 < t \le t\_f \end{matrix} \right\},\tag{1}$$

$$
\mu(\mathbf{x},0) = f(\mathbf{x}), \ u\_t(\mathbf{x},0) = \mathcal{g}(\mathbf{x}), \tag{2}
$$

$$u(0,t) = p(t), \ \int\_0^l u(\mathbf{x}, \mathbf{t})d\mathbf{x} = q(\mathbf{t}), \tag{3}$$

where *f*(*x*) and *g*(*x*) are initial conditions, *F*(*x*, *t*) is the given function, *q*(*t*) and *p*(*t*) are boundary conditions and the last condition is different from the conventional boundary condition on the right end. The data *f*(*x*), *g*(*x*), *q*(*t*) and *p*(*t*) must satisfy

$$\int\_{0}^{l} f(\mathbf{x})d\mathbf{x} = q(0), \int\_{0}^{l} \mathbf{g}(\mathbf{x})d\mathbf{x} = \dot{q}(0), \tag{4}$$

$$f(0) = p(0), \ g(0) = \dot{p}(0),\tag{5}$$

which are compatibility conditions derived from Equation (3) with *t* = 0 and Equation (2).

We first derive two main results to satisfy the specified conditions (2) and (3) and then use them to solve Equation (1) to Equation (3).

**Theorem 1.** *The function*

$$E^{0}(\mathbf{x},t) = w(\mathbf{x},t) - s\_1(\mathbf{x})[w(0,t) - p(t)] - s\_2(\mathbf{x})\left[\int\_0^l w(\mathbf{x},t)d\mathbf{x} - q(t)\right],\tag{6}$$

<sup>∀</sup> *<sup>w</sup>*(*x*, *<sup>t</sup>*) <sup>∈</sup> *<sup>C</sup>*1(Ω) *satisfies the conditions in Equation (3), where the nonlocal shape functions*

$$s\_1(\mathbf{x}) = 1 - \frac{2\mathbf{x}}{l},\ s\_2(\mathbf{x}) = 1 - \frac{2\mathbf{x}}{l^2} \tag{7}$$

*are derived from*

$$s\_1(0) = 1, \int\_0^l s\_1(\mathbf{x})d\mathbf{x} = 0, \ s\_2(0) = 0, \int\_0^l s\_2(\mathbf{x})d\mathbf{x} = 1. \tag{8}$$

**Proof.** Inserting *x* = 0 into Equation (6) and taking Equation (8) into account generates

$$\begin{split} E^{0}(0,t) &= w(0,t) - s\_1(0)[w(0,t) - p(t)] - s\_2(0) \left[ \int\_0^l w(\mathbf{x},t)d\mathbf{x} - q(t) \right] \\ &= w(0,t) - [w(0,t) - p(t)] = p(t). \end{split} \tag{9}$$

Then, we prove

$$\int\_{0}^{l} \mathbf{E}^{0}(\mathbf{x}, t) d\mathbf{x} = q(t),\tag{10}$$

by inserting Equation (6) for *E*0(*x*, *t*),

$$\begin{aligned} \int\_0^l E^0(\mathbf{x}, t) d\mathbf{x} &= \int\_0^l w(\mathbf{x}, t) d\mathbf{x} - \int\_0^l s\_1(\mathbf{x}) d\mathbf{x} [w(0, t) - p(t)] \\ &- \int\_0^l s\_2(\mathbf{x}) d\mathbf{x} \left[ \int\_0^l w(\mathbf{x}, t) d\mathbf{x} - q(t) \right], \end{aligned} \tag{11}$$

which, taking Equation (8) into account, yields

$$\int\_{0}^{l} E^{0}(\mathbf{x}, t) d\mathbf{x} = \int\_{0}^{l} w(\mathbf{x}, t) d\mathbf{x} - \left[ \int\_{0}^{l} w(\mathbf{x}, t) d\mathbf{x} - q(t) \right] = q(t). \tag{12}$$

Hence, this theorem is proved.

Notice that the nonlocal shape functions *s*1(*x*) and *s*2(*x*) were used by Dehghan and Saadatmandi [9] to transform Equation (1) to Equation (3) into a problem with a homogeneous boundary condition and a nonlocal condition for a new variable. Here, we give a different approach.

For *E*0(*x*, *t*), we have the following compatibility conditions:

$$\int\_{0}^{l} E^{0}(\mathbf{x}, \mathbf{0}) d\mathbf{x} = \int\_{0}^{l} f(\mathbf{x}) d\mathbf{x}, \int\_{0}^{l} E\_{t}^{0}(\mathbf{x}, \mathbf{0}) d\mathbf{x} = \int\_{0}^{l} \mathbf{g}(\mathbf{x}) d\mathbf{x}.\tag{13}$$

It follows from Equation (12) that

$$\int\_{0}^{l} E^{0}(\mathbf{x}, 0) d\mathbf{x} = q(0), \\ \int\_{0}^{l} E^{0}\_{t}(\mathbf{x}, 0) d\mathbf{x} = \dot{q}(0). \tag{14}$$

Upon comparing them with Equations (4) and (13), they are verified. It follows from Equations (9) and (5) that

$$E^0(0,0) = p(0) = f(0),\ \ E^0\_t(0,0) = \dot{p}(0) = \mathcal{g}(0). \tag{15}$$

Now, we prove that there exists a function *E*(*x*, *t*) which satisfies the specified conditions (2) and (3).

**Theorem 2.** *The function*

$$E(\mathbf{x},t) = E^0(\mathbf{x},t) - \left(1 - t^2\right) \left[E^0(\mathbf{x},0) - f(\mathbf{x})\right] - t\left[E^0\_t(\mathbf{x},0) - g(\mathbf{x})\right] \tag{16}$$

*satisfies the conditions (2) and (3).*

**Proof.** We first prove

$$E(\mathbf{x},0) = f(\mathbf{x}),\ E\_l(\mathbf{x},0) = \mathcal{g}(\mathbf{x}).\tag{17}$$

Inserting *t* = 0 into Equation (16), we have

$$E(\mathbf{x},0) = E^0(\mathbf{x},0) - \left[E^0(\mathbf{x},0) - f(\mathbf{x})\right] = f(\mathbf{x}).\tag{18}$$

Differentiating Equation (16) to *t* and inserting *t* = 0, one has

$$\begin{split} E\_t(\mathbf{x},0) &= \left\{ E\_t^0(\mathbf{x},t) + 2t \left[ E^0(\mathbf{x},0) - f(\mathbf{x}) \right] - \left[ E\_t^0(\mathbf{x},0) - g(\mathbf{x}) \right] \right\} \Big|\_{t=0} \\ &= E\_t^0(\mathbf{x},0) - \left[ E\_t^0(\mathbf{x},0) - g(\mathbf{x}) \right] = g(\mathbf{x}). \end{split} \tag{19}$$

Then, we prove

$$E(0,t) = p(t), \ \int\_0^l E(\mathbf{x}, t)d\mathbf{x} = q(t). \tag{20}$$

Inserting *x* = 0 into Equation (16) and using Equation (9) and the compatibility conditions *E*0(0, 0) = *f*(0) and *E*<sup>0</sup> *<sup>t</sup>* (0, 0) = *g*(0) in Equation (15) yields

$$E(0,t) = E^0(0,t) - \left(1 - t^2\right) \left[E^0(0,0) - f(0)\right] - t\left[E^0\_t(0,0) - g(0)\right] = E^0(0,t) = p(t). \tag{21}$$

It follows from Equation (16) that

$$\begin{split} \int\_{0}^{l} E(\mathbf{x}, t) d\mathbf{x} &= \int\_{0}^{l} E^{0}(\mathbf{x}, t) d\mathbf{x} - \left(1 - t^{2}\right) \int\_{0}^{l} [E^{0}(\mathbf{x}, 0) - f(\mathbf{x})] d\mathbf{x} \\ &- t \int\_{0}^{l} [E^{0}\_{t}(\mathbf{x}, 0) - g(\mathbf{x})] d\mathbf{x}, \end{split} \tag{22}$$

which, with the aid of Equations (12) and (13), becomes

$$\int\_{0}^{l} E(\mathbf{x}, t) d\mathbf{x} = \int\_{0}^{l} E^{0}(\mathbf{x}, t) d\mathbf{x} = q(t). \tag{23}$$

Consequently, this theorem is proved.

#### **3. Numerical Method for Forward Nonlocal Wave Problem**

Let

$$w\_{ij}(\mathbf{x}, t) := \mathbf{x}^{i-j} t^{j-1}, \ i = 1, \ldots, j = 1, \ldots, i,\tag{24}$$

be the Pascal triangle in terms of *x* and *t* [22]. We can reconstruct *wij*(*x*, *t*) to be the Pascal bases for *u*(*x*, *t*) in Equation (1) to Equation (3) based on Theorem 2.

**Theorem 3.** *For the Pascal polynomial wij*(*x*, *t*) *and*

$$E\_{i\bar{j}}^{0}(\mathbf{x},t) = w\_{i\bar{j}}(\mathbf{x},t) - s\_1(\mathbf{x}) \left[ w\_{i\bar{j}}(0,t) - p(t) \right] - s\_2(\mathbf{x}) \left[ \frac{l^{i-j+1}}{i-j+1} t^{j-1} - q(t) \right],\tag{25}$$

*the two-parameter functions*

$$E\_{\vec{i}\vec{j}}(\mathbf{x},t) = E\_{\vec{i}\vec{j}}^{0}(\mathbf{x},t) - \left(1 - t^{2}\right) \left[E\_{\vec{i}\vec{j}}^{0}(\mathbf{x},0) - f(\mathbf{x})\right] - t\left[E\_{\vec{i}\vec{j},t}^{0}(\mathbf{x},0) - \mathbf{g}(\mathbf{x})\right] \tag{26}$$

*are Pascal bases to match Equations (2) and (3)*.

**Proof.** In Theorem 2, *E*(*x*, *t*) replaced by *Eij*(*x*, *t*) and *E*0(*x*, *t*) by *E*<sup>0</sup> *ij*(*x*, *t*), inserting *wij*(*x*, *t*) for *w*(*x*, *t*) into Equation (6) and integrating *<sup>l</sup>* <sup>0</sup> *<sup>x</sup>i*−*<sup>j</sup> tj*−1*dx* out, this theorem is proved.

The two-parameter functions *Eij*(*x*, *t*) in Equation (26) are called the Pascal bases, which are used to solve the forward nonlocal wave Equation (1) to Equation (3) by

$$u(x,t) = \sum\_{j=1}^{m} \sum\_{k=1}^{j} a\_{jk} s\_{jk} E\_{jk}(x,t),\tag{27}$$

where *ajk* are subjected to

$$\sum\_{j=1}^{m} \sum\_{k=1}^{j} a\_{jk} = 1,\tag{28}$$

such that *u*(*x*, *t*) fulfills Equations (2) and (3).

Inserting Equation (27) into Equation (1) and including Equation (28), we determine *ajk* by

$$\sum\_{j=1}^{m} \sum\_{k=1}^{j} a\_{jk} s\_{jk} \left[ E\_{jk,tt} \left( \mathbf{x}\_{i\prime} \, y\_{j} \right) - E\_{jk,xx} \left( \mathbf{x}\_{i\prime} \, y\_{j} \right) \right] = F \left( \mathbf{x}\_{i\prime} \, t\_{j} \right), \tag{29}$$

where *n*<sup>1</sup> and *n*<sup>2</sup> are the grid numbers in the spatial and time direction *xi* = *i*(*l*/*n*<sup>1</sup> + 1), *tj* = *j tf* /*n*<sup>2</sup> and *N* = *n*<sup>1</sup> × *n*2. Consequently, the *N* + 1 linear Equations (28) and (29) are written as a matrix-vector form:

$$\mathbf{A}\mathbf{a} = \mathbf{b},\tag{30}$$

where **A** is the coefficient matrix, **b** is given source term and **a** is the vector form of *ajk*. Let *sk* be the *k*th component of the vectorization of *sjk* which has multiple scales given in [23] by

$$s\_k = \frac{R\_0}{||\!|\!|\mathbf{a}\_k\!\!|},\tag{31}$$

where **a***<sup>k</sup>* denotes the *k*th column of **A**, and *R*<sup>0</sup> is a characteristic length which can increase numerical stability and accuracy. Solving the linear system (30), we can obtain *ajk* and then *u*(*x*, *y*) is calculated from Equation (27).

#### **4. Examples for Forward Nonlocal Wave Problem**

**Example 1.** *Consider the exact solution as follows:*

$$u(\mathbf{x},t) = \mathbf{x}^2 + 2t - 3\mathbf{x}^2t - \mathbf{x}^4 + \sin(2\pi t). \tag{32}$$

*The data F*(*x*, *t*), *f*(*x*), *g*(*x*), *q*(*t*) *and p*(*t*) *can be obtained*

$$F(\mathbf{x},t) = u\_{tt}(\mathbf{x},t) - u\_{xx}(\mathbf{x},t) = 12\mathbf{x}^2 - 2 + 6t - 4\pi^2\sin(2\pi t),\tag{33}$$

$$\begin{aligned} f(\mathbf{x}) &= \mathbf{x}^2 - \mathbf{x}^4, \quad g(\mathbf{x}) = 2 + 2\pi - 3\mathbf{x}^2, \\ p(t) &= 2t + \sin(2\pi t), \quad q(t) = \frac{l^3}{5} - \frac{l^5}{5} + \left(2l - l^3\right)t + l\sin(2\pi t). \end{aligned} \tag{34}$$

We take *l* = 1, *tf* = 1, *m* = 5, *R*<sup>0</sup> = 0.1 and *N* = 5 × 5. Figure 1 shows an absolute maximum error (ME) of *u*(*x*, *t*) with respect to *t*. When the convergence criteria *ε* = 10−10, the total iteration number of the conjugate gradient method (CGM) is 10. Figure 2 displays ME(*u*) with respect to *x* at *tf* = 1. We can observe that the solution is very accurate with ME <sup>=</sup> 1.45 <sup>×</sup> <sup>10</sup>−13. In paper [24], by using the boundary consistent method for the usual wave equation with the Dirichlet boundary conditions, ME <sup>=</sup> 2.01 <sup>×</sup> <sup>10</sup>−<sup>8</sup> and is much larger than 1.45 <sup>×</sup> <sup>10</sup>−13. The current solution is much more accurate than that in [24].

**Figure 1.** For Example 1: ME(*u*) versus *t*.

**Figure 2.** For Example 1: ME(u) versus *x* at *tf* = 1.

**Example 2.** *In order to further display the accuracy of the presented method we consider the exact solution as follows:*

$$u(\mathbf{x},t) = \exp(\mathbf{x} + \sin t). \tag{35}$$

*Then, F*(*x*, *t*), *f*(*x*), *g*(*x*), *q*(*t*) *and p*(*t*) *can be expressed as follows:*

$$F(\mathbf{x},t) = u\_{lt}(\mathbf{x},t) - u\_{\mathbf{x}\mathbf{x}}(\mathbf{x},t) = \left(\cos^2 t - \sin t - 1\right) \exp(\mathbf{x} + \sin t). \tag{36}$$

$$f(\mathbf{x}) = \mathbf{g}(\mathbf{x}) = \mathbf{e}^{\mathbf{x}},$$

$$p(t) = \exp(\sin t), \quad q(t) = \begin{pmatrix} \mathbf{e}^{l} - 1 \end{pmatrix} \exp(\sin t). \tag{37}$$

We take *l* = 1, *tf* = 1, *m* = 14, *R*<sup>0</sup> = 0.1 and *N* = 12 × 12. When the convergence criteria *<sup>ε</sup>* <sup>=</sup> <sup>10</sup>−9, the total iteration number of the CGM is 7200, and ME <sup>=</sup> 2.93 <sup>×</sup> <sup>10</sup>−8. Further, when the iteration number is stopped at the 2000 step, the MEs of *u*(*x*, *t*) are plotted versus *t* in Figure 3. The ME(*u*) with respect to *x* at *tf* = 1 is shown in Figure 4. Obviously, the solution is quite accurate with ME <sup>=</sup> 4.54 <sup>×</sup> <sup>10</sup>−8. When considering the same setting as above and *l* = 8, in Figure 5, the solid line displays ME(*u*) with respect to *x*,

where ME <sup>=</sup> 4.56 <sup>×</sup> <sup>10</sup>−1, and max *<sup>u</sup>*(*x*, *<sup>t</sup>*) is 5689.34. Therefore, the solution of this method is acceptable.

**Figure 3.** For Example 2: ME(*u*) versus *t* in the time interval.

**Figure 4.** For Example 2 with l = 1: ME(u) versus *x* at *tf* = 1.

**Figure 5.** For Example 2 with l = 8: ME(u) versus *x* at *tf* = 1.

**Example 3** . *This example is for the linear Klein–Gordon equation:*

$$
\mu\_{tt}(\mathbf{x},t) - \mu\_{xx}(\mathbf{x},t) + \mathfrak{H}(\mathbf{x},t) = \mathbf{0}, \quad (\mathbf{x},t) \in \Omega. \tag{38}
$$

*We set the exact solution as follows:*

$$
\mu(x,t) = \sin(x - 2t). \tag{39}
$$

*Then, f*(*x*), *g*(*x*), *q*(*t*) *and p*(*t*) *can be expressed as follows:*

$$\begin{aligned} f(x) &= \sin x, & g(x) &= -2\cos x, \\ p(t) &= -\sin 2t, & q(t) &= \cos 2t - \cos(l - 2t). \end{aligned} \tag{40}$$

We take *l* = 1, *tf* = 1, *m* = 11, *R*<sup>0</sup> = 0.1 and *N* = 20 × 20. When convergence criteria *<sup>ε</sup>* <sup>=</sup> <sup>10</sup>−10, the total iteration number (TIN) of the CGM is 1221, and ME <sup>=</sup> 7.27 <sup>×</sup> <sup>10</sup>−8. Further, when the iteration number is at the 1000 step, the MEs of *u*(*x*, *t*) are plotted versus *t* and *x*, as shown in Figures 6 and 7. Obviously, the solution is quite accurate with ME <sup>=</sup> 5.06 <sup>×</sup> <sup>10</sup><sup>−</sup>8. For the different convergence criteria, the convergence results are shown in Table 1. Hence, this method can quickly obtain solutions without using higher-order polynomials or strict convergence conditions.

**Figure 6.** For Example 3: ME(*u*) versus *t* in the time interval.

**Figure 7.** For Example 3: ME(u) versus *x* at *tf* = 1.

**Table 1.** The iteration number and ME under the different convergence criteria.


**Example 4.** *This example is given in [8];*

$$u(x,t) = \cos(\pi x)\sin(\pi t)\tag{41}$$

*is the exact solution.*

$$\begin{aligned} f(\mathbf{x}) &= 0, & g(\mathbf{x}) &= \pi \cos(\pi \mathbf{x}), \\ p(t) &= \sin(\pi t), & q(t) &= 0. \end{aligned} \tag{42}$$

*Instead of the Pascal polynomials, for this example, we take*

$$w\_{ij}(\mathbf{x}, t) = \cos(i\pi\mathbf{x})\sin(j\pi t), \ 1 \le i, j \le m.$$

We take *l* = 1, *tf* = 0.5, *m* = 5, *R*<sup>0</sup> = 1 and *N* = 5 × 5. In this case, we use the Gaussian elimination to solve the linear system. Figure 8 displays the MEs of *u*(*x*, *t*) versus *<sup>t</sup>*, which are highly accurate with ME <sup>=</sup> 7.77 <sup>×</sup> <sup>10</sup>−<sup>16</sup> and are much more accurate than [8]. Figure 9 displays ME(*u*) with respect to *x* at *tf* = 0.5.

**Figure 8.** For Example 4: ME(*u*) versus *t* in the time interval.

**Figure 9.** For Example 4: ME(u) versus *x* at *tf* = 0.5.

#### **5. Numerical Method for Backward Nonlocal Wave Problem**

We consider the backward nonlocal wave problem and replace Equation (2) by the final time conditions:

$$
\mu\left(\mathbf{x}, \mathbf{t}\_f\right) = h(\mathbf{x}), \ \mathfrak{u}\_t\left(\mathbf{x}, \mathbf{t}\_f\right) = r(\mathbf{x}).\tag{43}
$$

The data *h*(*x*), *r*(*x*), *q*(*t*) and *p*(*t*) satisfy

$$\int\_{0}^{l} h(\mathbf{x})d\mathbf{x} = q\left(t\_f\right), \quad \int\_{0}^{l} r(\mathbf{x})d\mathbf{x} = \dot{q}\left(t\_f\right),\tag{44}$$

$$h(0) = p\left(t\_f\right), \ r(0) = \dot{p}\left(t\_f\right),\tag{45}$$

which are available from Equation (3) with *t* = *tf* and using Equation (43).

For the backward nonlocal wave problem, Theorem 2 is modified as follows.

**Theorem 4.** *The following NLBSF:*

$$E(\mathbf{x},t) = E^0(\mathbf{x},t) - \left[1 + \left(t - t\_f\right)^2\right] \left[E^0(\mathbf{x},t\_f) - h(\mathbf{x})\right] - \left(t - t\_f\right)\left[E^0\_t\left(\mathbf{x},t\_f\right) - r(\mathbf{x})\right] \tag{46}$$

*satisfies the conditions (43) and (3), where E*0(*x*, *t*) *is still given by Equation (6).*

**Proof.** Inserting *t* = *tf* into Equation (46), we have

$$E\left(\mathbf{x}, t\_f\right) = E^0\left(\mathbf{x}, t\_f\right) - \left[E^0\left(\mathbf{x}, t\_f\right) - h(\mathbf{x})\right] = h(\mathbf{x}).\tag{47}$$

Differentiating Equation (46) to *t* and inserting *t* = *tf* , one has

$$\begin{split} E\_t \left( \mathbf{x}, t\_f \right) &= \left\{ E\_t^0(\mathbf{x}, t) - 2 \left( \mathbf{t} - \mathbf{t}\_f \right) \left[ E^0(\mathbf{x}, t\_f) - h(\mathbf{x}) \right] - \left[ E\_t^0(\mathbf{x}, t\_f) - r(\mathbf{x}) \right] \right\} \Big|\_{t = t\_f} \\ &= E\_t^0(\mathbf{x}, t\_f) - \left[ E\_t^0(\mathbf{x}, t\_f) - r(\mathbf{x}) \right] = r(\mathbf{x}). \end{split} \tag{48}$$

Next, we prove the compatibility conditions for *E*0(*x*, *t*):

$$\int\_{0}^{l} E^{0} \left( \mathbf{x}, \mathbf{t}\_{f} \right) d\mathbf{x} = \int\_{0}^{l} h(\mathbf{x}) d\mathbf{x}, \quad \int\_{0}^{l} E\_{t}^{0} \left( \mathbf{x}, \mathbf{t}\_{f} \right) d\mathbf{x} = \int\_{0}^{l} r(\mathbf{x}) d\mathbf{x}, \tag{49}$$

It follows from Equation (12) that

$$\int\_{0}^{l} \mathbb{E}^{0} \left( \mathbf{x}, \mathbf{t}\_{f} \right) d\mathbf{x} = q \left( \mathbf{t}\_{f} \right), \quad \int\_{0}^{l} \mathbb{E}\_{t}^{0} \left( \mathbf{x}, \mathbf{t}\_{f} \right) d\mathbf{x} = \dot{q} \left( \mathbf{t}\_{f} \right). \tag{50}$$

Upon comparing them to Equation (44), we can derive Equation (49). Similarly, it follows from Equations (9) and (45) that

$$E^0(0, t\_f) = p\left(t\_f\right) = h(0), \quad E^0\_t\left(0, t\_f\right) = \dot{p}\left(t\_f\right) = r(0). \tag{51}$$

Finally, we prove that *E*(*x*, *t*) satisfies

$$E(0,t) = p(t), \ \int\_0^l E(\mathbf{x}, t)d\mathbf{x} = q(t). \tag{52}$$

Inserting *x* = 0 into Equation (46) and using Equation (9) and the compatibility conditions *E*<sup>0</sup> 0, *tf* = *h*(0) and *E*<sup>0</sup> *t* 0, *tf* = *r*(0) in Equation (51), one has

$$\begin{split} E(0,t) &= E^0(0,t) - \left[ 1 + \left(t - t\_f\right)^2 \right] \left[ E^0\left(0, t\_f\right) - h(0) \right] - \left(t - t\_f\right) \left[ E^0\_t\left(0, t\_f\right) - r(0) \right] \\ &= E^0(0,t) = p(t) .\end{split} \tag{53}$$

From Equation (46) it follows that

$$\begin{aligned} \int\_{0}^{l} E(\mathbf{x}, t) d\mathbf{x} &= \int\_{0}^{l} E^{0}(\mathbf{x}, t) d\mathbf{x} - \left[ 1 + \left( t - t\_{f} \right)^{2} \right] \int\_{0}^{l} \left[ E^{0} \left( \mathbf{x}, t\_{f} \right) - h(\mathbf{x}) \right] d\mathbf{x} \\ &- \left( t - t\_{f} \right) \int\_{0}^{l} \left[ E^{0}\_{t} \left( \mathbf{x}, t\_{f} \right) - r(\mathbf{x}) \right] d\mathbf{x}, \end{aligned} \tag{54}$$

which, with the aid of Equations (12) and (49), becomes

$$\int\_{0}^{l} E(\mathbf{x}, t) d\mathbf{x} = \int\_{0}^{l} E^{0}(\mathbf{x}, t) d\mathbf{x} = q(t). \tag{55}$$

The proof is ended.

Replacing *E*(*x*, *t*) and *E*0(*x*, *t*) in Theorem 4 by *Eij*(*x*, *t*) and *E*<sup>0</sup> *ij*(*x*, *t*), respectively, Theorem 3 is still applicable for the backward nonlocal wave problem but with

$$E\_{lj}(\mathbf{x},t) = E\_{lj}^{0}(\mathbf{x},t) - \left[1 + \left(t - t\_f\right)^2\right] \left[E\_{lj}^{0}(\mathbf{x},t\_f) - h(\mathbf{x})\right] - \left(t - t\_f\right) \left[E\_{lj,t}^{0}\left(\mathbf{x},t\_f\right) - r(\mathbf{x})\right],\tag{56}$$

which automatically satisfies the conditions (43) and (3),

To solve the backward nonlocal wave problem in Equations (1), (43) and (3), we take

$$\mu(\mathbf{x},t) = \sum\_{j=1}^{m} \sum\_{k=1}^{j} c\_{jk} s\_{jk} E\_{jk}(\mathbf{x}, t), \tag{57}$$

where

$$\sum\_{j=1}^{m} \sum\_{k=1}^{j} c\_{jk} = 1.\tag{58}$$

Other procedures are similar to that in Section 3.

#### **6. Numerical Examples for Backward Nonlocal Wave Problem**

To test the backward nonlocal wave problem, we add a noise *s* on

$$
\hat{h}(\mathbf{x}) = h(\mathbf{x}) + s\mathbf{R}, \ \mathbf{\dot{r}}(\mathbf{x}) = r(\mathbf{x}) + s\mathbf{R}, \tag{59}
$$

where *R* is a random number. We fix *s* = 0.1 for all testing examples given below.

**Example 5.** *For Example 1, we consider the final time conditions:*

$$h(\mathbf{x}) = \mathbf{x}^2 + \left(2 - 3\mathbf{x}^2\right)t\_f - \mathbf{x}^4 + \sin\left(2\pi t\_f\right), \quad r(\mathbf{x}) = 2 - 3\mathbf{x}^2 + 2\pi\cos\left(2\pi t\_f\right). \tag{60}$$

If no noise is added, i.e., *s* = 0 under *l* = 1, *tf* = 1, *m* = 5, *R*<sup>0</sup> = 0.1, TIN = 500 and *<sup>N</sup>* <sup>=</sup> <sup>10</sup> <sup>×</sup> 10, *<sup>u</sup>*(*x*, *<sup>t</sup>*) is very accurate with ME <sup>=</sup> 9.14 <sup>×</sup> <sup>10</sup>−13, which is slightly worse than 1.45 <sup>×</sup> <sup>10</sup>−<sup>13</sup> for the forward wave problem, as presented in Example 1.

Under *l* = 1, *tf* = 1, *m* = 5, *R*<sup>0</sup> = 0.1, TIN = 500 and *N* = 5 × 5, the solution is obtained very quickly. In Figure 10, the dashed line shows the ME of *u*(*x*, *t*) with respect to *<sup>x</sup>*, of which ME <sup>=</sup> 3.04 <sup>×</sup> <sup>10</sup>−3, where max *<sup>u</sup>*(*x*, *<sup>t</sup>*) is 1.9988. Then, we take *tf* <sup>=</sup> 10 and *N* = 20 × 20, and other parameters remain the same. In Figure 10, the solid line displays ME(*u*) with respect to *<sup>x</sup>*, where ME <sup>=</sup> 4.81 <sup>×</sup> <sup>10</sup>−3, and max *<sup>u</sup>*(*x*, *<sup>t</sup>*) is 19.98. Hence, the method can obtain a stable and accurate solution with *O* 10−<sup>3</sup> even for the final time with noise disturbance.

**Figure 10.** For Example 5 of the backward nonlocal wave problem: ME(*u*) versus *x* with different final times.

**Example 6.** *Then, we consider*

$$h(\mathbf{x}) = \exp(\mathbf{x} + \sin t\_f), \ r(\mathbf{x}) = \cos t\_f \exp(\mathbf{x} + \sin t\_f). \tag{61}$$

*Other data are given in Example 2.*

We take *l* = 2, *tf* = 3, *m* = 12, *R*<sup>0</sup> = 0.1, TIN = 2000 and *N* = 15 × 15. In Figure 11, the MEs of *u*(*x*, *t*) are plotted versus *x*. Although under a large noise with *s* = 0.1, the solution is with ME <sup>=</sup> 3.29 <sup>×</sup> <sup>10</sup>−2, where max *<sup>u</sup>*(*x*, *<sup>t</sup>*) is 19.13.

**Figure 11.** For Example 6 of the backward nonlocal wave problem: ME(*u*) versus *x* in the spatial interval.

**Example 7.** *According to Example 3, we consider the backward nonlocal wave problem for the linear Klein–Gordon equation with the final time data:*

$$h(\mathbf{x}) = \sin(\mathbf{x} - 2t\_f), \ r(\mathbf{x}) = -2\cos\left(\mathbf{x} - 2t\_f\right). \tag{62}$$

Under *l* = 1, *tf* = 2, *m* = 15, *R*<sup>0</sup> = 0.1, TIN = 2000 and *N* = 15 × 15, in Figure 12, the solid line displays the ME of *<sup>u</sup>*(*x*, *<sup>t</sup>*) with respect to x, of which ME <sup>=</sup> 4.03 <sup>×</sup> <sup>10</sup><sup>−</sup>3, and max *u*(*x*, *t*) is 1. Then, we take *tf* = 4 and *N* = 25 × 25, and other parameters remain the same. In Figure 12, the dashed line shows ME(*u*) with respect to *<sup>x</sup>*, where ME <sup>=</sup> 2.73 <sup>×</sup> <sup>10</sup>−3.

**Figure 12.** For Example 7 of the backward nonlocal wave problem of the Klein–Gordon equation: ME(*u*) versus *x* with different final times.

When we extend the domain to *<sup>l</sup>* <sup>=</sup> 3 and *tf* <sup>=</sup> 4, ME increases to 8.52×10<sup>−</sup>2. However, we can take *<sup>R</sup>*<sup>0</sup> <sup>=</sup> 0.001 and *<sup>N</sup>* <sup>=</sup> <sup>30</sup> <sup>×</sup> 30 and reduce ME to 6.16 <sup>×</sup> <sup>10</sup>−2. Therefore, it can be seen that increasing the grid number *N* and decreasing the characteristic length *R*<sup>0</sup> can increase the numerical accuracy.

#### **7. Complex Two-Side Nonlocal BCs**

The method presented in Section 5 is easily tailored to account for the backward nonlocal wave problem under complex two-side, nonlocal BCs:

$$u\_{tt}(\mathbf{x},t) - u\_{xx}(\mathbf{x},t) = F(\mathbf{x},t), \ (\mathbf{x},t) \in \Omega,\tag{63}$$

$$u\left(\mathbf{x}, \mathbf{t}\_f\right) = h(\mathbf{x}), \quad u\_t\left(\mathbf{x}, \mathbf{t}\_f\right) = r(\mathbf{x}), \tag{64}$$

$$a\_1 u(0, t) + a\_2 u\_x(0, t) + \int\_0^l a\_3(\mathbf{x}) u(\mathbf{x}, t) d\mathbf{x} = p(t),\tag{65}$$

$$b\_1 u(l, t) + b\_2 u\_x(l, t) + \int\_0^l b\_3(x) u(x, t) dx = q(t). \tag{66}$$

The key function *E*0(*x*, *t*) in Theorem 1 is modified to

$$\begin{aligned} E^0(\mathbf{x}, t) &= w(\mathbf{x}, t) - s\_1(\mathbf{x}) \left[ a\_1 w(0, t) + a\_2 w\_x(0, t) + \int\_0^l a\_3(\mathbf{x}) w(\mathbf{x}, t) d\mathbf{x} - p(t) \right] \\ &- s\_2(\mathbf{x}) \left[ b\_1 w(l, t) + b\_2 w\_x(l, t) + \int\_0^l b\_3(\mathbf{x}) w(\mathbf{x}, t) d\mathbf{x} - q(t) \right], \end{aligned} \tag{67}$$

where the nonlocal shape functions are derived from

$$\begin{aligned} a\_1 s\_1(0) + a\_2 s\_1'(0) + \int\_0^l a\_3(\mathbf{x}) s\_1(\mathbf{x}) d\mathbf{x} &= 1, \\ b\_1 s\_1(l) + b\_2 s\_1'(l) + \int\_0^l b\_3(\mathbf{x}) s\_1(\mathbf{x}) d\mathbf{x} &= 0, \end{aligned} \tag{68}$$

$$\begin{aligned} a\_1 s\_2(0) + a\_2 s\_2'(0) + \int\_0^l a\_3(\mathbf{x}) s\_2(\mathbf{x}) d\mathbf{x} &= 0, \\ b\_1 s\_2(l) + b\_2 s\_2'(l) + \int\_0^l b\_3(\mathbf{x}) s\_2(\mathbf{x}) d\mathbf{x} &= 1 \end{aligned} \tag{69}$$

. Inserting Equation (67) and *w*(*x*, *t*) = *xi*−*<sup>j</sup> tj*−<sup>1</sup> into Equation (46), we can generate the Pascal bases

$$E\_{lj}(\mathbf{x},t) = E^0(\mathbf{x},t) - \left[1 + \left(t - t\_f\right)^2\right] \left[E^0(\mathbf{x},t\_f) - h(\mathbf{x})\right] - \left(t - t\_f\right)\left[E^0\_t\left(\mathbf{x},t\_f\right) - r(\mathbf{x})\right],\tag{70}$$

$$\begin{split} E^{0}(\mathbf{x},t) &= \mathbf{x}^{i-j}t^{j-1} - s\_1(\mathbf{x}) \left[ a\_1 \mathbf{x}^{i-j}t^{j-1} \big|\_{\mathbf{x}=0} + a\_2(i-j)\mathbf{x}^{i-j-1}t^{j-1} \big|\_{\mathbf{x}=0} \right] \\ &- s\_1(\mathbf{x}) \left[ \int\_0^l a\_3(\mathbf{x}) \mathbf{x}^{i-j}t^{j-1} d\mathbf{x} - p(t) \right] \\ &- s\_2(\mathbf{x}) \left[ b\_1 \mathbf{x}^{i-j}t^{j-1} \big|\_{\mathbf{x}=l} + b\_2(i-j)\mathbf{x}^{i-j-1}t^{j-1} \big|\_{\mathbf{x}=l} + \int\_0^l b\_3(\mathbf{x}) \mathbf{x}^{i-j}t^{j-1} d\mathbf{x} - q(t) \right]. \end{split} \tag{71}$$

**Example 8.** *As an extension of Example 7, we consider the backward nonlocal wave problem for the linear Klein–Gordon equation with the final time data and two-side nonlocal BCs:*

$$h(\mathbf{x}) = \sin\left(\mathbf{x} - 2t\_f\right), \; r(\mathbf{x}) = -2\cos\left(\mathbf{x} - 2t\_f\right),\tag{72}$$

$$u(0,t) + u\_x(0,t) + \int\_0^l u(\mathbf{x},t)d\mathbf{x} = p(t), \quad u(l,t) + \int\_0^l \mathbf{x}u(\mathbf{x},t)d\mathbf{x} = q(t), \tag{73}$$

*where*

$$p(t) = (2 - \cos l)\cos(2t) - (1 + \sin l)\sin(2t),\tag{74}$$

$$q(t) = \sin(l - 2t) + (\sin l - l \cos l)\cos(2t) + (1 - \cos l - l \sin l)\sin(2t). \tag{75}$$

*For this problem, we can derive*

$$s\_1(\mathbf{x}) = \frac{12l + 4l^3 - (12 + 6l^2)\mathbf{x}}{l^4 + 4l^3 + 12l - 12}, \quad s\_2(\mathbf{x}) = \frac{12(1+l)\mathbf{x} - 12 + 6l^2}{l^4 + 4l^3 + 12l - 12}.\tag{76}$$

*Under l* = 4*,tf* = 1*, m* = 20*,R*<sup>0</sup> = 1, *TIN = 2000 andN* = 30 × 30, *in Figure 13, the solid line displays ME(u) with respect to x, of which ME* <sup>=</sup> 2.36 <sup>×</sup> <sup>10</sup>−3*. If tf* <sup>=</sup> <sup>2</sup>*, we obtain ME* <sup>=</sup> 7.02 <sup>×</sup> <sup>10</sup>−3*, the results of which are shown in Figure 13 by a dashed line. As the figure shows, this method still yields a stable solution even if the computation time increases*.

**Figure 13.** For Example 8 of the backward nonlocal wave problem of the Klein–Gordon equation under two-side nonlocal BCs: ME(u) versus *x* with different final times.

**Example 9** . *Let*

$$
\mu\_{\rm th}(\mathbf{x},t) - \mu\_{\rm xx}(\mathbf{x},t) - 3u(\mathbf{x},t) = 0, \ (\mathbf{x},t) \in \Omega,\tag{77}
$$

$$h(\mathbf{x}) = \exp\left(\mathbf{x} - 2t\_f\right), \ \mathbf{r}(\mathbf{x}) = -2\exp\left(\mathbf{x} - 2t\_f\right),\tag{78}$$

$$u(0,t) - u\_x(0,t) + \int\_0^l x u(\mathbf{x},t) d\mathbf{x} = p(t), \ u\_x(l,t) + \int\_0^l u(\mathbf{x},t) d\mathbf{x} = q(t), \tag{79}$$

*where*

$$p(t) = e^{-2t} \left( l e^l - e^l + 1 \right), \ q(t) = e^{-2t} \left( 2e^l - 1 \right), \tag{80}$$

*and u*(*x*, *t*) = exp(*x* − 2*t*) *is the exact solution*.

*For this problem, we can derive*

$$s\_1(\mathbf{x}) = \frac{12 + 6l^2 - 12l\mathbf{x}}{12 + 18l + 6l^2 - l^3}, \quad s\_2(\mathbf{x}) = \frac{12 - 4l^2 + (12 + 6l)\mathbf{x}}{12 + 18l + 6l^2 - l^3}. \tag{81}$$

*Under l* = 3*, m* = 10*, R*<sup>0</sup> = 1*, TIN = 2000 and N* = 20 × 20*, in Figure 14, the solid line displays ME(u) with respect to <sup>x</sup> for tf* <sup>=</sup> 0.5*, of which ME* <sup>=</sup> 3.92 <sup>×</sup> <sup>10</sup>−3*, and the dashed line displays ME(u) with respect to x for tf* <sup>=</sup> <sup>1</sup>*, of which ME* <sup>=</sup> 4.94 <sup>×</sup> <sup>10</sup>−2*. Notice max(u) = 17.76. When considering m* = 20*, l* = 5 *and tf* = 1*, ME(u) with respect to x is shown in Figure 15, where ME =* 3.292 <sup>×</sup> <sup>10</sup>−1*, and max u*(*x*, *<sup>t</sup>*) *is 124.97. The result shows that the solution of this method is acceptable. Hence, we successfully apply the NLBSF to resolve the wave problem with two-side nonlocal BCs, especially for the backward problem in time.*

**Figure 14.** For Example 9 of the backward nonlocal wave problem under two-side nonlocal BCs: ME(u) versus *x* with different final times.

**Figure 15.** For Example 9 of the backward nonlocal wave problem under two-side nonlocal BCs: ME(u) versus *x* in the spatial interval.

#### **8. Conclusions**

In this paper, the numerical solutions of the backward and forward non-homogeneous wave problems with nonlocal boundary conditions were developed. When boundary conditions are not set on the boundaries, the solution may have a large boundary error. For this reason, it is difficult to solve such nonlinear problems by conventional numerical methods, especially when addressing the backward nonlocal wave problem. To reduce the boundary error and increase numerical accuracy by the NLBSF method, we let the free function be the Pascal triangle and then the solution was a weighted superposition of the complete Pascal bases. These basis functions automatically satisfy a left-boundary condition, a nonlocal right-boundary condition and two initial conditions for the forward nonlocal wave problem or two final time conditions for the backward nonlocal wave problem. We gave four examples for the forward nonlocal wave problem to support that the nonlocal wave equation can be solved quickly and accurately. For the backward nonlocal wave problem with one-side or two-side nonlocal boundary conditions, we recovered accurate solutions in the entire domain; even a large time span and large noise were taken into account. From the nine examples, the results demonstrate that the presented method is more effective and stable than conventional numerical schemes. Hence, it can be concluded that the proposed method for the forward or backward problems in time is accurate, stable, effective and robust for addressing boundary conditions with noise level effects.

**Author Contributions:** C.-S.L. contributed to the conception and supervision of the work (conceptualization, resources, methodology, writing—original draft), collected and analyzed the data and interpreted the results. C.-W.C. contributed to the conception of the work (project administration, software). Y.-W.C. contributed to the writing, design and validation of the work (writing—editing, validating and visualizing the data) and the funding acquisition. J.-H.S. contributed to the writing and supervision of the work (writing—review and editing, software, project administration). All authors have read and agreed to the published version of the manuscript.

**Funding:** The third authors would like to thank the Ministry of Science and Technology, Taiwan, for their financial support (grant number MOST 111-2221-E-019-048).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare that there are no conflict of interest regarding the publication of this paper.

#### **References**


## *Article* **Parameter Uniform Numerical Method for Singularly Perturbed 2D Parabolic PDE with Shift in Space**

**V. Subburayan <sup>1</sup> and S. Natesan 2,\***


**Abstract:** Singularly perturbed 2D parabolic delay differential equations with the discontinuous source term and convection coefficient are taken into consideration in this paper. For the time derivative, we use the fractional implicit Euler method, followed by the fitted finite difference method with bilinear interpolation for locally one-dimensional problems. The proposed method is shown to be almost first-order convergent in the spatial direction and first-order convergent in the temporal direction. Theoretical results are illustrated with numerical examples.

**Keywords:** delay differential equations; 2D parabolic equations; fractional step method; convection diffusion problems

**MSC:** 34K26; 35B25; 65M22; 65M50; 65N22

#### **1. Introduction**

Differential equations with small or large parameters can be used to describe a variety of applied practical problems, including the theory of boundary layers. For example, the shock waves occurring in gas motions, edge effects when elastic plates deform, etc. These mathematical problems are very difficult (or even impossible) to solve exactly, so approximate solutions are necessary. It is possible to obtain an approximation of the solution through perturbation methods. Basically, these methods aim to solve a simpler problem (as a first approximation) and systematically improve the approximate solution.

When using finite difference or finite element methods on equally spaced grids and allowing the perturbation parameter tend to zero, boundary layers produce inaccurate numerical solutions. The most popular method for overcoming this difficulty is to construct uniformly valid numerical methods on layers adapted to the mesh. There are several uniformly valid methods available in the literature, for instance, to cite a few (see Refs. [1,2] and the references therein). As pointed out in Ref. [3], the direct discretization of the singularly perturbed 2D parabolic differential equations leads to a pentadiagonal linear system of equations. This problem is exceedingly complex to solve computationally. We use the fractional step method in order to reduce the computation cost. At each time level, the fractional step method leads to the tridiagonal system of algebraic equations. Several types of research have been conducted recently on the fractional step method, such as Refs. [4–6] and the references therein.

Singularly perturbed delay differential equations (SPDDEs) are a class of perturbation problems with at least one delay or deviating argument. This type of problem occurs frequently in the modelling of various types of physical and biological problems. For example, the neuronal variability and its theoretical analysis have been modelled as delay parabolic equations [7,8]. Asymptotic analyses for 1D stationery SPDDEs have been well studied by Lange and Miura [9]. Several numerical methods for SPDDEs of 1D stationery problems have been reported in the literature, such as Refs. [10–13] and the reference

**Citation:** Subburayan, V.; Natesan, S. Parameter Uniform Numerical Method for Singularly Perturbed 2D Parabolic PDE with Shift in Space. *Mathematics* **2022**, *10*, 3310. https:// doi.org/10.3390/math10183310

Academic Editors: Patricia J. Y. Wong and Luigi Rodino

Received: 10 August 2022 Accepted: 8 September 2022 Published: 12 September 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

therein. The numerical method for 1D parabolic equations was initiated by Ref. [14] and it gained the interest of many researchers. Das and Natesan [15] presented computing techniques for solving 2D time SPDDEs. Ref. [16] presented some applications and existence results for partial delay differential equations. The modelling of option pricing, to generalize the celebrated Black–Scholes equation with suitable weight, led to the 2D parabolic differential equations with space shift [17]. We consider discontinuous convection and source terms in 2D parabolic SPDDEs in this article, as mentioned in the abstract. This problem exhibits interior layers at *x* = *dx* and *y* = *dy* and, due to the presents of the shift in space, the boundary layers occurs at *x* = 1 and *y* = 1. The existence results pertaining to the parabolic equation with discontinuous coefficients are addressed in Ref. [18]. The method presented in this article is a combination of the layers adopted technique and linear interpolations. The interpolation term takes care of the delay arguments. The proposed method is validated theoretically and numerically to be uniformly convergent in both space and time by considering some numerical examples.

The constant *C* is generic positive, that is, it is independent of the perturbation parameter as well as the discretization parameters *N* and *M* throughout the paper. For convenience, it is assumed that the number of mesh points in the spatial domains Ω*<sup>x</sup>* and Ω*<sup>y</sup>* are same, that is, *N* and the index set I*N*<sup>0</sup> = {1, 2, 3, ··· , *N*0} for any positive integer *<sup>N</sup>*0. It is conventional to assume for the convection coefficient problem that *<sup>ε</sup>* <sup>≤</sup> *CN*−<sup>1</sup> for practical purposes. Further, to measure the error bounds and derivative bounds, we use the following norm *ψ<sup>D</sup>* <sup>=</sup> sup**x**∈*<sup>D</sup> ψ*(**x**), **<sup>x</sup>** = (*x*, *<sup>y</sup>*).

The article is organized as follows: the problem is considered in Section 2. The fractional implicit Euler method for time derivative and locally 1D problems are presented in Section 3. In the same section, the stability results and derivative estimates of the locally one-dimensional problems are presented. Section 4 presents the numerical method for the considered problem. The discretizations incurred by the errors are estimated in Section 5. Numerical validations through some test example problems are done in Section 6. Finally, in Section 7, some concluding remarks are made.

#### **2. Statement of Continuous Problem**

Motivated by the works of Refs. [7,17], we consider the following two-dimensional singularly perturbed parabolic differential equations: We find *u* such that

$$\mathfrak{L}u := u\_t - \varepsilon \Delta u + \nabla u \cdot \overline{p}(\mathbf{x}) + q(\mathbf{x})u(\mathbf{x} - \mathbf{d}, t) = \mathbf{g}(\mathbf{x}, t), \ (\mathbf{x}, t) \in \mathfrak{D}^\* \times (0, T], \tag{1}$$

$$u(\mathbf{x},0) = u\_0(\mathbf{x}), \ \mathbf{x} \in \mathfrak{D},\tag{2}$$

$$
u(\mathbf{x},t) = 0, \text{ on } \partial \mathfrak{D} \times [0,T], \tag{3}$$

$$u(\mathbf{x},t) = 0,\text{ on } [-d\_{\mathbf{x}},0] \times [-d\_{\mathbf{y}},1] \times [0,T] \cup [-d\_{\mathbf{x}},1] \times [-d\_{\mathbf{y}},0] \times [0,T],\tag{4}$$

where **x** = (*x*, *y*), **d** = (*dx*, *dy*), Ω*<sup>x</sup>* = (0, 1) = Ω*y*, D = Ω*<sup>x</sup>* × Ω*y*, D<sup>∗</sup> = Ω<sup>∗</sup> *<sup>x</sup>* × Ω<sup>∗</sup> *y*, Ω∗ *<sup>ν</sup>* = Ω<sup>−</sup> *<sup>ν</sup>* <sup>∪</sup> <sup>Ω</sup><sup>+</sup> *<sup>ν</sup>* , Ω<sup>−</sup> *<sup>ν</sup>* = (0, *dν*), Ω<sup>+</sup> *<sup>ν</sup>* = (*dν*, 1), *ν* = *x*, *y*, the functions *u*0, *q* are sufficiently differentiable and bonded on D, *p*1, *p*2, *g*1, *g*<sup>2</sup> are sufficiently differentiable and bounded on their respective domains D∗, D<sup>∗</sup> × [0, *T*]. In addition, we assume that,

$$\begin{split} &u\_{x}(\overline{d\_{x}},y,t) = u\_{x}(\overline{d\_{x}}^{+},y,t), \ u\_{y}(\mathbf{x},\overline{d\_{y}}^{-},t) = u\_{y}(\mathbf{x},\overline{d\_{y}}^{+},t), \\ &p(\mathbf{x}) = (p\_{1}(\mathbf{x}),p\_{2}(\mathbf{x}))^{T}, \ \nabla u = (u\_{x},u\_{y}), \\ &p\_{1}^{+} \geq p\_{1}(\mathbf{x}) \geq p\_{1}^{-} > 0, \ \mathbf{x} \in \Omega\_{x}^{-} \times \Omega\_{y'}^{\*}, p\_{1}^{+} \geq -p\_{1}(\mathbf{x}) \geq p\_{1}^{-} > 0, \ \mathbf{x} \in \Omega\_{x}^{+} \times \Omega\_{y'}^{\*}, \\ &p\_{2}^{+} \geq p\_{2}(\mathbf{x}) \geq p\_{2}^{-} > 0, \ \mathbf{x} \in \Omega\_{x}^{\*} \times \Omega\_{y'}^{-}, p\_{2}^{+} \geq -p\_{2}(\mathbf{x}) \geq p\_{2}^{-} > 0, \ \mathbf{x} \in \Omega\_{x}^{\*} \times \Omega\_{y'}^{+}, \\ &|p\_{1}(d\_{x}^{-},y) - p\_{1}(d\_{x'}^{+},y)| < \infty, \ |p\_{2}(\mathbf{x},d\_{y}^{-}) - p\_{2}(\mathbf{x},d\_{y}^{+})| < \infty, \\ &q(\mathbf{x}) = q\_{1}(\mathbf{x}) + q\_{2}(\mathbf{x}), \ 0 \geq q\_{1}, q\_{2} \geq \beta, \ g(\mathbf{x},t) = g\_{1}(\mathbf{x},t) + g\_{2}(\mathbf{x},t). \end{split}$$

Let <sup>L</sup>*<sup>x</sup>* :<sup>=</sup> <sup>−</sup>*<sup>ε</sup> <sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>x*<sup>2</sup> <sup>+</sup> *<sup>p</sup>*1(**x**) *<sup>∂</sup> <sup>∂</sup><sup>x</sup>* <sup>+</sup> *<sup>q</sup>*1(**x**)*I***<sup>d</sup>** and <sup>L</sup>*<sup>y</sup>* :<sup>=</sup> <sup>−</sup>*<sup>ε</sup> <sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>y*<sup>2</sup> <sup>+</sup> *<sup>p</sup>*2(**x**) *<sup>∂</sup> <sup>∂</sup><sup>y</sup>* + *q*2(**x**)*I***<sup>d</sup>** be two differential operators, *I***d***u*(**x**, *t*) = *u*(**x** − **d**, *t*), then the differential operator L defined in (1) can be written as L := *<sup>∂</sup> <sup>∂</sup><sup>t</sup>* + L*<sup>x</sup>* + L*y*.

#### **3. Time Domain Discretization and Stability Analysis**

#### *3.1. Discretization of Time Domain*

The time domain [0, *T*] is discretized uniformly with step length *ht* = *T*/*M*, where *<sup>M</sup>* is a positive integer. Then we have the uniform mesh in the temporal direction <sup>Ω</sup>*<sup>M</sup> <sup>t</sup>* = {*tk* <sup>=</sup> *<sup>k</sup>* <sup>×</sup> *ht*}*<sup>M</sup> <sup>k</sup>*=0.

#### *3.2. An Alternating Direction Implicit Method*

Let us assume that *<sup>u</sup>*ˆ0(**x**) = *<sup>u</sup>*0(**x**), **<sup>x</sup>** <sup>∈</sup> <sup>D</sup>. Now, we discretize the IBVP (1)–(3) using the fractional implicit Euler method and obtain the following semidiscrete scheme on the time levels *n* = 0, 1, ··· , *M* − 1:

let *y* ∈ Ω*y*, then

$$\begin{cases} \mathfrak{D}\_{\mathbf{x}}\mathfrak{A}^{n+\frac{1}{2}} = \mathfrak{A}^{n} + h\_{t}\mathfrak{g}\_{1}(\mathbf{x}, \mathbf{y}, t\_{n+1}), \; \mathbf{x} \in \Omega^{\*}\_{\mathbf{x}^{\prime}}\\ \mathfrak{A}^{n+\frac{1}{2}}(0, \mathbf{y}) = 0 = \mathfrak{A}^{n+\frac{1}{2}}(1, \mathbf{y}), \\ \mathfrak{A}^{n+\frac{1}{2}}\_{x}(d^{-}\_{\mathbf{x}^{\prime}}\mathbf{y}) = \mathfrak{A}^{n+\frac{1}{2}}\_{x}(d^{+}\_{\mathbf{x}^{\prime}}\mathbf{y}), \end{cases} \tag{5}$$

let *x* ∈ Ω*x*, then

$$\begin{cases} \mathfrak{D}\_y \mathfrak{h}^{n+1} = \mathfrak{h}^{n+\frac{1}{2}} + h\_t \mathfrak{g}\_2(\mathfrak{x}, y, t\_{n+1}), \; y \in \Omega\_y^\* \\ \hat{\mathfrak{h}}^{n+1}(\mathfrak{x}, 0) = 0 = \hat{\mathfrak{u}}^{n+1}(\mathfrak{x}, 1), \\ \hat{\mathfrak{u}}\_y^{n+1}(\mathfrak{x}, d\_y^-) = \hat{\mathfrak{u}}\_y^{n+1}(\mathfrak{x}, d\_y^+), \end{cases} \tag{6}$$

where *u*ˆ*n*(*x*, *y*) is the exact solution of *u* at the time level *t* = *tn*, D*<sup>x</sup>* := *I* + *ht*L*<sup>x</sup>* and D*<sup>y</sup>* := *I* + *ht*L*y*.

If the exact solution of the problem (1) is known at *t* = *tn*, then we have the following semi-discrete scheme: let *y* ∈ Ω*y*, then

$$\begin{cases} \mathfrak{D}\_{\mathbf{x}} \overline{u}^{n+\frac{1}{2}} = u(\mathbf{x}, y, t\_{\mathbf{n}}) + h\_{t} \underline{g}\_{1}(\mathbf{x}, y, t\_{\mathbf{n}+1}), \; \mathbf{x} \in \Omega\_{\mathbf{x}^{\star}}^{\*}\\ \overline{u}^{n+\frac{1}{2}}(0, y) = \overline{u}^{n+\frac{1}{2}}(1, y) = 0, \\ \overline{u}\_{\mathbf{x}}^{n+\frac{1}{2}}(d\_{\mathbf{x}^{-}}^{-}, y) = \overline{u}\_{\mathbf{x}}^{n+\frac{1}{2}}(d\_{\mathbf{x}^{+}}^{+}, y), \end{cases} \tag{7}$$

let *x* ∈ Ω*x*, then

$$\begin{cases} \mathfrak{D}\_{y}\overline{u}^{n+1} = \overline{u}^{n+\frac{1}{2}} + h\_{t}\mathfrak{g}\_{2}(\mathbf{x}, \mathbf{y}, t\_{n+1}), \; y \in \Omega\_{y}^{\*},\\ \overline{u}^{n+1}(\mathbf{x}, 0) = \overline{u}^{n+1}(\mathbf{x}, 1) = 0, \\ \overline{u}\_{y}^{n+1}(\mathbf{x}, d\_{y}^{-}) = \overline{u}\_{y}^{n+1}(\mathbf{x}, d\_{y}^{+}). \end{cases} \tag{8}$$

Solving the problem (1)–(4) is more computationally expensive than solving lowerdimensional problems. As a result, we used the ADI scheme to divide the two-dimensional problem into two sets of one-dimensional problems in order to decrease the computing cost and to have an efficient numerical solution.

#### *3.3. Stability Results and Derivative Estimates*

This section presents the maximum principles for the above-mentioned locally one dimensional problems. Further, with regard to the applications of the maximum principle, we estimate the solution derivative bounds and local and global truncation errors in the temporal direction.

The test functions

$$s(\mathbf{x}) = \begin{cases} \mathbf{x} + \mathbf{1}, & \mathbf{x} \in [\mathbf{x}, d\_{\mathbf{x}}],\\ d\_{\mathbf{x}} \frac{d\_{\mathbf{x}} - \mathbf{x}}{\mathbf{1} - d\_{\mathbf{x}}} + d\_{\mathbf{x}} + \mathbf{1}, & \mathbf{x} \in [d\_{\mathbf{x}}, 1] \end{cases} \quad \text{and } s(\mathbf{y}) = \begin{cases} y + \mathbf{1}, & \mathbf{y} \in [0, d\_{\mathbf{y}}],\\ d\_{\mathbf{y}} \frac{d\_{\mathbf{y}} - \mathbf{y}}{\mathbf{1} - d\_{\mathbf{y}}} + d\_{\mathbf{y}} + \mathbf{1}, & \mathbf{y} \in [d\_{\mathbf{y}}, 1] \end{cases}$$

are used in the following lemmas and sections.

**Lemma 1.** *Let <sup>ψ</sup>* <sup>∈</sup> *<sup>C</sup>*0(Ω*x*) <sup>∩</sup> *<sup>C</sup>*2(Ω<sup>∗</sup> *<sup>x</sup>*) *be a function satisfying ψ*(*x*) ≥ 0, *x* = 0, 1, D*xψ*(*x*) ≥ 0, *x* ∈ Ω<sup>∗</sup> *<sup>x</sup> and ψ* (*dx*−) − *ψ* (*dx*+) ≥ 0, *then ψ*(*x*) ≥ 0, *x* ∈ Ω*x*.

**Proof.** The proof is by construction and similar to Refs. [12,13]. It is shown that <sup>D</sup>*xs*(*x*) > 0, *x* = *dx* and *s* (*d*− *<sup>x</sup>* ) − *s* (*d*<sup>+</sup> *<sup>x</sup>* ) ≥ 0. By using the argument given by Ref. [12], Theorem 3.1, one can prove this lemma.

Similar to the above lemma and using the test function *s*(*y*), we can prove the following lemma.

**Lemma 2.** *Let <sup>ψ</sup>* <sup>∈</sup> *<sup>C</sup>*0(Ω*y*) <sup>∩</sup> *<sup>C</sup>*2(Ω<sup>∗</sup> *<sup>y</sup>*) *be a function satisfies ψ*(*y*) ≥ 0, *y* = 0, 1, D*yψ*(*y*) ≥ 0, *y* ∈ Ω<sup>∗</sup> *<sup>y</sup> and ψ* (*dy*−) − *ψ* (*dy*+) ≥ 0, *then ψ*(*y*) ≥ 0, *x* ∈ Ω*y*.

One can prove that the solutions of (5) and (6) are stable and unique if they exist. Further, they are bounded from Lemmas 1 and 2.

**Lemma 3.** *Assume that ∂i u ∂ti* <sup>≤</sup> *<sup>C</sup>*, 0 <sup>≤</sup> *<sup>i</sup>* <sup>≤</sup> 3. *Then en* ≤ *Ch*<sup>2</sup> *<sup>t</sup> where <sup>u</sup>*(*tn*) = *<sup>u</sup>n*(*x*, *<sup>y</sup>*) + *en, <sup>u</sup>*(*tn*) = *<sup>u</sup>*(*x*, *<sup>y</sup>*, *tn*). *In addition,* sup*n*≤*T*/*ht En*<sup>∞</sup> <sup>≤</sup> *C ht*, *where the global error En* <sup>=</sup> *<sup>u</sup>*(*tn*) <sup>−</sup> *u*ˆ*n*.

**Proof.** The proof is similar to that of Refs. [4,6]. For that, one can express

$$\begin{aligned} \boldsymbol{\mu}(t\_{n-1}) &= \mathfrak{D}\_{\mathbf{x}}[\mathfrak{D}\_{y}\boldsymbol{\mu}(t\_{n}) - \boldsymbol{h}\_{t}\mathfrak{G}\_{2}(\mathbf{x}\_{\prime}\boldsymbol{y}\_{\prime}\boldsymbol{t}\_{n})] - \boldsymbol{h}\_{t}\mathfrak{g}\_{1}(\mathbf{x}\_{\prime}\boldsymbol{y}\_{\prime}\boldsymbol{t}\_{n}) + O(\boldsymbol{h}\_{t}^{2}), \\ \boldsymbol{\mu}(t\_{n-1}) &= \mathfrak{D}\_{\mathbf{x}}[\mathfrak{D}\_{y}\overline{\mathfrak{a}}(t\_{n}) - \boldsymbol{h}\_{t}\mathfrak{g}\_{2}(\mathbf{x}\_{\prime}\boldsymbol{y}\_{\prime}\boldsymbol{t}\_{n})] - \boldsymbol{h}\_{t}\mathfrak{g}\_{1}(\mathbf{x}\_{\prime}\boldsymbol{y}\_{\prime}\boldsymbol{t}\_{n}), \\ \mathfrak{D}\_{\mathbf{x}}\mathfrak{D}\_{y}\boldsymbol{e}\_{n} &= O(\boldsymbol{h}\_{t}^{2}). \end{aligned}$$

First by the application of Lemma 1 then by Lemma 2, we have <sup>|</sup>*en*| ≤ *Ch*<sup>2</sup> *<sup>t</sup>* . To prove the second part, consider

$$\begin{aligned} E\_{\mathfrak{n}} &= \mathfrak{e}\_{\mathfrak{n}} + \overline{\mathfrak{u}}^{n} - \hat{\mathfrak{u}}^{n}, \\ \mathfrak{D}\_{\mathfrak{y}}(\overline{\mathfrak{u}}^{n} - \hat{\mathfrak{u}}^{n}) &= \overline{\mathfrak{u}}^{(n-1) + \frac{1}{2}} - \hat{\mathfrak{u}}^{(n-1) + \frac{1}{2}}, \\ \overline{\mathfrak{u}}^{n} - \mathfrak{u}^{n} &= \mathfrak{D}\_{\mathfrak{y}}^{-1} \mathfrak{D}\_{\mathfrak{x}}^{-1} E\_{n-1}. \end{aligned}$$

making use of the arguments given in Ref. [4], we have |*En*| ≤ *Cht*, which concludes the proof.

From the above lemma, we can conclude that the semidiscretization process is uniformly convergent of order *O*(*ht*). In the rest of the sections it is assumed that, *dx* = 0.5 = *dy*.

Let the solution *u*ˆ*n*<sup>+</sup> <sup>1</sup> <sup>2</sup> be decomposed as *u*ˆ*n*<sup>+</sup> <sup>1</sup> <sup>2</sup> = *vn*<sup>+</sup> <sup>1</sup> <sup>2</sup> + *wn*<sup>+</sup> <sup>1</sup> <sup>2</sup> for obtaining the sharp bounds on the derivatives. Further, let the decomposition of the regular component be *vn*<sup>+</sup> <sup>1</sup> <sup>2</sup> = ∑<sup>2</sup> *<sup>k</sup>*=<sup>0</sup> *<sup>ε</sup>kv n*+ <sup>1</sup> 2 *<sup>k</sup>* , leading the desired bounds on the derivatives. The functions *v n*+ <sup>1</sup> 2 *<sup>k</sup>* , *<sup>k</sup>* <sup>=</sup> 0, 1, 2, *<sup>w</sup>n*<sup>+</sup> <sup>1</sup> <sup>2</sup> satisfy the following problems:

$$\begin{cases} v\_0^{n+\frac{1}{2}} + h\_l \left( p\_1(\mathbf{x}) \frac{d}{dx} v\_0^{n+\frac{1}{2}} + q\_1(\mathbf{x}) I\_\mathbf{d} v\_0^{n+\frac{1}{2}} \right) = v\_0^n + h\_l g\_1(t\_{n+1}), \ \mathbf{x} \in \Omega\_{\mathbf{x}}^\* \\ v\_0^{n+\frac{1}{2}}(\mathbf{x}) = \mathfrak{a}^{n+\frac{1}{2}}(\mathbf{x}), \ \mathbf{x} \in [-d\_{\mathbf{x}}, 0], \ v\_0^{n+\frac{1}{2}}(1) = \mathfrak{a}^{n+\frac{1}{2}}(1), \end{cases} \tag{9}$$

$$\begin{cases} v\_1^{n+\frac{1}{2}} + h\_l \left( p\_1(\mathbf{x}) \frac{d}{dx} v\_1^{n+\frac{1}{2}} + q\_1(\mathbf{x}) I\_\mathbf{d} v\_1^{n+\frac{1}{2}} \right) = v\_1^n + h\_l \frac{d^2}{dx^2} \left( v\_0^{n+\frac{1}{2}} \right), \mathbf{x} \in \Omega\_{\mathbf{x}'} \\ v\_1^{n+\frac{1}{2}}(\mathbf{x}) = 0, \ x \in [-d\_\mathbf{x}, 0], \ v\_1^{n+\frac{1}{2}}(1) = 0, \end{cases} \tag{10}$$

$$\begin{cases} \mathfrak{D}\_x v\_2^{n + \frac{1}{2}} = v\_2^n + h\_t \frac{d^2}{dx^2} \left( v\_1^{n + \frac{1}{2}} \right), \; x \in \Omega\_{\mathbf{x}}^\*, \\ v\_2^{n + \frac{1}{2}}(\mathbf{x}) = 0, \; \mathbf{x} \in [-d\_{\mathbf{x}}, 0], v\_2^{n + \frac{1}{2}}(1) = 0, \\\\ \frac{d}{dx} v\_2^{n + \frac{1}{2}}(d\_{\mathbf{x}}^-) = \frac{d}{dx} v\_2^{n + \frac{1}{2}}(d\_{\mathbf{x}}^+), \end{cases} \tag{11}$$

and the functions *vn*<sup>+</sup> <sup>1</sup> <sup>2</sup> and *wn*<sup>+</sup> <sup>1</sup> <sup>2</sup> satisfy the following boundary-value problems (BVPs):

$$\begin{cases} \mathfrak{D}\_{\mathbf{x}} v^{n + \frac{1}{2}} = v^n + h\_l g\_1(t\_{n+1}), \; \mathbf{x} \in \Omega^\*\_{\mathbf{x}'} \\\\ v^{n + \frac{1}{2}}(\mathbf{x}) = \mathfrak{A}^{n + \frac{1}{2}}(\mathbf{x}), \; \mathbf{x} \in [-d\_{\mathbf{x}}, 0], \; v^{n + \frac{1}{2}}(1) = \mathfrak{A}^{n + \frac{1}{2}}(1), \\\\ \left[ v^{n + \frac{1}{2}}(d\_{\mathbf{x}}) \right] = \sum\_{k=0}^{2} [v^{n + \frac{1}{2}}\_k(d\_{\mathbf{x}})], \; \left[ \frac{d}{d\mathbf{x}} v^{n + \frac{1}{2}}(d\_{\mathbf{x}}) \right] = \sum\_{k=0}^{1} \mathfrak{c}^k \left[ \frac{d}{d\mathbf{x}} v^{n + \frac{1}{2}}\_k(d\_{\mathbf{x}}) \right], \end{cases} \tag{12}$$

and

$$\begin{cases} \mathfrak{D}\_{\mathbf{x}}w^{n+\frac{1}{2}} = w^n, \; \mathbf{x} \in \Omega\_{\mathbf{x}}^\* \\\\ w^{n+\frac{1}{2}}(\mathbf{x}) = 0, \; \mathbf{x} \in [-d\_{\mathbf{x}}, 0], \; w^{n+\frac{1}{2}}(1) = 0, \\\\ \left[w^{n+\frac{1}{2}}(d\_{\mathbf{x}})\right] = -\left[v^{n+\frac{1}{2}}(d\_{\mathbf{x}})\right], \; \left[\frac{d}{d\mathbf{x}}w^{n+\frac{1}{2}}(d\_{\mathbf{x}})\right] = -\left[\frac{d}{d\mathbf{x}}v^{n+\frac{1}{2}}(d\_{\mathbf{x}})\right], \end{cases} \tag{13}$$

where the square bracket operation denotes the jump discontinuity [*α*(*ζ*)] <sup>=</sup> *<sup>α</sup>*(*ζ*+) <sup>−</sup> *<sup>α</sup>*(*ζ*−). It is assumed that *v*<sup>0</sup> = *u*ˆ0, *w*<sup>0</sup> = 0.

**Theorem 1.** *Let u*ˆ*n*<sup>+</sup> <sup>1</sup> <sup>2</sup> *be the solution of the problem (5) and let k be a nonnegative integer, then the regular and singular components satisfy the following bounds on the derivatives*

$$\begin{aligned} & \left\| \frac{d^k v^{n + \frac{1}{2}}}{dx^k} \right\|\_{\Omega^\*} \leq C(\varepsilon^{-k + 2} + 1), \ 0 \leq k \leq 3, \\ & \left| \frac{d^k w^{n + \frac{1}{2}}(x)}{dx^k} \right| \leq C \varepsilon^{-k} \begin{cases} \exp\left(\frac{p\_1^- (x - d\_x)}{\varepsilon}\right), & x \in \Omega\_x^-, \\ \exp\left(\frac{p\_1^- (d\_x - x)}{\varepsilon}\right) + \varepsilon \exp\left(\frac{p\_1^- (x - 1)}{\varepsilon}\right), & x \in \Omega\_x^+. \end{cases} \end{aligned}$$

**Proof.** We show that by integrating the differential Equations (9)–(11), and using the argument presented in Refs. [13,19], and Lemma 1, we have *vn*<sup>+</sup> <sup>1</sup> <sup>2</sup> ≤ *C*. Successive differentiation of Equations (9)–(11), we have *<sup>d</sup><sup>k</sup> dx<sup>k</sup> <sup>v</sup>n*<sup>+</sup> <sup>1</sup> <sup>2</sup> ≤ *<sup>C</sup>*(*ε*2−*<sup>k</sup>* <sup>+</sup> <sup>1</sup>). From the Lemma 1, we see that *u*ˆ*n*<sup>+</sup> <sup>1</sup> <sup>2</sup> and *vn*<sup>+</sup> <sup>1</sup> <sup>2</sup> are bounded, hence *wn*<sup>+</sup> <sup>1</sup> <sup>2</sup> . Let us assume that <sup>|</sup>*wn*<sup>+</sup> <sup>1</sup> <sup>2</sup> (*dx*)| ≤ *γ*. Now define the barrier functions

$$\phi^{\pm}(\mathbf{x}) = \mathcal{C}\gamma \exp\left(\frac{p\_1^-(\mathbf{x} - d\_{\mathbf{x}})}{\varepsilon}\right) \pm w^{n + \frac{1}{2}}, \ \mathbf{x} \in \Omega\_{\mathbf{x}}^-.$$

It is easy to show that *φ*±(0) ≥ 0, *φ*±(*dx*) ≥ 0 and D*xφ*±(*x*) ≥ 0 on Ω<sup>−</sup> *<sup>x</sup>* . From the results of Ref. [19], we have <sup>|</sup>*wn*<sup>+</sup> <sup>1</sup> <sup>2</sup> (*x*)| ≤ *<sup>C</sup>* exp *<sup>p</sup>*<sup>−</sup> <sup>1</sup> (*x* − *dx*) *ε* , *x* ∈ Ω<sup>−</sup> *<sup>x</sup>* . Using the following barrier functions

$$\psi^{\pm} = \mathbb{C}\gamma(\varepsilon + \exp\left(\frac{p\_1^-(d\_\mathbf{x} - \mathbf{x})}{\varepsilon}\right) - \varepsilon \exp\left(\frac{p\_1^-(\mathbf{x} - \mathbf{1})}{\varepsilon}\right) \\ \qquad \pm w^{n + \frac{1}{2}}, \ \mathbf{x} \in \Omega\_\mathbf{x}^+.$$

we prove that <sup>|</sup>*wn*<sup>+</sup> <sup>1</sup> <sup>2</sup> (*x*)| ≤ *C* exp *<sup>p</sup>*<sup>−</sup> <sup>1</sup> (*dx* − *x*) *ε* <sup>+</sup> *<sup>ε</sup>* exp *<sup>p</sup>*<sup>−</sup> <sup>1</sup> (*x* − 1) *ε* , *<sup>x</sup>* <sup>∈</sup> <sup>Ω</sup><sup>+</sup> *<sup>x</sup>* . Further the successive differentiation's leads the desired results.

In a similar manner, one can decompose *u*ˆ*n*+<sup>1</sup> as *vn*+<sup>1</sup> + *wn*+<sup>1</sup> = *u*ˆ*n*+<sup>1</sup> and *vn*<sup>+</sup>1, *wn*+<sup>1</sup> satisfy the following BVPs:

$$\begin{cases} \mathcal{B}\_{y}v^{n+1} = v^{n+\frac{1}{2}} + h\_{l}g\_{2}(t\_{n+1}), \ y \in \Omega\_{y}^{\*} \\\\ v^{n+1}(y) = \mathcal{U}^{n+1}(y), \ y \in [-d\_{y}, 0], \ v^{n+1}(1) = \mathcal{U}^{n+1}(1), \\\\ \left[v^{n+1}(d\_{y})\right] = \sum\_{k=0}^{1} \epsilon^{k} \left[v\_{k}^{n+1}(d\_{y})\right], \ \left[\frac{d}{dy}v^{n+1}(d\_{y})\right] = \sum\_{k=0}^{1} \epsilon^{k} \left[\frac{d}{dy}v\_{k}^{n+1}(d\_{y})\right], \end{cases} \tag{14}$$

$$\begin{cases} \mathfrak{D}\_y w^{n+1} = w^{n+\frac{1}{2}}, \ y \in \Omega\_{y^\*}^\* \\\\ w^{n+1}(y) = 0, \ y \in [-d\_{\mathcal{Y}}, 0], \ w^{n+1}(1) = 0, \\\\ \left[w^{n+1}(d\_{\mathcal{Y}})\right] = -\left[v^{n+1}(d\_{\mathcal{Y}})\right], \ \left[\frac{d}{dy} w^{n+1}(d\_{\mathcal{Y}})\right] = -\left[\frac{d}{dy} v^{n+1}(d\_{\mathcal{Y}})\right] \end{cases} \tag{15}$$

and we have the following result.

**Theorem 2.** *Let u*ˆ*n*+<sup>1</sup> *be the solution to the problem (6), then its regular and singular components satisfy the following bounds on the derivatives*

$$\begin{aligned} \left| \left| \frac{d^k v^{n+1}}{dy^k} \right| \right|\_{\Omega^+} &\leq \mathbb{C} (1 + \varepsilon^{2-k}), \ k = 0, 1, 2, 3, \\\left| \frac{d^k w^{n+1} (x)}{dy^k} \right| &\leq \mathbb{C} \begin{cases} \varepsilon^{-k} \exp \left( \frac{p\_2^- (y - dy)}{\varepsilon} \right), y \in \Omega\_y^-, & k = 0, 1, 2, 3, \\\varepsilon^{-k} \exp \left( \frac{p\_2^- (dy - y)}{\varepsilon} \right) + \varepsilon^{-k+1} \exp \left( \frac{p\_2^- (y - 1)}{\varepsilon} \right), y \in \Omega\_y^+. & k \end{cases} \end{aligned}$$

#### **4. Discrete Problem**

#### *4.1. Spatial Domain Discretization*

From Theorems 1 and 2, we observe that the IBVP (1)–(3) exhibits twin interior layers along the lines (*dx*, *y*), *y* ∈ Ω*<sup>y</sup>* and (*x*, *dy*), *x* ∈ Ω*<sup>x</sup>* and weak boundary layers along *x* = 1 and *y* = 1. Let *N* be the number of mesh points in both spatial *x* and *y* directions. As the mesh defined in Ref. [13], we define the mesh points in both *x* and *y* directions, which is given in the following: let *<sup>τ</sup>*1,*<sup>x</sup>* <sup>=</sup> min{ *dx* <sup>2</sup> , <sup>2</sup> *p*− 1 *<sup>ε</sup>*ln *<sup>N</sup>*}, *<sup>τ</sup>*2,*<sup>x</sup>* <sup>=</sup> min{ <sup>1</sup>−*dx* <sup>4</sup> , <sup>2</sup> *p*− 1 *ε*ln *N*}, *<sup>τ</sup>*1,*<sup>y</sup>* <sup>=</sup> min{ *dy* <sup>2</sup> , <sup>2</sup> *p*− 2 *<sup>ε</sup>*ln *<sup>N</sup>*} and *<sup>τ</sup>*2,*<sup>y</sup>* <sup>=</sup> min{ <sup>1</sup>−*dy* <sup>4</sup> , <sup>2</sup> *p*− 2 *ε*ln *N*}. Using the transition parameters *τi*,*ν*, *i* = 1, 2, *ν* = *x*, *y*, we partitioned the domains Ω*<sup>x</sup>* and Ω*<sup>y</sup>* as follows:

$$\begin{split} \Pi\_{\mathbf{x}} &= \cup\_{i=1}^{5} \Omega\_{i,\mathbf{x}'} \, \, \Omega\_{\mathbf{l},\mathbf{x}} = [0, d\_{\mathbf{x}} - \mathfrak{r}\_{\mathbf{l},\mathbf{x}}], \, \Omega\_{\mathbf{2},\mathbf{x}} = [d\_{\mathbf{x}} - \mathfrak{r}\_{\mathbf{l},\mathbf{x}'}, d\_{\mathbf{x}}], \, \Omega\_{\mathbf{3},\mathbf{x}} = [d\_{\mathbf{x}}, d\_{\mathbf{x}} + \mathfrak{r}\_{\mathbf{2},\mathbf{x}}], \\ \Omega\_{\mathbf{4},\mathbf{x}} &= [d\_{\mathbf{x}} + \mathfrak{r}\_{\mathbf{2},\mathbf{x}}, 1 - \mathfrak{r}\_{\mathbf{2},\mathbf{x}}], \, \Omega\_{\mathbf{5},\mathbf{x}} = [1 - \mathfrak{r}\_{\mathbf{2},\mathbf{x}}, 1], \\ \Pi\_{\mathbf{5}} &= \cup\_{i=1}^{5} \Omega\_{\mathbf{i},\mathbf{y}'} \, \, \Omega\_{\mathbf{1},\mathbf{y}} = [0, d\_{\mathbf{y}} - \mathfrak{r}\_{\mathbf{1},\mathbf{y}}], \, \Omega\_{\mathbf{2},\mathbf{y}} = [d\_{\mathbf{y}} - \mathfrak{r}\_{\mathbf{1},\mathbf{y}'}, d\_{\mathbf{y}}], \, \Omega\_{\mathbf{3},\mathbf{y}} = [d\_{\mathbf{y}}, d\_{\mathbf{y}} + \mathfrak{r}\_{\mathbf{2},\mathbf{y}}], \\ \Omega\_{\mathbf{4},\mathbf{y}} &= [d\_{\mathbf{y}} + \mathfrak{r}\_{\mathbf{2},\mathbf{y}}, 1 - \mathfrak{r}\_{\mathbf{2},\mathbf{y}}], \, \Omega\_{\mathbf{5},\mathbf{y}} = [1 - \mathfrak{r}\_{\mathbf{2},\mathbf{y}}, 1]. \end{split}$$

On each sub-domains Ω*i*,*x*, *i* = 1, 2, 3, 4, 5, respectively, we place *<sup>N</sup>* <sup>4</sup> , *<sup>N</sup>* <sup>4</sup> , *<sup>N</sup>* <sup>8</sup> , *<sup>N</sup>* <sup>4</sup> , *<sup>N</sup>* <sup>8</sup> mesh points with mesh sizes <sup>4</sup>(*dx*−*τ*1,*x*) *<sup>N</sup>* , <sup>4</sup>*τ*1,*<sup>x</sup> <sup>N</sup>* , <sup>8</sup>*τ*2,*<sup>x</sup> <sup>N</sup>* , <sup>4</sup>(1−2*τ*2,*x*−*dx*) *<sup>N</sup>* , <sup>8</sup>*τ*2,*<sup>x</sup> <sup>N</sup>* . In the same manner the mesh points in Ω*i*,*y*, *i* = 1, 2, 3, 4, 5 are defined. Now let us denote the mesh sizes to be *hx*(*i*) = *xi* <sup>−</sup> *xi*−1, *<sup>i</sup>* ∈ I*<sup>N</sup>* and *hy*(*i*) = *yi* <sup>−</sup> *yi*−1, *<sup>i</sup>* ∈ I*<sup>N</sup>* and define the mesh <sup>Ω</sup>*<sup>N</sup> <sup>x</sup>* <sup>=</sup> {*xi*}*i*=*<sup>N</sup> <sup>i</sup>*=<sup>0</sup> , *<sup>x</sup>*<sup>0</sup> <sup>=</sup> 0, *xi* <sup>=</sup> *xi*−<sup>1</sup> <sup>+</sup> *hx*(*i*), *<sup>i</sup>* ∈ I*<sup>N</sup>* and <sup>Ω</sup>*<sup>N</sup> <sup>y</sup>* <sup>=</sup> {*yi*}*i*=*<sup>N</sup> <sup>i</sup>*=<sup>0</sup> , *<sup>y</sup>*<sup>0</sup> <sup>=</sup> 0, *yi* <sup>=</sup> *yi*−<sup>1</sup> <sup>+</sup> *hy*(*i*), *<sup>i</sup>* ∈ I*N*. The mesh distribution is depicted in the Figure 1.

**Figure 1.** Mesh points distribution.

*4.2. The Finite Difference Schemes*

On the meshes <sup>Ω</sup>*<sup>N</sup> <sup>x</sup>* and <sup>Ω</sup>*<sup>N</sup> <sup>y</sup>* , we define the following finite difference schemes.

$$\begin{aligned} \text{fix } y &= y\_j, \\\\ \begin{cases} \mathcal{U}^{n+\frac{1}{2}}\_{i,j} + (-\varepsilon \delta^2\_{\mathbf{x}} \mathcal{U}^{n+\frac{1}{2}}\_{i,j} + p\_{1i,j} D^-\_{\mathbf{x}} \mathcal{U}^{n+\frac{1}{2}}\_{i,j} + q\_{1i,j} \mathcal{U}^{N}\_{\mathbf{d}} \mathcal{U}^{n+\frac{1}{2}}\_{i,j}) h\_l &= \mathcal{U}^n\_{i,j} \\ &+ h\_l g\_{1}(x\_i, y\_j, t\_{n+1}), \ i \in \mathcal{Z}\_{\frac{N}{2}-1}, \\\ D^-\_{\mathbf{x}} \mathcal{U}^{n+\frac{1}{2}}\_{N/2, j} &= D^+\_{\mathbf{x}} \mathcal{U}^{n+\frac{1}{2}}\_{N/2, j}, \qquad i = \frac{N}{2}, \\\ \mathcal{U}^{n+\frac{1}{2}}\_{i,j} + (-\varepsilon \delta^2\_{\mathbf{x}} \mathcal{U}^{n+\frac{1}{2}}\_{i,j} + p\_{1i,j} D^+\_{\mathbf{x}} \mathcal{U}^{n+\frac{1}{2}}\_{i,j} + q\_{1i,j} \mathcal{U}^{N}\_{\mathbf{d}} \mathcal{U}^{n+\frac{1}{2}}\_{i,j}) h\_l &= \mathcal{U}^n\_{i,j} \\ &+ h g\_1(x\_i, y\_j, t\_{n+1}), i \in \mathcal{Z}\_{N-1} \mid \mathcal{Z}\_{\frac{N}{2}} \end{cases} \end{aligned} \tag{16}$$

$$\iota \mathcal{U}\_{0,j}^{n+\frac{1}{2}} = \widehat{\iota}^{n+\frac{1}{2}}(0, y\_j); \; \mathcal{U}\_{N,j}^{n+\frac{1}{2}} = \widehat{\iota}^{n+\frac{1}{2}}(1, y\_j)\_{\prime}$$

fix *x* = *xi*,

$$\mathcal{D}\_{y}^{N}\mathcal{U}\_{i,j}^{n+1} := \begin{cases} \mathcal{U}\_{i,j}^{n+1} + (-\varepsilon \delta\_{y}^{2}\mathcal{U}\_{i,j}^{n+1} + p\_{2,j}D\_{y}^{-}\mathcal{U}\_{i,j}^{n+1} + q\_{2,j}\mathcal{U}\_{i}^{N}\mathcal{U}\_{i,j}^{n+1})h\_{t} = \mathcal{U}\_{i,j}^{n+\frac{1}{2}} \\ \\ \qquad + h\varrho\mathcal{Z}(\mathbf{x}\_{i}, y\_{j}, t\_{n+1}), \; j \in \mathcal{Z}\_{\frac{N}{2}-1} \\\\ \mathcal{D}\_{y}^{-}\mathcal{U}\_{i,N/2}^{n+1} = \mathcal{D}\_{y}^{+}\mathcal{U}\_{i,N/2'}^{n+1} & j = \frac{N}{2}, \\\\ \mathcal{U}\_{i,j}^{n+1} + (-\varepsilon \delta\_{y}^{2}\mathcal{U}\_{i,j}^{n+1} + p\_{2,j}D\_{y}^{+}\mathcal{U}\_{i,j}^{n+1} + q\_{2,j}I\_{i}^{N}\mathcal{U}\_{i,j}^{n+1})h\_{t} = \mathcal{U}\_{i,j}^{n+\frac{1}{2}} \\ \qquad + h\varrho\mathcal{Z}(\mathbf{x}\_{i}, y\_{j}, t\_{n+1}), \; j \in \mathcal{Z}\_{N-1} \; \mathcal{Z}\_{\frac{N}{2}'} \\\\ \mathcal{U}\_{0,j}^{n+1} = \mathcal{U}^{n+1}(\mathbf{x}\_{i}, \mathbf{0}), \; \mathcal{U}\_{i,N}^{n+1} = \mathcal{u}^{n+1}(\mathbf{x}\_{i}, \mathbf{1}), \end{cases}$$

where *δ*<sup>2</sup> *<sup>ζ</sup>* , *D*<sup>−</sup> *<sup>ζ</sup>* and *<sup>D</sup>*<sup>+</sup> *<sup>ζ</sup>* , *ζ* = *x*, *y* are the standard finite difference operators,

*IN* **<sup>d</sup>** *<sup>U</sup>n*<sup>+</sup> <sup>1</sup> 2 *<sup>i</sup>*,*<sup>j</sup>* = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ 0, *i* ∈ I *<sup>N</sup>* <sup>2</sup> <sup>−</sup>1, *<sup>U</sup>n*<sup>+</sup> <sup>1</sup> 2 *<sup>η</sup>*,*<sup>ξ</sup> <sup>l</sup>η*,*x*(*xi* <sup>−</sup> *dx*)*lξ*,*y*(*yj* <sup>−</sup> *dy*) + *<sup>U</sup>n*<sup>+</sup> <sup>1</sup> 2 *<sup>η</sup>*+1,*<sup>ξ</sup> lη*+1,*x*(*xi* − *dx*)*lξ*,*y*(*yj* − *dy*) <sup>+</sup>*Un*<sup>+</sup> <sup>1</sup> 2 *<sup>η</sup>*,*ξ*+1*lη*,*x*(*xi* − *dx*)*lξ*+1,*y*(*yj* − *dy*) <sup>+</sup>*Un*<sup>+</sup> <sup>1</sup> 2 *<sup>η</sup>*+1,*ξ*+1*lη*+1,*x*(*xi* − *dx*)*lξ*+1,*y*(*yj* − *dy*), *i* ∈ I*N*−<sup>1</sup> \ I *<sup>N</sup>* 2 *IN* **<sup>d</sup>** *<sup>U</sup>n*+<sup>1</sup> *<sup>i</sup>*,*<sup>j</sup>* = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎩ 0, *j* ∈ I *<sup>N</sup>* <sup>2</sup> <sup>−</sup>1, *Un*+<sup>1</sup> *<sup>η</sup>*,*<sup>ξ</sup> <sup>l</sup>η*,*x*(*xi* <sup>−</sup> *dx*)*lξ*,*y*(*yj* <sup>−</sup> *dy*) + *<sup>U</sup>n*+<sup>1</sup> *<sup>η</sup>*+1,*<sup>ξ</sup> lη*+1,*x*(*xi* − *dx*)*lξ*,*y*(*yj* − *dy*) +*Un*+<sup>1</sup> *<sup>η</sup>*,*ξ*+1*lη*,*x*(*xi* − *dx*)*lξ*+1,*y*(*yj* − *dy*) +*Un*+<sup>1</sup> *<sup>η</sup>*+1,*ξ*+1*lη*+1,*x*(*xi* − *dx*)*lξ*+1,*y*(*yj* − *dy*), *j* ∈ I*N*−<sup>1</sup> \ I *<sup>N</sup>* 2 ,

*<sup>l</sup>η*,*x*(*xi* <sup>−</sup> *dx*) = *<sup>x</sup>η*+<sup>1</sup> <sup>−</sup> (*xi* <sup>−</sup> *dx*) *hx*(*<sup>η</sup>* <sup>+</sup> <sup>1</sup>) , *<sup>l</sup>η*+1,*x*(*xi* <sup>−</sup> *dx*) = (*xi* <sup>−</sup> *dx*) <sup>−</sup> *<sup>x</sup><sup>η</sup> hx*(*<sup>η</sup>* <sup>+</sup> <sup>1</sup>) , *<sup>l</sup>ξ*,*y*(*yj* <sup>−</sup> *dy*) = *<sup>y</sup>ξ*+<sup>1</sup> <sup>−</sup> (*yj* <sup>−</sup> *dy*) *hy*(*<sup>ξ</sup>* <sup>+</sup> <sup>1</sup>) , *<sup>l</sup>ξ*+1,*y*(*yj* <sup>−</sup> *dy*) = (*yj* <sup>−</sup> *dy*) <sup>−</sup> *<sup>y</sup><sup>ξ</sup> hy*(*<sup>ξ</sup>* <sup>+</sup> <sup>1</sup>) , *<sup>x</sup>η*, *<sup>x</sup>η*<sup>+</sup>1, *<sup>y</sup><sup>ξ</sup>* , *<sup>y</sup>ξ*+<sup>1</sup> are the nodal points such that *xi* − *dx* ∈ [*xη*, *xη*+1] and *yj* − *dy* ∈ [*y<sup>ξ</sup>* , *yξ*+1]. The above two difference operators D*<sup>N</sup> <sup>x</sup>* and D*<sup>N</sup> <sup>y</sup>* satisfy the following discrete maximum principles.

**Note:** Let us denote the difference operators *D*∗ *<sup>x</sup>* = *D*− *<sup>x</sup>* , *<sup>i</sup>* <sup>&</sup>lt; *<sup>N</sup>* 2 , *D*<sup>+</sup> *<sup>x</sup>* , *<sup>i</sup>* <sup>&</sup>gt; *<sup>N</sup>* <sup>2</sup> , and *<sup>D</sup>*<sup>∗</sup> *<sup>y</sup>* = *D*− *<sup>y</sup>* , *<sup>j</sup>* <sup>&</sup>lt; *<sup>N</sup>* 2 , *D*<sup>+</sup> *<sup>y</sup>* , *<sup>j</sup>* <sup>&</sup>gt; *<sup>N</sup>* 2 . In the following we use the above difference operators. Further, the test functions *s*(*xi*) = ⎧ ⎨ ⎩ *xi* + 1, *i* ≤ *N*/2, *dx dx* − *xi* 1 − *dx* <sup>+</sup> *dx* <sup>+</sup> 1, *<sup>i</sup>* <sup>&</sup>gt; *<sup>N</sup>*/2, and *s*(*yj*) = ⎧ ⎪⎨ ⎪⎩ *yj* + 1, *j* ≤ *N*/2, *dy dy* − *yj* 1 − *dy* <sup>+</sup> *dy* <sup>+</sup> 1, *<sup>j</sup>* <sup>&</sup>gt; *<sup>N</sup>*/2,

are also used.

#### *4.3. Discrete Stability Results*

**Lemma 4.** *Let the mesh function be* <sup>Ψ</sup>*i*,*j, satisfies* <sup>Ψ</sup>0,*<sup>j</sup>* <sup>≥</sup> 0, <sup>Ψ</sup>*N*,*<sup>j</sup>* <sup>≥</sup> <sup>0</sup>*,* <sup>D</sup>*<sup>N</sup> <sup>x</sup>* <sup>Ψ</sup>*i*,*<sup>j</sup>* <sup>≥</sup> <sup>0</sup> *and* [*D*<sup>+</sup> *x* − *D*− *<sup>x</sup>* ]Ψ*N*/2,*<sup>j</sup>* ≤ 0*, then* Ψ*i*,*<sup>j</sup>* ≥ 0 *for all i*.

**Proof.** Making use of the test mesh function *s*(*xi*) and the arguments given in Ref. [13], Lemma 6.1, the lemma can be proved.

**Lemma 5.** *Let the mesh function be* <sup>Ψ</sup>*i*,*j, satisfies* <sup>Ψ</sup>*i*,0 <sup>≥</sup> 0, <sup>Ψ</sup>*i*,*<sup>N</sup>* <sup>≥</sup> <sup>0</sup>*,* <sup>D</sup>*<sup>N</sup> <sup>y</sup>* <sup>Ψ</sup>*i*,*<sup>j</sup>* <sup>≥</sup> <sup>0</sup> *and* [*D*<sup>+</sup> *y* − *D*− *<sup>y</sup>* ]Ψ*i*,*N*/2 ≤ 0*, then* Ψ*i*,*<sup>j</sup>* ≥ 0 *for all j*.

Using the above two Lemmas 4 and 5, we can have the following discrete stability results.

**Lemma 6.** *Let Un*<sup>+</sup> <sup>1</sup> 2 *<sup>i</sup>*,*<sup>j</sup> be a numerical solution defined by (16), then*

$$|\mathsf{U}\_{i,j}^{n+\frac{1}{2}}| \leq \mathbb{C} \max\left\{ |\mathsf{U}\_{0,j}^{n+\frac{1}{2}}|, |\mathsf{U}\_{N,j}^{n+\frac{1}{2}}|, \sup\_{i} |\mathsf{D}\_{x}^{N}\mathsf{U}\_{i,j}^{n+\frac{1}{2}}| \right\}| \quad \text{ for all } i.$$

**Lemma 7.** *Let Un*+<sup>1</sup> *<sup>i</sup>*,*<sup>j</sup> be a numerical solution defined by (17), then*

$$|\mathcal{U}\_{i,j}^{n+1}| \le \mathbb{C} \max \left\{ |\mathcal{U}\_{i,0}^{n+1}|, |\mathcal{U}\_{i,N}^{n+1}|, \sup\_j |\mathfrak{D}\_y^N \mathcal{U}\_{i,j}^{n+1}| \right\}\_{\prime} \quad \text{for all } j.$$

**Remark 1.** *From Lemmas 6 and 7, we can see that, the numerical solutions defined in (16) and (17) are stable. Further, by the results of Ref. [20], the matrices associated with the difference schemes (16) and (17) are M-matrices.*

#### **5. Error Computation**

Analogous to the continuous solution, the numerical solution is decomposed into smooth and singular components. The solution *Un*<sup>+</sup> <sup>1</sup> <sup>2</sup> is decomposed as *Un*<sup>+</sup> <sup>1</sup> <sup>2</sup> = *Vn*<sup>+</sup> <sup>1</sup> <sup>2</sup> + *Wn*<sup>+</sup> <sup>1</sup> <sup>2</sup> satisfy the following difference equations:

$$\begin{cases} \mathfrak{D}\_{\mathbf{x}}^{N} V\_{i,j}^{n+\frac{1}{2}} = V\_{i,j}^{n} + h\_{t} g\_{1}(\mathbf{x}\_{i}, y\_{j}, t\_{n+1}), \; i \in \mathcal{Z}\_{N} \\\\ D\_{\mathbf{x}}^{+} V\_{N/2,j}^{n+\frac{1}{2}} - D\_{\mathbf{x}}^{-} V\_{N/2,j}^{n+\frac{1}{2}} = \left[ v^{n+\frac{1}{2}'} (d\_{\mathbf{x}}) \right]\_{\prime} V\_{0,j}^{n+\frac{1}{2}} = 0, \; V\_{N,j}^{n+\frac{1}{2}} = 0, \end{cases} \tag{18}$$

$$\begin{cases} \mathfrak{D}\_{\mathbf{x}}^{N} \mathcal{W}\_{i,j}^{n+\frac{1}{2}} = \mathcal{W}\_{i,j}^{n} \; i \in \mathbb{Z}\_{N} \; \backslash \; \{N, \frac{N}{2}, 0\}, \\\\ D\_{\mathbf{x}}^{+} \mathcal{W}\_{N/2,j}^{n+\frac{1}{2}} - D\_{\mathbf{x}}^{-} \mathcal{W}\_{N/2,j}^{n+\frac{1}{2}} = -\left[v^{n+\frac{1}{2}'} (d\_{\mathbf{x}})\right]\_{\prime} \mathcal{W}\_{0,j}^{n+\frac{1}{2}} = 0 \; \mathcal{W}\_{N,j}^{n+\frac{1}{2}} = 0. \end{cases} \tag{19}$$

Similarly, the solution *Un*+<sup>1</sup> is decomposed as *Un*+<sup>1</sup> = *Vn*+<sup>1</sup> + *Wn*+<sup>1</sup> and they satisfy the following difference equations:

$$\begin{cases} \mathfrak{D}\_y^N V\_{i,j}^{n+1} = V\_{i,j}^{n+\frac{1}{2}} + h\_t g\_2(\mathfrak{x}\_{i\prime} y\_j, t\_{n+1}), \; j \in \mathbb{Z}\_N \; \backslash \; \{\mathcal{N}, \frac{\mathcal{N}}{2}, 0\}, \\ D\_y^+ V\_{i,N/2}^{n+1} - D\_y^- V\_{i,N/2}^{n+1} = \left[ v^{n+1'} (d\_y) \right], \; V\_{i,0}^{n+1} = 0, \; V\_{i,N}^{n+1} = 0, \end{cases} \tag{20}$$

$$\begin{cases} \mathfrak{D}\_{y}^{N} \mathcal{W}\_{i,j}^{n+1} = \mathcal{W}\_{i,j}^{n+\frac{1}{2}}, \; j \in \mathbb{Z}\_{N} \; \backslash \; \{N, \frac{N}{2}, 0\}, \\\\ D\_{y}^{+} \mathcal{W}\_{i,N/2}^{n+1} - D\_{y}^{-} \mathcal{W}\_{i,N/2}^{n+1} = -\left[v^{n+1'}(d\_{y})\right], \; \mathcal{W}\_{i,0}^{n+1} = 0, \; \mathcal{W}\_{i,N}^{n+1} = 0. \end{cases} \tag{21}$$

Note: The error estimate in each time level is proved in the following way:

**Step 1 :**First we estimate the absolute difference of *U* and *V*;

**Step 2 :**We estimate the error bound of the regular component, that is |*v* − *V*|;


**Lemma 8.** *Let <sup>U</sup>* <sup>1</sup> 2 *<sup>i</sup>*,*<sup>j</sup> and V* 1 2 *<sup>i</sup>*,*<sup>j</sup> be numerical solutions of (16) and (18), respectively, when n* = 0*, then*

$$|\mathcal{U}\_{i,j}^{\frac{1}{2}} - V\_{i,j}^{\frac{1}{2}}| \le \mathcal{C} \begin{cases} N^{-1}, & i \in \mathcal{I}\_{\frac{N}{4}}, \\ \zeta + N^{-1}, & i \in \mathcal{I}\_{\frac{5N}{8}} \backslash \mathcal{I}\_{\frac{N}{4}}, \\ N^{-1}, & i \in \mathcal{I}\_{N-1} \backslash \mathcal{I}\_{\frac{5N}{8}}. \end{cases}$$

*ζ is constant.*

**Proof.** Fix *j*. Let us consider the mesh function

$$\Psi^{\pm}(\mathbf{x}\_{i}) = \mathcal{C}[\mathbf{N}^{-1}\mathbf{s}(\mathbf{x}\_{i}) + \Psi(\mathbf{x}\_{i})] \pm [\mathcal{U}^{\frac{1}{2}}\_{i,j} - V^{\frac{1}{2}}\_{i,j}]\_{\prime\prime} \,\,\forall i,j$$

where *ζ* = max *<sup>N</sup>* <sup>4</sup> <sup>+</sup>1≤*i*,*j*<sup>≤</sup> <sup>5</sup>*<sup>N</sup>* <sup>8</sup> <sup>−</sup><sup>1</sup> <sup>|</sup>*<sup>U</sup>* <sup>1</sup> 2 *<sup>i</sup>*,*<sup>j</sup>* − *V* 1 2 *i*,*j* |, and *ψ*(*xi*) = ⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ *xi* <sup>−</sup> (*dx* <sup>−</sup> *<sup>τ</sup>*1,*x*) *τ*1,*<sup>x</sup> ζ*, *i* ∈ I *<sup>N</sup>* 2 \ I *<sup>N</sup>* 1 + *dx* − *xi τ*2,*<sup>x</sup> ζ*, *i* ∈ I<sup>5</sup>*<sup>N</sup>* <sup>8</sup> <sup>−</sup><sup>1</sup> \ I *<sup>N</sup>* 2

0, otherwise.

It is easy to show that Ψ±(*xi*) ≥ 0, *i* = 0, *N*, and by the arguments of [13], we have

4

$$\begin{aligned} \mathfrak{D}\_{\underline{x}}^{N} \Psi^{\pm}(\mathbf{x}\_{i}) &= \mathfrak{D}\_{\underline{x}}^{N} (\mathsf{CN}^{-1} \mathbf{s}(\mathbf{x}\_{i}) + \psi(\mathbf{x}\_{i})) \pm \mathfrak{D}\_{\underline{x}}^{N} (\mathsf{U}\_{i,j}^{\frac{1}{2}} - V\_{i,j}^{\frac{1}{2}}) \ge 0, \; i \ne \frac{N}{2}, \\\\ (D\_{\underline{x}}^{+} - D\_{\underline{x}}^{-}) \Psi^{\pm}(\mathbf{x}\_{\underline{i}}) &\le 0, \; i = \frac{N}{2}. \end{aligned}$$

By the Lemma 4, we have Ψ±(*xi*) ≥ 0. Hence the proof of the lemma.

**Lemma 9.** *Let v* 1 <sup>2</sup> *and V* <sup>1</sup> <sup>2</sup> *be two solutions of (12) and (18), respectively, the* |*v* 1 <sup>2</sup> (*xi*, *y*) − *V* 1 2 *i*,*y*| ≤ *CN*−1, <sup>∀</sup>*i*.

**Proof.** Now, we see that

$$\mathfrak{D}\_{\mathbf{x}}^{N}(\upsilon^{\frac{1}{2}}(\mathbf{x}\_{i},\boldsymbol{y})-V\_{i,\boldsymbol{y}}^{\frac{1}{2}})=\mathfrak{D}\_{\mathbf{x}}^{N}\upsilon^{\frac{1}{2}}(\mathbf{x}\_{i},\boldsymbol{y})-\mathfrak{D}\_{\mathbf{x}}^{N}V\_{i,\boldsymbol{y}}^{\frac{1}{2}}=\mathfrak{D}\_{\mathbf{x}}^{N}\upsilon^{\frac{1}{2}}(\mathbf{x}\_{i},\boldsymbol{y})-\mathfrak{D}\_{\mathbf{x}}\upsilon^{\frac{1}{2}}(\mathbf{x}\_{i},\boldsymbol{y})$$

$$=h\_{t}\left[-\varepsilon\left(\delta\_{\mathbf{x}}^{2}-\frac{d^{2}}{d\mathbf{x}^{2}}\right)+p\_{1,i}\left(D\_{\mathbf{x}}^{\*}-\frac{d}{d\mathbf{x}}\right)+q\_{1,i}[I\_{\mathbf{d}}^{N}-I\_{\mathbf{d}}]\right]\upsilon^{\frac{1}{2}}(\mathbf{x}\_{i},\boldsymbol{y}),$$

from the results given in Refs. [2,21,22], we have <sup>|</sup>D*<sup>N</sup> <sup>x</sup>* (*v* 1 <sup>2</sup> (*xi*, *y*) − *V* 1 2 *<sup>i</sup>*,*y*)| ≤ *ChtN*−1. Using the following barrier function

$$\psi^{\pm}(\mathfrak{x}\_{i}) = \mathcal{C}N^{-1}\mathfrak{s}(\mathfrak{x}\_{i}) \pm (v^{\frac{1}{2}}(\mathfrak{x}\_{i}, \mathfrak{y}) - V^{\frac{1}{2}}\_{i, \mathfrak{y}})\_{\prime}$$

we can see that *<sup>ψ</sup>*±(*xi*) <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 0, *<sup>N</sup>*, <sup>D</sup>*<sup>N</sup> <sup>x</sup> <sup>ψ</sup>*±(*xi*) <sup>≥</sup> 0 and (*D*<sup>+</sup> *<sup>x</sup>* − *D*<sup>−</sup> *<sup>x</sup>* )*ψ*±(*x <sup>N</sup>* 2 ) ≤ 0. From the Lemma 4, we have the desired result.

**Lemma 10.** *Let w*<sup>1</sup> <sup>2</sup> *and W* <sup>1</sup> <sup>2</sup> *be the solutions of (13) and (19), respectively, then* <sup>|</sup>*w*<sup>1</sup> <sup>2</sup> (*xi*, *y*) − *W* 1 2 *<sup>i</sup>*,*y*| ≤ *CN*−<sup>1</sup> ln *<sup>N</sup>*, <sup>∀</sup>*i*.

**Proof.** By the triangle inequality, Theorem 1, Lemmas 8 and 9, we have


where *ζ* = max *<sup>N</sup>* <sup>4</sup> <sup>+</sup>1≤*i*,*j*<sup>≤</sup> <sup>5</sup>*<sup>N</sup>* <sup>8</sup> <sup>−</sup><sup>1</sup> <sup>|</sup>*<sup>U</sup>* <sup>1</sup> 2 *<sup>i</sup>*,*<sup>j</sup>* − *V* 1 2 *i*,*j* |. Hence |*u*ˆ 1 <sup>2</sup> (*xi*, *<sup>y</sup>*) <sup>−</sup> *<sup>U</sup>* <sup>1</sup> 2 *<sup>i</sup>*,*y*| ≤ *CN*−1, *<sup>i</sup>* <sup>=</sup> 0, 1, ··· , *<sup>N</sup>* 4 , 5*N* <sup>8</sup> , ··· , *<sup>N</sup>*. Therefore <sup>|</sup>*w*<sup>1</sup> <sup>2</sup> (*xi*, *<sup>y</sup>*) <sup>−</sup> *<sup>W</sup>* <sup>1</sup> 2 *<sup>i</sup>*,*y*| ≤ *CN*−1, *<sup>i</sup>* <sup>=</sup> 0, 1, ··· , *<sup>N</sup>* <sup>4</sup> , <sup>5</sup>*<sup>N</sup>* <sup>8</sup> , ··· , *N*. To prove the result inside the inner region, we consider the following mesh function

$$\boldsymbol{\Psi}^{\pm}(\mathbf{x}\_{i}) = \mathbb{C}N^{-1}\boldsymbol{\Phi}(\mathbf{x}\_{i}) \pm (\boldsymbol{w}^{\frac{1}{2}} - \boldsymbol{W}^{\frac{1}{2}}),\\\mathbf{x}\_{i} \in (d\_{\mathbf{x}} - \boldsymbol{\tau}\_{1,\mathbf{x}}, d\_{\mathbf{x}}) \cup (d\_{\mathbf{x}}, d\_{\mathbf{x}} + \boldsymbol{\tau}\_{2,\mathbf{x}}) \cap \overline{\boldsymbol{\Omega}}^{N}\_{\mathbf{x}}.$$

where *φ*(*xi*) = ⎧ ⎪⎪⎨ ⎪⎪⎩ (1 + *xi*) + *<sup>τ</sup><sup>x</sup> <sup>ε</sup>*<sup>2</sup> (*xi* <sup>−</sup> (*dx* <sup>−</sup> *<sup>τ</sup>*1,*x*)), *xi* <sup>∈</sup> [*dx* <sup>−</sup> *<sup>τ</sup>*1,*x*, *dx*) <sup>∩</sup> <sup>Ω</sup>*<sup>N</sup> x* , 1 + *dx* + *dx dx* − *xi* 1 − *dx* + *<sup>τ</sup><sup>x</sup> <sup>ε</sup>*<sup>2</sup> (*dx* <sup>+</sup> *<sup>τ</sup>*2,*<sup>x</sup>* <sup>−</sup> *xi*), *xi* <sup>∈</sup> [*dx*, *dx* <sup>+</sup> *<sup>τ</sup>*2,*x*] <sup>∩</sup> <sup>Ω</sup>*<sup>N</sup> x* , , *<sup>τ</sup><sup>x</sup>* <sup>=</sup> min{*τ*1,*x*, *<sup>τ</sup>*2,*x*}. Then we have, *<sup>ψ</sup>*±(*xi*) <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> *<sup>N</sup>* <sup>4</sup> , <sup>5</sup>*<sup>N</sup>* <sup>8</sup> . Further <sup>|</sup>D*<sup>N</sup> <sup>x</sup>* (*w*<sup>1</sup> <sup>2</sup> <sup>−</sup> *<sup>W</sup>* <sup>1</sup> <sup>2</sup> )| ≤ *C*1*htε*−2*N*−1, *i* = *<sup>N</sup>* <sup>4</sup> <sup>+</sup> 1, ··· , *<sup>N</sup>* <sup>2</sup> <sup>−</sup> 1, *<sup>N</sup>* <sup>2</sup> <sup>+</sup> 1, ··· , <sup>5</sup>*<sup>N</sup>* <sup>8</sup> . Now, D*<sup>N</sup> <sup>x</sup> ψ*±(*xi*) = *CN*−1D*<sup>N</sup> <sup>x</sup> <sup>φ</sup>*(*xi*) <sup>±</sup> <sup>D</sup>*<sup>N</sup> <sup>x</sup>* (*w*<sup>1</sup> <sup>2</sup> <sup>−</sup> *<sup>W</sup>* <sup>1</sup> <sup>2</sup> ), *xi* <sup>∈</sup> (*dx* <sup>−</sup> *<sup>τ</sup>*1,*x*, *dx*) <sup>∪</sup> (*dx*, *dx* <sup>+</sup> *<sup>τ</sup>*2,*x*) <sup>∩</sup> <sup>Ω</sup>*<sup>N</sup> x* <sup>≥</sup> *CN*−<sup>1</sup> ⎧ ⎪⎨ ⎪⎩ 1 + *ht p*<sup>−</sup> <sup>1</sup> <sup>+</sup> *<sup>τ</sup><sup>x</sup> <sup>ε</sup>*<sup>2</sup> *ht p*<sup>−</sup> <sup>1</sup> , *xi* <sup>∈</sup> (*dx* <sup>−</sup> *<sup>τ</sup>*1,*x*, *dx*) <sup>∩</sup> <sup>Ω</sup>*<sup>N</sup> x* 1 + *ht*(*p*<sup>−</sup> 1 *dx* <sup>1</sup>−*dx* <sup>+</sup> *<sup>β</sup>*1) + *<sup>τ</sup><sup>x</sup> <sup>ε</sup>*<sup>2</sup> *ht*(*p*<sup>−</sup> <sup>1</sup> <sup>+</sup> *<sup>β</sup>*1), *xi* <sup>∈</sup> (*dx*, *dx* <sup>+</sup> *<sup>τ</sup>*2,*x*) <sup>∩</sup> <sup>Ω</sup>*<sup>N</sup> x* ∓ *C*1*htε* <sup>−</sup>2*N*−<sup>1</sup> <sup>≥</sup> <sup>0</sup>

for a suitable choice of *<sup>C</sup>* > 0. At the point *xN*/2, we have (*D*<sup>+</sup> *<sup>x</sup>* − *D*<sup>−</sup> *<sup>x</sup>* )*ψ*±(*xN*/2) ≤ 0. From the Lemma 4, we have <sup>|</sup>*w*<sup>1</sup> <sup>2</sup> <sup>−</sup> *<sup>W</sup>* <sup>1</sup> <sup>2</sup> | ≤ *CN*−<sup>1</sup> ln *<sup>N</sup>*, *<sup>i</sup>* <sup>=</sup> *<sup>N</sup>* <sup>4</sup> , ··· , <sup>5</sup>*<sup>N</sup>* <sup>8</sup> . Therefore <sup>|</sup>*w*<sup>1</sup> <sup>2</sup> (*xi*, *y*) − *W* 1 2 *<sup>i</sup>*,*y*| ≤ *CN*−<sup>1</sup> ln *<sup>N</sup>*, <sup>∀</sup>*i*.

**Lemma 11.** *Let u*ˆ 1 <sup>2</sup> *and U* <sup>1</sup> <sup>2</sup> *be the solution of (5) and (16), respectively, then u*ˆ 1 <sup>2</sup> <sup>−</sup> *<sup>U</sup>* <sup>1</sup> <sup>2</sup> ≤ *CN*−<sup>1</sup> ln *N*.

**Proof.** The proof follows from the above two lemmas.

**Lemma 12.** *Let v*1*, w*1*, u*ˆ1*, V*1*, W*1*, and U*<sup>1</sup> *be the solutions of (14), (15), (6), (20), (21), and (17), respectively, then*

$$\|v^1 - V^1\| \le \mathcal{CN}^{-1}{}\_{\prime} \; \|w^1 - W^1\| \le \mathcal{CN}^{-1} \ln N,$$

$$\|\hat{u}^1 - U^1\| \le \mathcal{CN}^{-1} \ln N.$$

**Proof.** We see that, *u*ˆ1(*x*, 0) = *U*<sup>1</sup> *<sup>x</sup>*,0 and *<sup>u</sup>*ˆ1(*x*, 1) = *<sup>U</sup>*<sup>1</sup> *<sup>x</sup>*,*N*.

Similar to the proof of Lemma 8, we can prove the following,

$$|\mathcal{U}^1 - V^1| \le C \begin{cases} N^{-1}, & i, j \in \mathcal{I}\_{\frac{N}{4}}, \\ \zeta + N^{-1}, & i, j \in \mathcal{I}\_{\frac{5N}{8}} \backslash \mathcal{I}\_{\frac{N}{4}}, \\ N^{-1}, & i, j \in \mathcal{I}\_{N-1} \backslash \mathcal{I}\_{\frac{5N}{8}} \end{cases} \\ \zeta = \max\_{\frac{N}{4} + 1 \le i, j \le \frac{5N}{8} - 1} |\mathcal{U}^1\_{i,j} - V^1\_{i,j}|.$$

Let *v*<sup>1</sup> and *V*<sup>1</sup> be the solutions of (14) and (20), then similar to Lemma 9, we have

$$\mathfrak{D}\_{y}^{N}(\upsilon^{1}(\mathbf{x},y\_{j})-V^{1}\_{\mathbf{x},j}) = \mathfrak{D}\_{y}^{N}\upsilon^{1}(\mathbf{x},y\_{j}) - \mathfrak{D}\_{y}^{N}V^{1}\_{\mathbf{x},j} = \mathfrak{D}\_{y}^{N}\upsilon^{1}(\mathbf{x},y\_{j}) - \mathfrak{D}\_{y}\upsilon^{1}(\mathbf{x},y\_{j})$$

$$= h\_{l}\left[-\mathfrak{c}\left(\delta\_{y}^{2}-\frac{d^{2}}{dy^{2}}\right) + p\_{2\underline{\imath},j}\left(D\_{y}^{\*}-\frac{d}{dy}\right) + q\_{2\underline{\imath},j}\left[I^{N}\_{\mathbf{d}}-I\_{\mathbf{d}}\right]\right]\upsilon^{1}(\mathbf{x},y\_{j}),$$

and <sup>|</sup>D*<sup>N</sup> <sup>y</sup>* (*v*1(*x*, *yj*) <sup>−</sup> *<sup>V</sup>*<sup>1</sup> *x*,*j* )| ≤ *ChtN*−1. Then by a suitable barrier function one can prove that *v*<sup>1</sup> <sup>−</sup> *<sup>V</sup>*1 ≤ *CN*−1. Similar to the Lemma 11, we estimate *w*<sup>1</sup> <sup>−</sup> *<sup>W</sup>*1,

$$\leq \mathbb{C} \begin{cases} |\mathring{\mathfrak{a}}^{1}(\mathbf{x}, y\_{j}) - \boldsymbol{\mathcal{U}}^{1}\_{\mathbf{x}, j}| \leq |\boldsymbol{\mathcal{U}}^{1}\_{\mathbf{x}, j} - \boldsymbol{V}^{1}\_{\mathbf{x}, j}| + |\boldsymbol{v}^{1}(\mathbf{x}, y\_{j}) - \boldsymbol{V}^{1}\_{\mathbf{x}, j}| + |\mathring{\mathfrak{a}}^{1}(\mathbf{x}, y\_{j}) - \boldsymbol{v}(\mathbf{x}, y\_{j})| \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \boldsymbol{i}\_{i} \in \mathbb{Z}\_{\frac{N}{8}}, \\ \boldsymbol{\zeta} + N^{-1} + \exp\left(\frac{p\_{2}^{-}(y\_{j} - \boldsymbol{d}\_{y})}{\varepsilon}\right), \ i, j \in \mathbb{Z}\_{\frac{N}{2}} \backslash \mathbb{Z}\_{\frac{N}{4}}, \\ \boldsymbol{\zeta} + N^{-1} + \exp\left(\frac{p\_{2}^{-}(\boldsymbol{d}\_{y} - y\_{j})}{\varepsilon}\right), \ i, j \in \mathbb{Z}\_{\frac{N}{8}} \backslash \mathbb{Z}\_{\frac{N}{2}}, \\ N^{-1}, \qquad \quad \quad \quad \boldsymbol{i}\_{i} \in \mathbb{Z}\_{N} \backslash \mathbb{Z}\_{\frac{N}{8}}. \end{cases}$$

Hence <sup>|</sup>*u*ˆ1(*x*, *yj*) <sup>−</sup> *<sup>U</sup>*<sup>1</sup> *x*,*j* | ≤ *CN*−1, *<sup>j</sup>* <sup>=</sup> 0, 1, ··· , *<sup>N</sup>* <sup>4</sup> , <sup>5</sup>*<sup>N</sup>* <sup>8</sup> , ··· , *<sup>N</sup>* and <sup>|</sup>*w*1(*x*, *yj*) <sup>−</sup> *<sup>W</sup>*<sup>1</sup> *x*,*j* | ≤ *CN*−1, *<sup>j</sup>* <sup>=</sup> 0, 1, ··· , *<sup>N</sup>* <sup>4</sup> , <sup>5</sup>*<sup>N</sup>* <sup>8</sup> , ··· , *N*. Using the barrier function

*<sup>ψ</sup>*±(*yj*) = *CN*−1*φ*(*yj*) <sup>±</sup> (*w*<sup>1</sup> <sup>−</sup> *<sup>W</sup>*1), *yj* <sup>∈</sup> (*dy* <sup>−</sup> *<sup>τ</sup>*1,*y*, *dy*) <sup>∪</sup> (*dy*, *dy* <sup>+</sup> *<sup>τ</sup>*2,*y*) <sup>∩</sup> <sup>Ω</sup>*<sup>N</sup> y* , where *φ*(*yj*) = ⎧ ⎪⎪⎨ ⎪⎪⎩ 1 + *yj* + *<sup>τ</sup><sup>y</sup> <sup>ε</sup>*<sup>2</sup> (*yj* <sup>−</sup> (*dy* <sup>−</sup> *<sup>τ</sup>*1,*y*)), *yj* <sup>∈</sup> [*dy* <sup>−</sup> *<sup>τ</sup>*1,*y*, *dy*) <sup>∩</sup> <sup>Ω</sup>*<sup>N</sup> y* , 1 + *dy* + *dy dy* − *yj* 1 − *dy* + *<sup>τ</sup><sup>y</sup> <sup>ε</sup>*<sup>2</sup> (*dy* <sup>+</sup> *<sup>τ</sup>*2,*<sup>y</sup>* <sup>−</sup> *yj*), *yj* <sup>∈</sup> [*dy*, *dy* <sup>+</sup> *<sup>τ</sup>*2,*y*] <sup>∩</sup> <sup>Ω</sup>*<sup>N</sup> y* ,

,

*<sup>τ</sup><sup>y</sup>* <sup>=</sup> min{*τ*1,*y*, *<sup>τ</sup>*2,*y*} we prove that *w*<sup>1</sup> <sup>−</sup> *<sup>W</sup>*1 ≤ *CN*−<sup>1</sup> ln *<sup>N</sup>*, *<sup>j</sup>* <sup>=</sup> *<sup>N</sup>* <sup>4</sup> <sup>+</sup> 1, ··· , <sup>5</sup>*<sup>N</sup>* <sup>8</sup> − 1. Hence the proof.

**Theorem 3.** *Let u*ˆ*n*<sup>+</sup> <sup>1</sup> <sup>2</sup> *, <sup>u</sup>*ˆ*n*+1*, <sup>U</sup>n*<sup>+</sup> <sup>1</sup> 2 *<sup>i</sup>*,*<sup>j</sup> and <sup>U</sup>n*+<sup>1</sup> *<sup>i</sup>*,*<sup>j</sup> be the solutions of (5), (6), (16), and (17), respectively, then*

$$\|\hat{\mu}^{n+\frac{1}{2}} - \mathcal{U}^{n+\frac{1}{2}}\| \le \mathbb{C}N^{-1}\ln N, \quad \text{and} \quad \|\hat{\mu}^{n+1} - \mathcal{U}^{n+1}\| \le \mathbb{C}N^{-1}\ln N.$$

**Proof.** We prove the theorem on each time level *<sup>t</sup>* = *tn*. We know that *<sup>u</sup>*ˆ*n*<sup>+</sup> <sup>1</sup> <sup>2</sup> (0, *<sup>y</sup>*) <sup>−</sup> *<sup>U</sup>n*<sup>+</sup> <sup>1</sup> 2 0,*<sup>y</sup>* = 0, *u*ˆ*n*<sup>+</sup> <sup>1</sup> <sup>2</sup> (1, *<sup>y</sup>*) <sup>−</sup> *<sup>U</sup>n*<sup>+</sup> <sup>1</sup> 2 *<sup>N</sup>*,*<sup>y</sup>* <sup>=</sup> 0, *<sup>u</sup>*ˆ*n*+1(*x*, 0) <sup>−</sup> *<sup>U</sup>n*+<sup>1</sup> *<sup>x</sup>*,0 <sup>=</sup> 0 and *<sup>u</sup>*ˆ*n*+1(*x*, 1) <sup>−</sup> *<sup>U</sup>n*+<sup>1</sup> *<sup>x</sup>*,*<sup>N</sup>* <sup>=</sup> 0.

D*<sup>N</sup> <sup>x</sup>* (*Un*<sup>+</sup> <sup>1</sup> <sup>2</sup> <sup>−</sup> *<sup>V</sup>n*<sup>+</sup> <sup>1</sup> <sup>2</sup> ) = D*<sup>N</sup> <sup>x</sup> <sup>U</sup>n*<sup>+</sup> <sup>1</sup> <sup>2</sup> <sup>−</sup> <sup>D</sup>*<sup>N</sup> <sup>x</sup> <sup>V</sup>n*<sup>+</sup> <sup>1</sup> <sup>2</sup> <sup>=</sup> *<sup>U</sup><sup>n</sup>* <sup>−</sup> *<sup>V</sup>n*, D*<sup>N</sup> <sup>x</sup>* (*Un*<sup>+</sup> <sup>1</sup> <sup>2</sup> <sup>−</sup> *<sup>V</sup>n*<sup>+</sup> <sup>1</sup> <sup>2</sup> ) ≤ *C* ⎧ ⎪⎪⎨ ⎪⎪⎩ *<sup>N</sup>*−1, *<sup>i</sup>*, *<sup>j</sup>* ∈ I *<sup>N</sup>* 4 , *<sup>ζ</sup>* <sup>+</sup> *<sup>N</sup>*−1, *<sup>i</sup>*, *<sup>j</sup>* ∈ I<sup>5</sup>*<sup>N</sup>* 8 \ I *<sup>N</sup>* 4 , *<sup>N</sup>*−1, *<sup>i</sup>*, *<sup>j</sup>* ∈ I*N*−<sup>1</sup> \ I<sup>5</sup>*<sup>N</sup>* 8 , D*<sup>N</sup> <sup>y</sup>* (*Un*+<sup>1</sup> <sup>−</sup> *<sup>V</sup>n*+1) = <sup>D</sup>*<sup>N</sup> <sup>y</sup> <sup>U</sup>n*+<sup>1</sup> <sup>−</sup> <sup>D</sup>*<sup>N</sup> <sup>y</sup> <sup>V</sup>n*+<sup>1</sup> <sup>=</sup> *<sup>U</sup>n*<sup>+</sup> <sup>1</sup> <sup>2</sup> <sup>−</sup> *<sup>V</sup>n*<sup>+</sup> <sup>1</sup> 2 , D*<sup>N</sup> <sup>x</sup>* (*Un*+<sup>1</sup> <sup>−</sup> *<sup>V</sup>n*+1) ≤ *<sup>C</sup>* ⎧ ⎪⎪⎨ ⎪⎪⎩ *<sup>N</sup>*−1, *<sup>i</sup>*, *<sup>j</sup>* ∈ I *<sup>N</sup>* 4 , *<sup>ζ</sup>* <sup>+</sup> *<sup>N</sup>*−1, *<sup>i</sup>*, *<sup>j</sup>* ∈ I<sup>5</sup>*<sup>N</sup>* 8 \ I *<sup>N</sup>* 4 , *<sup>N</sup>*−1, *<sup>i</sup>*, *<sup>j</sup>* ∈ I*N*−<sup>1</sup> \ I<sup>5</sup>*<sup>N</sup>* 8 , *<sup>ζ</sup>* <sup>=</sup> max *<sup>n</sup>* max *<sup>N</sup>* <sup>4</sup> <sup>+</sup>1≤*i*,*j*<sup>≤</sup> <sup>5</sup>*<sup>N</sup>* <sup>8</sup> −1 <sup>|</sup>*U<sup>n</sup> <sup>i</sup>*,*<sup>j</sup>* <sup>−</sup> *<sup>V</sup><sup>n</sup> i*,*j* |,

with the successive applications of Lemmas 6 and 7 and the iteration in *n*, we prove that

$$\|\mathcal{U}^{\mu} - V^{\mu}\| \le \mathcal{C} \begin{cases} N^{-1}, & i, j \in \mathcal{Z}\_{\frac{N}{8}}, \\ \zeta + N^{-1}, & i, j \in \mathcal{Z}\_{\frac{5N}{8}} \backslash \mathcal{Z}\_{\frac{N}{4}}, \\ \quad N^{-1}, & i, j \in \mathcal{Z}\_{N-1} \nwarrow \mathcal{Z}\_{\frac{5N}{8}}. \end{cases} \mu = n + 1 \text{ \& } \mu = n + \frac{1}{2}.$$

Using the following barrier functions

$$\Psi\_1^{\pm}(\mathbf{x}\_i) = \mathbf{C}N^{-1}\mathbf{s}(\mathbf{x}\_i) \pm [v^{n+\frac{1}{2}}(\mathbf{x}\_{i\prime}y\_j) - V\_{i,j}^{n+\frac{1}{2}}], \forall i,$$

$$\Psi\_2^{\pm}(y\_j) = \mathbf{C}N^{-1}\mathbf{s}(y\_j) \pm [v^{n+1}(\mathbf{x}\_{i\prime}y\_j) - V\_{i,j}^{n+1}], \forall j,$$

and from Lemmas 6 and 7, we can prove that

$$\left\| \left| v^{n + \frac{1}{2}} - V^{n + \frac{1}{2}} \right| \right\| \le \mathbb{C} N^{-1}, \ \left\| v^{n+1} - V^{n+1} \right\| \le \mathbb{C} N^{-1}.$$

It is observed that <sup>|</sup>*wμ*(*xi*, *yj*) <sup>−</sup> *<sup>W</sup><sup>μ</sup> i*,*j* | ≤ *CN*−1, *<sup>μ</sup>* <sup>=</sup> *<sup>n</sup>* <sup>+</sup> <sup>1</sup> <sup>2</sup> , *<sup>n</sup>* <sup>+</sup> 1, *<sup>i</sup>*, *<sup>j</sup>* <sup>=</sup> 0, 1, ··· , *<sup>N</sup>* 4 , 5*N* <sup>8</sup> , ··· , *N*. Using the following barrier functions

$$\Phi\_1^{\pm}(\mathbf{x}\_i) = \text{CN}^{-1}\phi\_1(\mathbf{x}\_i) \pm (w^{n+\frac{1}{2}} - W^{n+\frac{1}{2}}), \; \mathbf{x}\_i \in [\Omega\_{2,x} \cup \Omega\_{3,x}] \cap \overline{\Pi}\_x^N,$$

$$\Phi\_2^{\pm}(y\_j) = \text{CN}^{-1}\phi\_2(y\_j) \pm (w^{n+1} - W^{n+1}), \; y\_j \in [\Omega\_{2,y} \cup \Omega\_{3,y}] \cap \overline{\Pi}\_y^N,$$

$$\text{where } \phi\_1(\mathbf{x}\_i) = \begin{cases} (1 + \mathbf{x}\_i) + \frac{\tau\_y}{\epsilon^2} (\mathbf{x}\_i - (\mathbf{d}\_x - \boldsymbol{\tau}\_{1,x})), & \mathbf{x}\_i \in \Omega\_{2,x} \cap \overline{\Omega}\_x^N, \\\\ \left(1 + d\_x + d\_x \frac{d\_x - \mathbf{x}\_i}{1 - d\_x}\right) + \frac{\tau\_y}{\epsilon^2} (d\_x + \tau\_{2,x} - \mathbf{x}\_i), \; \mathbf{x}\_i \in \Omega\_{3,x} \cap \overline{\Omega}\_x^N, \\\\ \left(1 + y\_j\right) + \frac{\tau\_y}{\epsilon^2} (y\_j - (d\_y - \tau\_{1,y})), & y\_j \in \Omega\_{2,y} \cap \overline{\Omega}\_y^N, \\\\ \left(1 + d\_y + d\_y \frac{d\_y - y\_j}{1 - d\_y}\right) + \frac{\tau\_y}{\epsilon^2} (d\_y + \tau\_{2,y} - y\_j), \; y\_j \in \Omega\_{3,y} \cap \overline{\Omega}\_y^N, \\\\ \cdots & \cdots \end{cases}$$

*τμ* <sup>=</sup> min{*τ*1,*μ*, *<sup>τ</sup>*2,*μ*}, *<sup>μ</sup>* <sup>=</sup> *<sup>x</sup>*, *<sup>y</sup>* we prove that <sup>|</sup>*wμ*(*xi*, *yj*) <sup>−</sup> *<sup>W</sup><sup>μ</sup> i*,*j* | ≤ *CN*−1, *<sup>μ</sup>* <sup>=</sup> *<sup>n</sup>* <sup>+</sup> 1 <sup>2</sup> , *<sup>n</sup>* <sup>+</sup> 1, and *<sup>i</sup>*, *<sup>j</sup>* <sup>=</sup> *<sup>N</sup>* <sup>4</sup> <sup>+</sup> 1, ··· , <sup>5</sup>*<sup>N</sup>* <sup>8</sup> − 1. By the triangle inequality, we have the desired results.

**Theorem 4.** *Let u*(*xi*, *yj*, *tn*) *and U<sup>n</sup> <sup>i</sup>*,*<sup>j</sup> be the solutions of (1) and (17), then*

$$||\mu - U|| \le \mathcal{C}(h\_t + N^{-1} \ln N).$$

**Proof.** The error can be obtained from the following

$$\mu(\mathbf{x}\_i, \mathbf{y}\_j, t\_n) - \mathcal{U}^n\_{i,j} = \mathfrak{h}^n(\mathbf{x}\_i, \mathbf{y}\_j) - \mathcal{U}^n\_{i,j} + \mathfrak{u}(\mathbf{x}\_i, \mathbf{y}\_j, t\_n) - \overline{\mathfrak{u}}^n(\mathbf{x}\_i, \mathbf{y}\_j) + \overline{\mathfrak{u}}^n(\mathbf{x}\_i, \mathbf{y}\_j) - \mathfrak{u}^n(\mathbf{x}\_i, \mathbf{y}\_j),$$

$$||\mu(t\_n) - \mathcal{U}^n|| \le ||\overline{\mathfrak{u}}^n - \hat{\mathfrak{u}}^n|| + ||\hat{\mathfrak{u}}^n - \mathcal{U}^n|| + ||\mu(t\_n) - \overline{\mathfrak{u}}^n||.$$

From Lemma 3, Theorem 3 and Ref. [6], Theorem 1, we have

$$\|\|\mu(t\_n) - \mathcal{U}^\pi\|\| \le \|\mu(t\_n) - \overline{\mu}^\pi\| + \|\overline{\mu}^\pi - \mathcal{U}^\pi\| + \|\mathcal{U}^\pi - \mathcal{U}^\pi\| \le \mathcal{C}h\_t + \mathcal{C}N^{-1}\ln N\_t$$

which completes the proof.

#### **6. Numerical Validation**

Two examples are presented in this section to validate the theoretical results presented in this article. The exact analytical solutions to the test problems are unknown, therefore we use the double mesh principle to calculate the maximum point-wise error and computational order of convergence. For fixed *M*, we define

$$\begin{aligned} E\_{\varepsilon}^{N} &= \max\_{i,j} \mid \mathcal{U}\_{i,j}^{N}(h\_{\mathbf{x}}, h\_{\mathbf{y}}, h\_{\mathbf{t}}) - \mathcal{U}\_{i,j}^{N}(\frac{h\_{\mathbf{x}}}{2}, \frac{h\_{\mathbf{y}}}{2}, \frac{h\_{\mathbf{t}}}{2}) \mid, \; 0 \le i, j \le N\\ D\_{\mathbf{x},\mathbf{y}}^{N} &= \max\_{\varepsilon} E\_{\varepsilon}^{N}, \; \rho^{N} = \log\_{2} \left( \frac{D\_{\mathbf{x},\mathbf{y}}^{N}}{\overline{D\_{\mathbf{x},\mathbf{y}}^{2N}}} \right), \end{aligned}$$

where *U<sup>N</sup> i*,*j* (*hx*, *hy*, *ht*) and *U<sup>N</sup> i*,*j* ( *hx* 2 , *hy* <sup>2</sup> , *ht* <sup>2</sup> ) are the numerical solutions at the node (*xi*, *yj*, *tn*) with mesh sizes (*hx*, *hy*, *ht*) and ( *hx* 2 , *hy* <sup>2</sup> , *ht* <sup>2</sup> ), respectively, *<sup>D</sup><sup>N</sup> <sup>x</sup>*,*<sup>y</sup>* is maximum over *ε* for fixed *N*.

**Example 1.** *Consider the 2D parabolic PDE (1) with discontinuous source and convection coefficients with the following data:*

$$\frac{\partial u}{\partial t} - \varepsilon \Delta u + \overline{p}(\mathbf{x}) \cdot \nabla u + q(\mathbf{x})u(\mathbf{x} - \mathbf{d}\_r t) = g(\mathbf{x}, t), \ (\mathbf{x}, t) \in \mathfrak{D}^\* \times (0, T]$$

$$p\_1(\mathbf{x}) = \begin{cases} 1 + \mathbf{x}(1 - \mathbf{x}), \ \mathbf{x} \in (0, d\_x), \ \forall y, \\ -(1 + \mathbf{x}(1 - \mathbf{x})), \ \mathbf{x} \in (d\_x, 1), \end{cases} \quad p\_2(\mathbf{x}) = \begin{cases} 1 + y(1 - y), \ y \in (0, d\_y), \ \forall x, \\ -(1 + \mathbf{x}(1 - \mathbf{x})), \ y \in (d\_y, 1), \end{cases}$$

$$q\_1(\mathbf{x}) = -0.5 - \mathbf{x}(1 - \mathbf{x}), \ q\_2(\mathbf{x}) = -0.5 - y(1 - y), \ d\_x = 0.5 = d\_y,$$

$$g\_1(\mathbf{x}, t) = \begin{cases} -\mathbf{x}^2 y(1 - \mathbf{x})(1 - y)^2 \exp\left(t^2 - \frac{\mathbf{x} y}{1 + \mathbf{x}^2 + y^2}\right), \ \mathbf{x} \in (0, d\_x), \\ \mathbf{x} y(1 - \mathbf{x})^2 (1 - y) \exp\left(t^2 - \frac{\mathbf{x}^2 y^2}{1 + \mathbf{x}^2 - y^2}\right), \ \mathbf{x} \in (d\_x, 1), \end{cases}$$

$$\{-\mathbf{x}^3 y, \ y \in (0, d\_x), \mathbf{x}^2 (1 - \mathbf{x}), \$$

,

$$\log \mathbf{z}(\mathbf{x}, t) = \begin{cases} -\mathbf{x}^3 y^2, \ y \in (0, d\_\mathbf{y}), \\ (1 - \mathbf{x})^5 \sqrt{1 - y}, \ y \in (d\_\mathbf{y}, 1), \end{cases} \\ u\_0 = \frac{\mathbf{x} y (1 - \mathbf{x}) (1 - y)}{1 + \mathbf{x}^2 + y^2}.$$

Table 1 presents the maximum pointwise error and the order of convergence corresponding to Example 1. Figures 2 and 3 depict the numerical solution and pointwise maximum error of the problem studied in Example 1, respectively.

**Table 1.** Maximum error and order of convergence for the Example 1 with *M* = 27.


**Figure 2.** Numerical solution of Example 1 for fixed *M* = 25, *N* = 27,*ε* = 10−5.

**Figure 3.** Maximum error of Example 1.

**Example 2.** *Consider the 2D parabolic PDE (1) with discontinuous source and convection coefficients with the following data:*

$$\begin{aligned} p\_1(\mathbf{x}) &= \begin{cases} 1 + \mathbf{x}(1-\mathbf{x}) + y^2, \ x \in (0, d\_x), \ \forall y, \\ -(1 + \mathbf{x}(1-\mathbf{x}) + \exp(-y)), \ x \in (d\_x, 1), \end{cases} \\ p\_2(\mathbf{x}) &= \begin{cases} 1 + y(1-y) + \sqrt{\mathbf{x}}, \ y \in (0, d\_y), \ \forall \mathbf{x}, \\ -(1 + y(1-y) + \mathbf{x}^2), \ y \in (d\_y, 1), \end{cases} \\\\ c\_1(\mathbf{x}) &= -0.5 - \mathbf{x}(1-\mathbf{x}), c\_2(\mathbf{x}) = -0.5 - y^2(1-y), \\\\ g\_1(\mathbf{x}, t) &= \begin{cases} 4txy \exp(\mathbf{x}^2 + y^2), \ x \in (0, d\_x), \\ 4t(1-\mathbf{x})(1-y), \ x \in (d\_x, 1), \end{cases} \\\\ u\_0 &= \frac{xy(1-\mathbf{x})(1-y)}{1 + \mathbf{x}^2 + y^2}. \end{aligned}$$

The maximum pointwise error and the order of convergence corresponding to Example 2 are given in Table 2. Figures 4 and 5 display the numerical solution and pointwise maximum error of Example 2, respectively.

**Figure 4.** Numerical solution of Example 2 for fixed *M* = 25, *N* = 27, *ε* = 10−5.

**Figure 5.** Maximum error of Example 2.

**Table 2.** Maximum error and order of convergence for the Example 2 with *M* = 27.


#### **7. Concluding Remarks**

This article discusses singularly perturbed 2D parabolic delay differential equations with discontinuous convection coefficients and source terms. As pointed out in Ref. [3], the fractional step method results in low-cost computation for 2D problems. Therefore, we first apply the fractional implicit Euler method for the time derivative. Then the higher dimensional problem is reduced to lower dimensional problems. In fact, we get 2*N* system of uncoupled equations. Each equation is a singularly perturbed differential equation with a discontinuous convection coefficient and source term. As discussed in Ref. [13], we discretized the spatial domains <sup>Ω</sup>*μ*, *<sup>μ</sup>* <sup>=</sup> *<sup>x</sup>*, *<sup>y</sup>* in the same manner, such as <sup>Ω</sup>*<sup>N</sup> <sup>μ</sup>* , *μ* = *x*, *y*. On each mesh we apply the difference scheme D*<sup>N</sup> <sup>μ</sup> Ui*,*j*, *μ* = *x*, *y*. It is proved that the present method is of almost first-order convergence in space and time. Figures 2 and 4 represent the test problems solutions stated in Examples 1 and 2, respectively, we see that, the layers occurs at the points *dx* and *dy*. Tables 1 and 2 present the maximum pointwise errors of the test example problems. It is also worth noting that when the parameter *ε* drops, the maximum pointwise error grows and stabilizes. It is assumed that the number of mesh points in the time direction is *M* = 128. From Figures 3 and 5 we see that the maximum pointwise error decreases as *N* increases. The present method works for the problems with any delay arguments of size 0 << *<sup>d</sup><sup>μ</sup>* <sup>≤</sup> 1, *<sup>μ</sup>* <sup>=</sup> *<sup>x</sup>*, *<sup>y</sup>*. In Example 1 we assumed that *dx* = 0.5 = *dy*,, whereas in Example 2 we assumed that *dx* = 0.5, *dy* = 0.25.

**Author Contributions:** Methodology, V.S.; software, V.S.; formal analysis, S.N.; investigation, S.N.; writing—original draft preparation, V.S.; writing—review and editing, S.N. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors are thankful for the DST-SERB for providing the fund under the scheme TARE, File No. TAR/2021/000053. The authors wish to acknowledge the referees for their valuable comments and suggestions, which helped to improve the presentation.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Ben Mansour Dia 1,\*, Mouhamadou Samsidy Goudiaby <sup>2</sup> and Oliver Dorn <sup>3</sup>**


**Abstract:** This paper treats a water flow regularization problem by means of local boundary conditions for the two-dimensional viscous shallow water equations. Using an *a-priori* energy estimate of the perturbation state and the Faedo–Galerkin method, we build a stabilizing boundary feedback control law for the volumetric flow in a finite time that is prescribed by the solvability of the associated Cauchy problem. We iterate the same approach to build by cascade a stabilizing feedback control law for infinite time. Thanks to a positive arbitrary time-dependent stabilization function, the control law provides an exponential decay of the energy.

**Keywords:** shallow water flow; Faedo–Galerkin method; feedback control; PDE's stabilization

**MSC:** 76D55; 93D15; 65M60; 93B18

#### **1. Introduction**

Regularization of free-surface fluid flows is a problem of practical interest for environmental and budgetary purposes in the current situation of climate change that rarefies fresh water sources worldwide. Depending on the specific application, the regularization of fluid flows is performed through control methodologies of the Navier–Stokes equations or a system of partial differential equations derived from them, which describe a particular setting and/or physical properties. Several mechanisms of controlling fluid flows have been designed in the recent past, see [1–5].

Control and stabilization of fluid flows governed by the Navier–Stokes equations have been extensively studied in the literature using various approaches. In the threedimensional setting, a local stabilization around an unstable stationary state is performed in [6] by means of a feedback control law. In [7], the existence of time-points values of boundary feedback laws is achieved by an optimal control problem to alleviate the high regularity required for the velocity components. One of the widely adopted approaches formulates the associate optimal control problem in infinite dimensional spaces, which gives rise to a Riccati equation [8]. Stabilization of the Navier-Stokes equations from the boundary or from a portion of the boundary is mainly investigated by means of feedback control laws. In addition to the series of papers [9–13], the inclusive examination in [14] lists a number of approaches for control by means of feedback laws. It also discusses the associated challenges, such as start-control, impulse-control and distributed-control laws, which have been studied for the Oseen and Navier-Stokes equations. Recently, global solution as well as an optimality system and a second-order sufficient optimality condition were obtained for the stationary two-dimensional Stokes equations [15,16], while an optimal controllability of a stationary two-dimensional non-Newtonian fluid in a pipeline network is studied in [17].

**Citation:** Dia, B.M.; Goudiaby, M.S.; Dorn, O. Boundary Feedback Stabilization of Two-Dimensional Shallow Water Equations with Viscosity Term. *Mathematics* **2022**, *10*, 4036. https://doi.org/10.3390/ math10214036

Academic Editor: Patricia J. Y. Wong

Received: 26 August 2022 Accepted: 15 October 2022 Published: 31 October 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

123

When the horizontal length scale of the physical domain is much greater than the vertical one, the flow movement can be captured by the water height and the horizontal velocity field. In that particular setting of shallow water flows, great advances have been made in the mechanism of controlling free-surface flow parameters by local boundary conditions, despite the challenges associated with the nonlinearity of the governing equations, see [18,19] for detailed and comprehensive reviews. Global stability of the two-dimensional water flow has been achieved in *L*2-norm [20] using the symmetrization of the flux matrices, in *H*2-norm [21], by acting on the tangential velocity. The approach of tracking the flow energy through the Riemann invariant variables is adopted in [22,23] in the one-dimensional setting and is explored in [24] for the two-dimensional channel flow. Besides the provided flexibility in practical experiments, the adding of the viscosity influence provides regularizing effects in estimating the flow energy, see [25].

In this paper, we address the stabilization of two-dimensional viscous shallow water around a steady-state, that is, the problem of driving the flow-state variables of viscous incompressible fluid inside a bounded container to a desired steady state. The control law acts on the volumetric flow vector along a portion of the boundary. Due to the challenges inherent in dealing with the nonlinear advection, we alleviate the nonlinearity issues by processing to the linearization around the steady state for small perturbations of the flow state. The resulting system of linear partial differential equations is referred to as the linearized shallow-water model, for which the existence and the uniqueness of solution is addressed by combining some notions of compactness and an a-priori energy estimate using the Faedo-Galerkin method. Subsequently, the stabilization of the nonlinear model around the steady state is rearranged as the stabilization of the linearized model around zero. In a short time, prescribed by the existence of a solution to the Cauchy problem associated with the weak formulation in an Hilbertian basis, the control building process explores only the estimation of the non-viscous energy of the linearized model and relies on a continuous time-dependent stabilization rate. The global-time stabilization result is established by cascading over a sequence of intervals.

The content of this paper is organized as follows: Section 2 introduces the equations governing the flow of a viscous shallow water in a three-dimensional domain with a given bathymetry. In Section 3, we detail the problem setting: we present the steady-state model, discuss the linearization, set the notations and the assumptions of the function spaces, and state the stabilization problem. Section 4 is devoted to the design of the small-time feedback control law. The main result of the stabilization of the linearized shallow-water model through the exponential decay of the energy is presented in Section 5. We conclude by giving some perspective directions of improvement of the presented method in Section 6.

#### **2. 2-D Viscous Shallow-Water Equations**

Consider a three-dimensional domain with a non-flat bottom in which a viscous water flows with a free-surface denoted by Ω, a bounded subset of R2, with boundary BΩ " Γ<sup>1</sup> Y Γ2. The SWE (shallow-water equations) are a set of partial differential equations derived by depth integrating the Navier–Stokes equations, see [26–29], with the assumption that the horizontal length scale of the domain is much greater than the vertical one. In the absence of Coriolis, frictional, and wind effects, the 2D viscous SWE with a viscosity coefficient *μ* in r*m*2¨*s*´1s are given by

$$
\hat{c}\_1 H + \hat{c}\_3 \left( H V\_1 \right) + \hat{c}\_{\mathcal{Y}} (H V\_2) = 0 \tag{1}
$$

$$
\hat{c}\_t \left( HV\_1 \right) + \hat{c}\_x \left( HV\_1^2 \right) + \mathcal{g}H \hat{c}\_x (H + \eta) + \hat{c}\_y \left( HV\_1 V\_2 \right) = \mu \, \Delta \left( HV\_1 \right) \quad \text{in } \mathbb{Q}\_\prime
$$

$$
\hat{c}\_t(HV\_2) + \hat{c}\_x\{HV\_1V\_2\} + \hat{c}\_y\{HV\_2^2\} + \text{g} \\
H\hat{c}\_y(H+\eta) = \mu\Lambda(HV\_2) \quad \text{in } \mathcal{Q}\_\prime \tag{1}
$$

$$\left(H, V\_1, V\_2\right)(0, \cdot, \cdot) = \left(H^0, V\_1^0, V\_2^0\right)(\cdot, \cdot) \tag{10.17}$$

boundary conditions to be specified,

\$

'''''''''''''&

'''''''''''''%

where Q " p0, *T*q ˆ Ω, *T* ą 0 denotes the duration of the study, *H* the height of the water column, ` *V*<sup>1</sup> , *V*<sup>2</sup> ˘ the velocity vector with reference to p*Ox*, *Oy*q, *η* the bathymetry describing the bottom elevation, and *g* is the constant of the acceleration due to the gravity force. The symbol B*<sup>t</sup>* designates the time derivative while B*<sup>x</sup>* and B*<sup>y</sup>* are the space derivatives in the *x*-direction and *y*-direction, respectively. The differential operator Δ represents the diffusion field Δ " B*xx* ` B*yy*. The triplet ` *H*, *V*<sup>1</sup> , *V*<sup>2</sup> ˘ varies with p*t*, *x*, *y*q and forms the solution of (1) while the bathymetry *η*p*x*, *y*q is independent of the time variable because there is no sediment transport. For the unidirectional propagation, an alternative approach to describing the waves at the free surface of shallow water under the influence of gravity is to consider the Korteweg–de Vries equation, see [30], where notions in differential geometry help to establish the existence of global solutions, see [31].

The diffusion effects have been modeled in several ways in the literature. It is shown in [32] that the formulation *μH*Δ*Vi* on the right hand side of (1) is not consistent with the primitive form of the equations for the energy norm and an energetically consistent formulation is given therein. This is deeply analyzed through the existence of weak solutions to the SWE in [33], where, by looking for p*V*<sup>1</sup> , *V*<sup>2</sup> q bounded in *L*2p0, *T*, *H*1pΩqq, *H* P *L*8p0, *T*, *L*1pΩqq and *H* log *H* P *L*8p0, *T*, *L*1pΩqq to induce the dissipation, the term *HVi* stands as an obstacle for the existence of solutions. That is why there is a constraint of small data to guarantee the existence of time-local weak solutions.

For the diffusion formulation *μ*Δp*HVi* q, as used here on the right hand side of (1), the existence of weak solutions and its stability are described in [26]. In that case, the diffusion provides regularizing effects due to an entropic inequality on the height variable *H*. It is important to notice that the stability result is restricted to the models where capillarity and friction are taken into account. For the 1-D model, a clearer result for the existence of weak global solutions can be elaborated with much less restrictive data [33].

Using the volumetric flow variable vector **Q** " p*Q*<sup>1</sup> , *Q*<sup>2</sup> q"p*HV*<sup>1</sup> , *HV*<sup>2</sup> q, the system (1) is rewritten for further analysis in the following conservative form:

$$\begin{aligned} \hat{c}\_{l}H + \text{div}\mathbf{Q} &= 0, & \text{in } &\mathcal{Q}\_{\prime} \\ \hat{c}\_{l}\mathbf{Q} + \text{div}\,\mathcal{F}(H,\mathbf{Q}) + \text{g}H\nabla(H+\eta) - \mu\Delta\mathbf{Q} &= 0, & \text{in } &\mathcal{Q}\_{\prime} \end{aligned} \tag{2}$$

where **Q** " ` *Q*<sup>1</sup> , *Q*<sup>2</sup> ˘<sup>J</sup> (the superscript <sup>J</sup> is the transpose operator), the matrix <sup>F</sup>p*H*, **<sup>Q</sup>**q " **Q** ¨ **Q**J{*H*, the differential operator ∇ is the gradient field, and div(¨) stands for the divergence operator, div(*f*) = <sup>∇</sup> ¨ *f* for a sufficiently regular vector function. Although the non-conservative formulation, see [20], is known to provide a better mass conservation of the volumetric quantity of water, it does not hold across a shock or a hydraulic jump since velocities do not generate fundamental conservation equations. On the other hand, the conservative formulation (2) supports front discontinuities such as shock waves at a fluid's interface and irregular source terms, and appeals to Riemann solver for numerical resolution, see [22,23,34,35]. Therefore, the conservative form (2) is well-suited for our stabilization problem, which we state in the next section.

#### **3. Statement of the Problem**

In this section, we lay out the stabilization problem from the linearization around the steady state to the setting of finding the boundary feedback control law.

#### *3.1. Steady-State, Linearization*

The objective of the stabilization is to bring the variables of the flow to a given steady state that is the reference state. In practice, this consists of finding a controller (inflows, outflows) allowing to adjust the flow parameters to keep the flow state variables near this reference state. We denote the steady state by *U***¯** " p¯ *h*, *q*¯1 , *q*¯2 q<sup>J</sup> and it is stationary and given by:

$$\begin{cases} \text{div}\overline{q} = 0, & \text{in } \Omega, \\ \text{gl}\nabla\left(\overline{h} + \eta\right) - \frac{1}{\hbar} \mathcal{F}(\overline{h}, \overline{q}) \cdot \nabla \overline{h} + \frac{1}{\hbar} (\nabla \overline{q}) \cdot \overline{q} - \mu \Delta \overline{q} = 0 & \text{in } \Omega. \end{cases} \tag{3}$$

The hydrodynamical variable vector p*H*, *Q*<sup>1</sup> , *Q*<sup>2</sup> q is formed by the equilibrium state p¯ *h*, *q*¯1 , *q*¯2 q and a perturbation state denoted by p*h*, *q*<sup>1</sup> , *q*<sup>2</sup> q. Hence, the linearization consists of using

$$\begin{aligned} H(t, \mathbf{x}, \mathbf{y}) &= \bar{h}(\mathbf{x}, \mathbf{y}) + h(t, \mathbf{x}, \mathbf{y}), \\ Q\_1(t, \mathbf{x}, \mathbf{y}) &= \bar{q}\_1(\mathbf{x}, \mathbf{y}) + q\_1(t, \mathbf{x}, \mathbf{y}), \\ Q\_2(t, \mathbf{x}, \mathbf{y}) &= \bar{q}\_2(\mathbf{x}, \mathbf{y}) + q\_2(t, \mathbf{x}, \mathbf{y}). \end{aligned} \tag{4}$$

We proceed to the linearization of the system (2) by replacing the state variables *H*, *Q*<sup>1</sup> and *<sup>Q</sup>*<sup>2</sup> by their above expressions. In addition, we consider the following assumption <sup>|</sup>*h*| ! ¯ *h*, ˇ ˇ*q*1 ˇ <sup>ˇ</sup> ! <sup>ˇ</sup> ˇ*q*¯1 ˇ <sup>ˇ</sup> and <sup>|</sup>*q*<sup>2</sup> |!|*q*¯2 <sup>|</sup> to justify keeping only the first-order terms in the perturbation state because we neglect higher order terms. We denote by

$$
\vec{\sigma} = \begin{pmatrix} \vec{\upsilon}\_1 \\ \\ \vec{\upsilon}\_2 \end{pmatrix}, \quad \mathfrak{a}\_0 = \begin{pmatrix} a\_0^1 \\\\ a\_0^2 \end{pmatrix}, \quad A = \begin{pmatrix} \beta\_0^1 & \gamma\_0^1 \\\\ \beta\_0^2 & \gamma\_0^2 \end{pmatrix}, \quad \text{and} \quad \mathbf{B} = \begin{pmatrix} a\_1^1 & a\_2^1 \\\\ a\_1^2 & a\_2^2 \end{pmatrix},
$$

where the coefficients are given by:

$$\begin{aligned} a\_0^1 &= \mathcal{g}\hat{c}\_x \bar{h} + \mathcal{g}\hat{c}\_x \eta\_\* & a\_0^2 &= \mathcal{g}\hat{c}\_y \bar{h} + \mathcal{g}\hat{c}\_y \eta\_\* \\ \beta\_0^1 &= \frac{1}{\bar{l}\hbar} \left( 2\hat{c}\_x \bar{q}\_1 - 2\hat{v}\_1 \hat{c}\_x \bar{h} + \hat{c}\_y \bar{q}\_2 - \bar{v}\_2 \hat{c}\_y \bar{h} \right), & \beta\_0^2 &= \frac{1}{\bar{l}\hbar} \left( \hat{c}\_x \bar{q}\_2 - \bar{v}\_2 \hat{c}\_x \bar{h} \right), \\ \gamma\_0^1 &= \frac{1}{\bar{l}\hbar} \left( \hat{c}\_y \bar{q}\_1 - \bar{v}\_1 \hat{c}\_y \bar{h} \right), & \gamma\_0^2 &= \frac{1}{\bar{l}\hbar} \left( 2\hat{c}\_y \bar{q}\_2 - 2\bar{v}\_2 \hat{c}\_y \bar{h} + \hat{c}\_x \bar{q}\_1 - \bar{v}\_1 \hat{c}\_x \bar{h} \right), \\ a\_1^1 &= c^2 - \bar{v}\_1^2, & a\_1^1 &= -\bar{v}\_1 \bar{v}\_2, \\ a\_1^2 &= -\bar{v}\_1 \bar{v}\_2, & a\_2^2 &= c^2 - \bar{v}\_2^2. \end{aligned}$$

The constant *c* " b *g*¯ *h* is the wave speed at the equilibrium. The linearization gives rise to the model governing the evolution of the residual state. This is the following linearized 2D shallow-water system

$$\begin{aligned} \hat{c}\_l h + (\text{div}\boldsymbol{q}) \vec{\boldsymbol{v}} + \nabla \boldsymbol{q} \cdot \vec{\boldsymbol{v}} - \mu \Delta \boldsymbol{q} + \mathbf{B} \cdot \nabla h + \boldsymbol{A} \cdot \boldsymbol{q} + h \boldsymbol{\alpha}\_0 &= \boldsymbol{0} \text{ in } \boldsymbol{Q}, \\ \boldsymbol{q}(t=0) &= \boldsymbol{q}^0 \text{ in } \Omega, \\ h(t=0) &= \boldsymbol{h}^0 \text{ in } \Omega. \end{aligned} \tag{5}$$

With given initial state ` *<sup>h</sup>*0, *q*<sup>0</sup> ˘ , the control problem consists of providing suitable boundary conditions *<sup>V</sup>* " pV1, <sup>V</sup>2<sup>q</sup> on a portion of the boundary, <sup>Γ</sup>1, so that the state <sup>p</sup>*h*, *<sup>q</sup>*<sup>q</sup> converges in time towards p0, 0, 0q with the assumption that the physical domain is uniformly convex with a Lipschitz boundary. Note that the advection, in the linearized system (5), runs with constant flux matrices depending only on the steady state <sup>B</sup>*x*Fp*U*¯ <sup>q</sup> and <sup>B</sup>*y*Fp*U*¯ <sup>q</sup>. The controlled boundary portion Γ<sup>1</sup> is defined in the next section.

#### *3.2. Notations and Function Spaces*

Physically, the domain Ω is a regular (it provides the required smoothness for a Lipschitz boundary) open-bounded subset of <sup>R</sup><sup>2</sup> with boundary <sup>B</sup>Ω. We remind the reader that there is no sediment movement; the bathymetry is, therefore, a time-invariant function, see Figure 1.

**Figure 1.** Domain representation.

For the sake of clarity, we specify the following two statements:

(S1) The boundary portion, where the control action is applied, is given by

$$
\Gamma\_1 = \left\{ (\mathfrak{x}, \mathfrak{y}) \in \hat{\mathcal{O}}\Omega \, : \, 2\bar{\upsilon}\_1 n\_{\mathfrak{x}} + \bar{\upsilon}\_2 n\_{\mathfrak{y}} < 0 \, \text{ and } \; \bar{\upsilon}\_1 n\_{\mathfrak{x}} + 2\bar{\upsilon}\_2 n\_{\mathfrak{y}} < 0 \right\}.
$$

The boundary portion Γ<sup>1</sup> exists (is nonempty) and is included in the boundary portion given by <sup>p</sup>*v*¯1 , *<sup>v</sup>*¯2 <sup>q</sup><sup>J</sup> ¨**<sup>n</sup>** <sup>ă</sup> 0, where the vector **<sup>n</sup>** " p*nx*, *ny*q<sup>J</sup> is the external normal unit vector at the boundary. The uncontrolled boundary portion Γ<sup>2</sup> " BΩzΓ1.

(S2) The flux variation is bounded at the boundary BΩ. This follows naturally because of the sub-critical flow regime considered here, and is stated for the sake of clarity. It means that the limit when <sup>p</sup>*x*, *<sup>y</sup>*<sup>q</sup> tends to the boundary <sup>B</sup><sup>Ω</sup> of the term }∇*q*}*L*2pΩ<sup>q</sup> is bounded, that is,

$$\max \left\{ \lim\_{(\mathbf{x}, \mathbf{y}) \to (\mathbf{x}\_b, y\_b)} \| \nabla q(\mathbf{x}, \mathbf{y}) \|\_{L^2(\Omega)} \text{ for } (\mathbf{x}\_b, y\_b) \in \partial \Omega \right\} \text{ is finite.} $$

The regularity of the steady-state <sup>p</sup>¯ *h*, *v*¯1 , *v*¯2 q depends on the nature of the bathymetry *η*; for instance, ¯ *h*, *v*¯1 and *v*¯2 are constant if the bathymetry is constant (flat bottom tomography). We, therefore, consider a sufficiently regular bathymetry *η*, such that ¯ *h*, *v*¯1 and *v*¯2 are differentiable in Ω: ¯ *h* P *H*<sup>1</sup> <sup>0</sup> <sup>p</sup>Ωq, *<sup>v</sup>*¯1 <sup>P</sup> *<sup>H</sup>*1pΩ<sup>q</sup> and *<sup>v</sup>*¯2 <sup>P</sup> *<sup>H</sup>*1pΩq.

For <sup>Q</sup> " p0, *<sup>T</sup>*q ˆ <sup>Ω</sup>, we consider the space *<sup>L</sup>*<sup>2</sup> ` 0, *T*; *H*1pΩq ˘ . In the same setting, we introduce also the space *L*<sup>2</sup> ´ 0, *T*; *H*<sup>1</sup> Γ1 pΩq ¯ , where the Hilbert space *H*<sup>1</sup> Γ1 pΩq is given by

$$H^1\_{\Gamma\_1}(\Omega) = \left\{ \mathfrak{u} \in L^2(\Omega) \,:\, \nabla \mathfrak{u} \in L^2(\Omega) \text{ and } \mathfrak{u}\_{\vert\_{\Gamma\_2}} = 0 \right\}.$$

The space *H*1pΩq and its subspace *H*<sup>1</sup> Γ1 <sup>p</sup>Ω<sup>q</sup> are equipped with the norm }¨}*H*1pΩ<sup>q</sup> defined for a function *u* by }*u*}<sup>2</sup> *<sup>H</sup>*1pΩ<sup>q</sup> " }*u*}<sup>2</sup> *<sup>L</sup>*2pΩ<sup>q</sup> ` }∇*u*}<sup>2</sup> *L*2pΩq . In the rest of this paper, we denote by *W* the space given by *W* " *L*<sup>2</sup> ` 0, *T*; *H*1pΩq ˘ ˆ *L*<sup>2</sup> ´ 0, *T*; *H*<sup>1</sup> Γ1 pΩq ¯ ˆ *L*<sup>2</sup> ´ 0, *T*; *H*<sup>1</sup> Γ1 pΩq ¯ .

#### *3.3. The Stabilization Problem*

With the conditions (4), the task of stabilizing the nonlinear state p*H*, *Q*1, *Q*<sup>2</sup> q around the steady state ` ¯ *h*, *q*¯1 , *q*¯2 ˘ is reformulated as a stabilization problem of the perturbation state <sup>p</sup>*h*, *<sup>q</sup>*<sup>q</sup> around <sup>p</sup>0, 0, 0q. The objective is to find suitable local boundary conditions on *q* that take the flow-state variables as quickly as possible to the steady-state equilibrium. We formulate this goal as a stabilization problem in the following way

$$\begin{aligned} \hat{c}\_{l}\boldsymbol{\eta} + (\text{div}\boldsymbol{q})\vec{\boldsymbol{v}} + \nabla\boldsymbol{q} \cdot \vec{\boldsymbol{v}} - \mu\Delta\boldsymbol{q} + \mathbf{B} \cdot \nabla\boldsymbol{h} + \mathbf{A} \cdot \boldsymbol{q} + h\mathbf{a}\_{\boldsymbol{\phi}} &= \quad \text{in} \quad \Omega, \\ \boldsymbol{q}(t=0) &= \quad \boldsymbol{q}^{0} \text{ in } \Omega, \\ \boldsymbol{h}(t=0) &= \quad \boldsymbol{h}^{0} \text{ in } \Omega, \\ \boldsymbol{q} &= \quad \boldsymbol{\mathcal{V}} \text{ on } (0,\infty) \times \Gamma\_{\text{i}}, \\ \boldsymbol{q} &= \quad \boldsymbol{0} \text{ on } (0,\infty) \times \Gamma\_{\text{2}}. \end{aligned} \tag{6}$$

Concretely, we look for *<sup>V</sup>* such that <sup>p</sup>*h*, *q*<sup>q</sup> from (6) converges to <sup>p</sup>0, 0, 0q. In that sequel, we state the weak formulation associated with (6) that consists in writing (6) as a system of ordinary differential equations depending only on the variable *t* by using the Green's formula of integration by parts: for all p*ϕ*, *φ*, *ψ*q P *W*, find p*h*, *q*<sup>1</sup> , *q*<sup>2</sup> q in *W* satisfying

$$\int\_{\Omega} \oint\_{\Omega} \boldsymbol{q} \hat{v}\_l \boldsymbol{l} \boldsymbol{l} d\Omega - \int\_{\Omega} \boldsymbol{q} \nabla \boldsymbol{q} d\Omega = -\int\_{\Gamma\_1} \boldsymbol{q} \boldsymbol{q} \cdot \mathbf{n} d\sigma\_{\prime} \tag{7}$$

ż Ω B*tq*1*φ d*Ω ` ż Ω <sup>p</sup>div*qv*¯1 <sup>q</sup>*<sup>φ</sup> <sup>d</sup>*<sup>Ω</sup> ´ ż Ω *<sup>q</sup>*1divp*φv*¯<sup>q</sup> *<sup>d</sup>*<sup>Ω</sup> ` *<sup>μ</sup>* ż Ω ∇*q*1∇*φ d*Ω ´ ż Ω *<sup>h</sup>*divp*φB*<sup>1</sup>¨<sup>q</sup> *<sup>d</sup>*<sup>Ω</sup> ` ż Ω *β*1 <sup>0</sup> *q*1*φ d*Ω ` ż Ω *γ*1 <sup>0</sup> *q*2*φ d*Ω ` ż Ω *hα*<sup>1</sup> <sup>0</sup>*φ d*Ω " ´ <sup>ż</sup> Γ1 <sup>p</sup>*q*1*φ*q*v*¯ ¨ **<sup>n</sup>** *<sup>d</sup><sup>σ</sup>* ` *<sup>μ</sup>* ż Γ1 p∇*q*<sup>1</sup> ¨ **n**q*φ dσ* ´ ż Γ1 *<sup>h</sup>φ*p*B*<sup>1</sup>¨ ¨ **<sup>n</sup>**<sup>q</sup> *<sup>d</sup>σ*, (8) ż Ω B*tq*2*ψ d*Ω ` ż Ω <sup>p</sup>div*qv*¯2 <sup>q</sup>*<sup>ψ</sup> <sup>d</sup>*<sup>Ω</sup> ´ ż Ω *<sup>q</sup>*2divp*ψv*¯<sup>q</sup> *<sup>d</sup>*<sup>Ω</sup> ` *<sup>μ</sup>* ż Ω ∇*q*2∇*ψ d*Ω ´ ż Ω *<sup>h</sup>*divp*ψB*<sup>2</sup>¨<sup>q</sup> *<sup>d</sup>*<sup>Ω</sup> ` ż Ω *β*2 <sup>0</sup> *q*1*ψ d*Ω ` ż Ω *γ*2 <sup>0</sup> *q*2*ψ d*Ω ` ż Ω *hα*<sup>2</sup> <sup>0</sup>*ψ d*Ω " ´ <sup>ż</sup> Γ1 <sup>p</sup>*q*2*ψ*q*v*¯ ¨ **<sup>n</sup>** *<sup>d</sup><sup>σ</sup>* ` *<sup>μ</sup>* ż Γ1 p∇*q*<sup>2</sup> .*n*q*ψ dσ* ´ ż Γ1 *<sup>h</sup>ψ*p*B*<sup>2</sup>¨ ¨ **<sup>n</sup>**<sup>q</sup> *<sup>d</sup>σ*. (9)

From now, the weak formulation of the stabilization problem (6) refers to (7)–(9), which are obtained by integration by parts of (6) multiplied with the test function p*ϕ*, *φ*, *ψ*q.

#### **4. Preliminary Result: Small-Time Control Design**

The stabilization problem (6) is constrained by the existence of a solution. In this section, we examine the existence of a solution to the dynamical system resulting from the representation of the weak formulation in an Hilbertian basis and we address the short-time stabilization problem.

**Lemma 1** (Existence of small-time weak solutions)**.** *There exists a time T*<sup>1</sup> *such that the ordinary differential Equations* (7)*–*(9) *with the Cauchy condition admit a solution on the time interval* r0, *T*1s*.*

The above lemma addresses the existence of local weak solutions. The proof is elaborated in two steps: the first one consists of writing the weak form as a system of differential equations using a Hilbertian basis of finite dimensions of *W*. The second step deals with

the existence of a solution of the resulting system of differential equations thanks to the Cauchy–Peano theorem, see [36,37].

**Proof.** The existence of a local weak solution is elaborated using the Cauchy–Peano theorem. Let <sup>t</sup>*ai*u*i*ě<sup>1</sup> (respectively, by <sup>t</sup>*ei*u*i*ě1) denote an Hilbertian basis of the space *<sup>H</sup>*1pΩ<sup>q</sup> (respectively, *H*<sup>1</sup> Γ1 pΩq). We consider a positive integer *n* such that the finite dimensional space *span*t*ai* : 1 ď *i* ď *n*u contains the term *h*<sup>0</sup> *<sup>n</sup>* of a sequence of functions p*h*<sup>0</sup> *<sup>n</sup>*q*n*ě<sup>1</sup> that converges toward the initial condition *h*<sup>0</sup> (respectively, *span*t*ei* : 1 ď *i* ď *n*u) containing the terms *q*<sup>0</sup> <sup>1</sup>*<sup>n</sup>* and *<sup>q</sup>*<sup>0</sup> <sup>2</sup>*<sup>n</sup>* sequences of functions <sup>p</sup>*q*<sup>0</sup> <sup>1</sup>*<sup>n</sup>* <sup>q</sup>*n*ě<sup>1</sup> and <sup>p</sup>*q*<sup>0</sup> <sup>2</sup>*<sup>n</sup>* q*n*ě<sup>1</sup> converging, respectively, to *q*0 <sup>1</sup> and *<sup>q</sup>*<sup>0</sup> <sup>2</sup> ). For a sufficiently large *n*, the projection of *h*, *q*<sup>1</sup> and *q*<sup>2</sup> allows us to write

$$h \approx \sum\_{i=1}^{n} a\_i(t) a\_i(\mathbf{x}, \mathbf{y}), \quad q\_1 \approx \sum\_{i=1}^{n} \beta\_i(t) e\_i(\mathbf{x}, \mathbf{y}), \quad \text{and} \quad q\_2 \approx \sum\_{i=1}^{n} \gamma\_i(t) e\_i(\mathbf{x}, \mathbf{y}),$$

where p*α<sup>i</sup>* p*t*qq1ď*i*ď*n*, p*β<sup>i</sup>* p*t*qq1ď*i*ď*<sup>n</sup>* and p*γ<sup>i</sup>* p*t*qq1ď*i*ď*<sup>n</sup>* are, respectively, the unknown coordinates of *h*, *q*<sup>1</sup> and *q*<sup>2</sup> at time *t*. Replacing test functions p*ϕ*, *φ*, *ψ*q by the p*aj* ,*ej* ,*ej* q for the *j th* dimension, the weak formulation becomes:

ÿ*n i*"1 B*tα<sup>i</sup>* p*t*q ż Ω *ai ej <sup>d</sup>*<sup>Ω</sup> ´ <sup>ÿ</sup>*<sup>n</sup> i*"1 *βi* p*t*q ż Ω *ai* B*xej <sup>d</sup>*<sup>Ω</sup> ´ <sup>ÿ</sup>*<sup>n</sup> i*"1 *γi* p*t*q ż Ω *gai* B*yej <sup>d</sup>*<sup>Ω</sup> " ´ <sup>ż</sup> Γ1 *ej* V ¨ **n** *dσ*, ÿ*n i*"1 B*tβ<sup>i</sup>* p*t*q ż Ω *ei ej <sup>d</sup>*<sup>Ω</sup> ` <sup>ÿ</sup>*<sup>n</sup> i*"1 *βi* p*t*q ż Ω *v*¯1 *ej* B*xei <sup>d</sup>*<sup>Ω</sup> ` <sup>ÿ</sup>*<sup>n</sup> i*"1 *γi* p*t*q ż Ω *v*¯1 *ej* B*yei d*Ω ´ <sup>ÿ</sup>*<sup>n</sup> i*"1 *βi* p*t*q ż Ω *ei* divp*ej v*¯q*d*<sup>Ω</sup> ` <sup>ÿ</sup>*<sup>n</sup> i*"1 *βi* p*t*q*μ* ż Ω ∇*ei* ∇*ej d*Ω ´ <sup>ÿ</sup>*<sup>n</sup> i*"1 *αi* p*t*q ż Ω *ai* divp*ej B*<sup>1</sup>¨q*d*<sup>Ω</sup> ` <sup>ÿ</sup>*<sup>n</sup> i*"1 *βi* p*t*q ż Ω *<sup>A</sup>*<sup>11</sup> *ei ej d*Ω ` <sup>ÿ</sup>*<sup>n</sup> i*"1 *γi* p*t*q ż Ω *<sup>A</sup>*<sup>12</sup> *ei ej <sup>d</sup>*<sup>Ω</sup> ` <sup>ÿ</sup>*<sup>n</sup> i*"1 *αi* p*t*q ż Ω *α*1 0 *ai ej d*Ω " ´ <sup>ż</sup> Γ1 pV<sup>1</sup> *ej* <sup>q</sup>*v*¯ ¨ **<sup>n</sup>***d<sup>σ</sup>* ` *<sup>μ</sup>* ÿ*n i*"1 *βi* p*t*q ż Γ1 p∇*ei* ¨ **n**q*ej dσ* ´ ż Γ1 *hej* ` *<sup>B</sup>*<sup>1</sup>¨ ¨ **<sup>n</sup>** ˘ *dσ*,

$$\begin{split} \sum\_{i=1}^{n} \partial\_{i} \gamma\_{i}(t) \int\_{\Omega} e\_{i} e\_{j} d\Omega &+ \sum\_{i=1}^{n} \beta\_{i}(t) \int\_{\Omega} \overline{o}\_{2} e\_{j} \overline{o}\_{2} e\_{i} d\Omega + \sum\_{i=1}^{n} \gamma\_{i}(t) \int\_{\Omega} \overline{o}\_{2} e\_{j} \overline{o}\_{j} e\_{i} d\Omega \\ &- \sum\_{i=1}^{n} \gamma\_{i}(t) \int\_{\Omega} e\_{i} \mathrm{div}(e\_{i} \overline{\sigma}) d\Omega + \sum\_{i=1}^{n} \gamma\_{i}(t) \mu \int\_{\Omega} \nabla e\_{i} \nabla e\_{j} d\Omega \\ &- \sum\_{i=1}^{n} a\_{i}(t) \int\_{\Omega} a\_{i} \mathrm{div}(e\_{j} \mathbf{B}\_{2}) d\Omega + \sum\_{i=1}^{n} \beta\_{i}(t) \int\_{\Omega} A\_{2i} e\_{i} e\_{j} d\Omega \\ &+ \sum\_{i=1}^{n} \gamma\_{i}(t) \int\_{\Omega} A\_{2i} e\_{i} e\_{j} d\Omega + \sum\_{i=1}^{n} a\_{i}(t) \int\_{\Omega} a\_{0}^{2} a\_{i} e\_{j} d\Omega \\ &= \ - \int\_{\Gamma\_{1}} (\mathcal{V}\_{2} e\_{j}) \bar{\mathbf{v}} \cdot \mathbf{n} d\sigma + \mu \sum\_{i=1}^{n} \gamma\_{i}(t) \int\_{\Gamma\_{1}} (\nabla e\_{i} \cdot \mathbf{n}) e\_{j} d\sigma - \int\_{\Gamma\_{1}} h e\_{j} (\mathbf{B}\_{2} \cdot \mathbf{n}) d\sigma. \end{split}$$

Let us introduce the matrices *<sup>M</sup><sup>k</sup>* for *<sup>k</sup>* " 1, ¨¨¨ , 11 as well as the vectors *<sup>m</sup><sup>l</sup>* for *l* " 1, ¨¨¨ , 3 as follows

*<sup>M</sup>*1*ji* " ż Ω *ai ej <sup>d</sup>*Ω, *<sup>M</sup>*2*ji* " ż Ω *ai* B*xej <sup>d</sup>*Ω, *<sup>M</sup>*3*ji* " ż Ω *ai* B*yej d*Ω *<sup>M</sup>*4*ji* " ż Ω *ei ej <sup>d</sup>*Ω, *<sup>M</sup>*5*ji* " ż Ω *v*¯1 *ej* B*xei d*Ω ´ ż Ω *ei* divp*ej v*¯q*d*<sup>Ω</sup> ` *<sup>μ</sup>* ż Ω ∇*ei* ∇*ej d*Ω ` ż Ω *<sup>A</sup>*<sup>11</sup> *ei ej d*Ω, *<sup>M</sup>*6*ji* " ż Ω *v*¯1 *ej* B*yei d*Ω ` ż Ω *<sup>A</sup>*<sup>12</sup> *ei ej <sup>d</sup>*Ω, *<sup>M</sup>*7*ji* " ´ <sup>ż</sup> Ω *ei* divp*ej B*<sup>1</sup>¨q*d*<sup>Ω</sup> ` ż Ω *α*1 0 *ei ej d*Ω, *<sup>M</sup>*8*ji* " ż Ω *v*¯2 *ej* B*xei d*Ω ` ż Ω *<sup>A</sup>*<sup>21</sup> *ei ej d*Ω, *<sup>M</sup>*9*ji* " ż Ω *v*¯2 *ej* B*yei d*Ω ´ ż Ω *ei* divp*ej v*¯q*d*<sup>Ω</sup> ` *<sup>μ</sup>* ż Ω ∇*ei* ∇*ej d*Ω ` ż Ω *<sup>A</sup>*<sup>22</sup> *ei ej d*Ω *<sup>M</sup>*10*ji* " ´ <sup>ż</sup> Ω *ei* divp*ej B*<sup>2</sup>¨q*d*<sup>Ω</sup> ` ż Ω *α*2 0 *ei ej <sup>d</sup>*Ω, *<sup>M</sup>*11*ji* " *<sup>μ</sup>* ż Γ1 p∇*ei* ¨ **n**q*ej dσ*,

and

$$\begin{aligned} \begin{aligned} \mathfrak{m}\_{1\_{\tilde{j}}} &=& -\int\_{\Gamma\_{1}} e\_{\cdot} n\_{x} d\sigma, & \mathfrak{m}\_{2\_{\tilde{j}}} &=& -\int\_{\Gamma\_{1}} e\_{\cdot} n\_{y} d\sigma, & \mathfrak{m}\_{3\_{\tilde{j}}} &=& -\int\_{\Gamma\_{1}} e\_{\cdot} \vec{\sigma} \cdot \mathbf{n} d\sigma, \\ \mathfrak{m}\_{4\_{\tilde{j}}} &=& -\int\_{\Gamma\_{1}} h e\_{\cdot} (\mathbf{B}\_{1\_{\tilde{\cdot}}} \cdot \mathbf{n}) d\sigma & \mathfrak{m}\_{5\_{\tilde{j}}} &=& -\int\_{\Gamma\_{1}} h e\_{\cdot} (\mathbf{B}\_{2\_{\tilde{\cdot}}} \cdot \mathbf{n}) d\sigma. \end{aligned} \end{aligned}$$

We now count the *n* components; this yields the following system of matrix equations

$$\begin{cases} \mathcal{M}\_1 \hat{c}\_l a(t) - \mathcal{M}\_2 b(t) - \mathcal{M}\_3 \mathbf{c}(t) = \mathcal{V}\_1 \mathfrak{m}\_1 + \mathcal{V}\_2 \mathfrak{m}\_2, \\\\ \mathcal{M}\_4 \hat{c}\_l b(t) + \mathcal{M}\_5 b(t) + \mathcal{M}\_6 \mathbf{c}(t) + \mathcal{M}\_7 \mathfrak{a}(t) = \mathcal{V}\_1 \mathfrak{m}\_3 + \mathcal{M}\_{11} b(t) + \mathfrak{m}\_4, \\\\ \mathcal{M}\_4 \hat{c}\_l \mathfrak{c}(t) + \mathcal{M}\_8 b(t) + \mathcal{M}\_9 \mathfrak{c}(t) + \mathcal{M}\_{10} \mathfrak{a}(t) = \mathcal{V}\_2 \mathfrak{m}\_3 + \mathcal{M}\_{11} \mathfrak{c}(t) + \mathfrak{m}\_5, \end{cases}$$

where the vectors *<sup>a</sup>*, *<sup>b</sup>* and *<sup>c</sup>* are given by the coordinates of the state variables *<sup>h</sup>*, *<sup>q</sup>*<sup>1</sup> and *<sup>q</sup>*<sup>2</sup> , respectively, i.e., *<sup>a</sup>* " p*α*<sup>1</sup> , ¨¨¨ , *<sup>α</sup><sup>n</sup>* <sup>q</sup>, *<sup>b</sup>* " p*β*<sup>1</sup> , ¨¨¨ , *<sup>β</sup><sup>n</sup>* <sup>q</sup>, and *<sup>c</sup>* " p*γ*<sup>1</sup> , ¨¨¨ , *<sup>γ</sup><sup>n</sup>* <sup>q</sup>. To introduce the matrices:

$$P\_1 = \begin{pmatrix} M\_1 & 0 & 0 \\ 0 & M\_4 & 0 \\ 0 & 0 & M\_4 \end{pmatrix}, \ P\_2 = \begin{pmatrix} 0 & M\_2 & M\_3 \\ M\_7 & M\_5 & M\_6 \\ M\_{10} & M\_8 & M\_9 \end{pmatrix}, \ \text{and} \ \ P\_3 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & M\_{11} & 0 \\ 0 & 0 & M\_{11} \end{pmatrix}.$$

and the vectors

$$\begin{array}{c} \begin{pmatrix} a \\ b \\ c \end{pmatrix}, \ p\_1 = \begin{pmatrix} m\_1 \\ m\_3 \\ 0 \end{pmatrix}, \ p\_2 = \begin{pmatrix} m\_2 \\ 0 \\ m\_3 \end{pmatrix} \text{ and } \ p\_3 = \begin{pmatrix} 0 \\ m\_4 \\ m\_5 \end{pmatrix}. \end{array}$$

we write the weak form of Equations (7)–(9) together as an ordinary differential equation, with the initial condition *<sup>y</sup>*<sup>0</sup> associated with <sup>p</sup>*h*0, *<sup>q</sup>*0q, in the form

$$\begin{cases} \ y'(t) = f(t, y), \\ y(t = 0) = y\_{0'} \end{cases} \tag{10}$$

where

$$f(t, y(t)) = -P\_1^{-1} P\_2 y(t) + P\_1^{-1} P\_3 y(t) + \mathcal{V}\_1(t) P\_1^{-1} p\_1 + \mathcal{V}\_2(t) P\_1^{-1} p\_2 + P\_1^{-1} p\_3(t).$$

It remains now to prove the continuity of the functional *f*. For that, we consider the matrix norm }¨}<sup>2</sup> given by:

$$\|M\|\_2 = \sup\_{\mathfrak{L}} \frac{\|M\mathfrak{L}\|\_2}{\|\mathfrak{L}\|\_2}, \text{ where } \|\mathfrak{L}\|\_2 = \sqrt{\sum\_{i}^{n} |\mathfrak{L}\_i|^2}$$

We have

$$\begin{split} \|f\|\_{2} &= \left\|-P\_{1}^{-1}\mathcal{P}\_{2}\mathcal{y}(t) + \mathcal{P}\_{1}^{-1}\mathcal{P}\_{3}\mathcal{y}(t) + \mathcal{V}\_{1}(t)\mathcal{P}\_{1}^{-1}\mathcal{p}\_{1} + \mathcal{V}\_{2}(t)\mathcal{P}\_{1}^{-1}\mathcal{p}\_{2} + \mathcal{P}\_{1}^{-1}\mathcal{p}\_{3}(t)\right\|\_{2} \\ &\leqslant \left\|-P\_{1}^{-1}\mathcal{P}\_{2}\mathcal{y}(t)\right\|\_{2} + \left\|\mathcal{P}\_{1}^{-1}\mathcal{P}\_{3}\mathcal{y}(t)\right\|\_{2} + \left\|\mathcal{V}\_{1}(t)\mathcal{P}\_{1}^{-1}\mathcal{p}\_{1}\right\|\_{2} + \left\|\mathcal{V}\_{2}(t)\mathcal{P}\_{1}^{-1}\mathcal{p}\_{2}\right\|\_{2} + \left\|\mathcal{P}\_{1}^{-1}\mathcal{p}\_{3}(t)\right\|\_{2}. \end{split}$$

Yet, we know that

$$\begin{aligned} \left\| -P\_1^{-1} P\_2 y \right\|\_2 &= \left\| -P\_1^{-1} P\_2 (y - y\_0 + y\_0) \right\|\_2 \\ &\lesssim \left\| -P\_1^{-1} P\_2 (y - y\_0) \right\|\_2 + \left\| -P\_1^{-1} P\_2 y\_0 \right\|\_2 \\ &\lesssim \left\| -P\_1^{-1} P\_2 \right\|\_2 (\left\| y - y\_0 \right\|\_2 + \left\| y\_0 \right\|\_2). \end{aligned}$$

Similarly, we bound the quantity › › ›´*P*´<sup>1</sup> <sup>1</sup> *P*3*y* › › › 2 . The terms › › ›V<sup>1</sup> <sup>p</sup>*t*q*P*´<sup>1</sup> <sup>1</sup> *p*<sup>1</sup> › › › 2 and › › ›V<sup>2</sup> <sup>p</sup>*t*q*P*´<sup>1</sup> <sup>1</sup> *p*<sup>2</sup> › › › 2 contain the control actions pV<sup>1</sup> , V<sup>2</sup> q; their majoration follows from the statement (S2). As the perturbation state for the height is supposed to be small compared to ¯ *h* and given the definition of the space *W*, the term › › › *P*´1 <sup>1</sup> *<sup>p</sup>*3p*t*<sup>q</sup> › › › <sup>2</sup> is bounded. Therefore, there exists a constant *K*<sup>1</sup> such that

$$\|f\|\_{2} \leqslant \mathcal{K}\_{\mathfrak{l}}.\tag{11}$$

.

It is clear that }*<sup>y</sup>* ´ *<sup>y</sup>*0}<sup>2</sup> <sup>ď</sup> *<sup>E</sup>*p0<sup>q</sup> because the control law acts to decrease the initial total energy *<sup>E</sup>*p0q. Denoting *<sup>T</sup>*<sup>1</sup> " *<sup>E</sup>*p0q{*K*<sup>1</sup> , it yields that *<sup>f</sup>* is bounded according to (11). Moreover, *f* is continuous because it is a composition of a linear function followed by a translation. Therefore, the Cauchy–Peano theorem ensures the existence of solutions to (10) in the time interval r0, *T*<sup>1</sup> s.

The proof below is enough for the infinite dimensional setting in Lemma 1 because for the drift function *<sup>f</sup>* satisfying the Lipschitz condition in the variable *<sup>y</sup>*, the Cauchy– Picard theorem, see [38], transmits the result from the finite dimensional case to an infinite dimensional case. Yet, the drift *f* fulfills the Lipschitz condition because of (11) and its affine structure.

#### *4.1. Energy Estimate*

Here, we define and estimate the energy of the perturbation state at a time *t* P r0, *T*1s.

**Definition 1** (Energy)**.** *For t* P r0, *T*<sup>1</sup> s*, we consider the energy defined as :*

$$\begin{aligned} E(t) &= \left\| \sqrt{g\dot{h}}h \right\|\_{L^2(\Omega)}^2 + \left\| q\_1 \right\|\_{L^2(\Omega)}^2 + \left\| q\_2 \right\|\_{L^2(\Omega)}^2 + \mu \int\_0^t \left\| \nabla q(\tau) \right\|\_{L^2(\Omega)}^2 d\tau \\ &= \left. E^1(t) + \mu \int\_0^t \left\| \nabla q(\tau) \right\|\_{L^2(\Omega)}^2 d\tau \right. \end{aligned} \tag{12}$$

*where E*1p*t*q *is the non-viscous energy and T*<sup>1</sup> *is the time bound given in Lemma 1.*

To establish an estimate on the energy, we replace the test functions <sup>p</sup>*ϕ*, *<sup>φ</sup>*, *<sup>ψ</sup>*<sup>q</sup> by <sup>p</sup>*g*¯ *hh*, *q*<sup>q</sup> in the variational form (7)–(9), to obtain

$$\frac{1}{2} \int\_{\Omega} \, \_{\mathcal{S}} \bar{h} \hat{c} \hat{\imath} \, h^2 \, d\Omega - \int\_{\Omega} \, \_{\mathcal{S}} \bar{g} \boldsymbol{q} \cdot \nabla (\bar{h} \boldsymbol{h}) \, d\Omega = - \int\_{\Gamma\_1} \, \_{\mathcal{S}} \bar{h} \bar{h} \boldsymbol{q} \cdot \mathbf{n} \, d\sigma\_{\mathcal{A}}$$

$$\begin{split} \frac{1}{2} \int\_{\Omega} \left\| \begin{pmatrix} q\_1^2 + q\_2^2 \end{pmatrix} d\Omega \right. &+ \left. \mu \int\_{\Omega} \| \nabla q \| \_{2}^{2} d\Omega + \int\_{\Omega} (\text{div} \boldsymbol{q}) \vec{\boldsymbol{\sigma}} \cdot \boldsymbol{q} \, d\Omega - \int\_{\Omega} q\_1 \text{div} (q\_1 \vec{\boldsymbol{\sigma}}) \, d\Omega \\ &- \int\_{\Omega} q\_2 \text{div} (q\_2 \vec{\boldsymbol{\sigma}}) \, d\Omega - \int\_{\Omega} h \text{div} (\mathbf{B} \cdot \boldsymbol{q}) \, d\Omega + \int\_{\Omega} (\boldsymbol{A} \cdot \boldsymbol{q}) \cdot \boldsymbol{q} \, d\Omega + \int\_{\Omega} h \mathbf{n}\_0 \cdot \boldsymbol{q} \, d\Omega \\ &= \quad - \int\_{\Gamma\_1} (q\_1^2 + q\_2^2) \vec{\boldsymbol{\sigma}} \cdot \mathbf{n} \, d\sigma + \mu \int\_{\Gamma\_1} (q\_1 \nabla q\_1 + q\_2 \nabla q\_2) \cdot \mathbf{n} \, d\sigma - \int\_{\Gamma\_1} h (\mathbf{B} \cdot \boldsymbol{q}) \cdot \mathbf{n} \, d\sigma. \end{split}$$

Adding the two equalities yields

$$\begin{aligned} \frac{1}{2}\hat{c}\_{t}\mathbb{E}^{1}(t) + \mu \int\_{\Omega} \|\nabla q\|\_{2}^{2} d\Omega &= \left.I\_{1} + I\_{2} + I\_{3} + I\_{4} - \int\_{\Gamma\_{1}} g\bar{h}h\boldsymbol{q} \cdot \mathbf{n} \, d\sigma - \int\_{\Gamma\_{1}} (q\_{1}^{2} + q\_{2}^{2})\vec{\boldsymbol{\sigma}} \cdot \mathbf{n} \, d\sigma \right| \\ &+ \mu \int\_{\Gamma\_{1}} (q\_{1}\nabla q\_{1} + q\_{2}\nabla q\_{2}) \cdot \mathbf{n} \, d\sigma - \int\_{\Gamma\_{1}} h(\mathbf{B}\cdot\boldsymbol{q}) \cdot \mathbf{n} \, d\sigma, \end{aligned}$$

where the quantities *Ii* are given by

$$\begin{aligned} I\_1 &= \int\_{\Omega} \mathcal{g}\boldsymbol{q} \cdot \nabla (\bar{h}h) \, d\Omega + \int\_{\Omega} h \text{div} (\mathcal{B} \cdot \boldsymbol{q}) \, d\Omega, \quad I\_2 = -\int\_{\Omega} \text{div} \, q\vec{v} \cdot \boldsymbol{q} \, d\Omega, \\\ I\_3 &= \int\_{\Omega} q\_1 \text{div} (q\_1 \vec{v}) \, d\Omega + \int\_{\Omega} q\_2 \text{div} (q\_2 \vec{v}) \, d\Omega, \quad I\_4 = -\int\_{\Omega} (\boldsymbol{A} \cdot \boldsymbol{q}) \cdot \boldsymbol{q} \, d\Omega - \int\_{\Omega} h \mathbf{a}\_0 \cdot \boldsymbol{q} \, d\Omega. \end{aligned}$$

We now investigate how to isolate the nonlinear terms in each *Ii* with the purpose of having no derivative terms on the boundary. For that, we apply the Green formula in an adaptive manner. Afterward, we take the maximum bound over the steady-state variables, which allows us to obtain a bound estimate for each quantity. For the quantity *I*<sup>1</sup> , it comes that

*I*<sup>1</sup> ď max Ω ´ *g* › ›∇¯ *h* › › ?¯ *h* ¯ ż Ω a¯ *<sup>h</sup>*}*hq*}*d*<sup>Ω</sup> ` max Ω ´ *<sup>v</sup>*¯<sup>2</sup> ?1 ¯ *h* ¯ ż Ω a¯ *h* ˇ <sup>ˇ</sup>*h*B*xq*<sup>1</sup> ˇ <sup>ˇ</sup>*d*<sup>Ω</sup> ` max Ω ´ˇ ˇB*xv*¯<sup>2</sup> 1 ˇ ˇ ?¯ *h* ¯ ż Ω a¯ *h* ˇ <sup>ˇ</sup>*hq*<sup>1</sup> ˇ <sup>ˇ</sup>*d*<sup>Ω</sup> ` max Ω ´ *<sup>v</sup>*¯<sup>2</sup> ?2 ¯ *h* ¯ ż Ω a¯ *h* ˇ <sup>ˇ</sup>*h*B*yq*<sup>2</sup> ˇ <sup>ˇ</sup>*d*<sup>Ω</sup> ` max Ω ´ˇ ˇB*yv*¯<sup>2</sup> 2 ˇ ˇ ?¯ *h* ¯ ż Ω a¯ *h*|*hq*<sup>2</sup> |*d*Ω ` max Ω ´ˇ <sup>ˇ</sup>*v*¯1 *<sup>v</sup>*¯2 ˇ ˇ ?¯ *h* ¯ ż Ω a¯ *h*|*h*B*xq*<sup>2</sup> |*d*Ω ` max Ω ´ˇ <sup>ˇ</sup>B*x*p*v*¯1 *<sup>v</sup>*¯2 <sup>q</sup> ˇ ˇ ?¯ *h* ¯ ż Ω a¯ *h*|*hq*<sup>2</sup> |*d*Ω ` max Ω ´ˇ <sup>ˇ</sup>*v*¯1 *<sup>v</sup>*¯2 ˇ ˇ ?¯ *h* ¯ ż Ω a¯ *h* ˇ <sup>ˇ</sup>*h*B*yq*<sup>1</sup> ˇ <sup>ˇ</sup>*d*<sup>Ω</sup> ` max Ω ´ˇ <sup>ˇ</sup>B*y*p*v*¯1 *<sup>v</sup>*¯2 <sup>q</sup> ˇ ˇ ?¯ *h* ¯ ż Ω a¯ *h* ˇ <sup>ˇ</sup>*hq*<sup>1</sup> ˇ <sup>ˇ</sup>*d*<sup>Ω</sup> ` ż BΩ *g*¯ *hhq* ¨ **<sup>n</sup>** *<sup>d</sup>σ*. (13)

Similarly, we bound the quantities *I*<sup>2</sup> , *I*<sup>3</sup> , and *I*<sup>4</sup> as follows

$$\begin{split} I\_{2} &\ll \quad \frac{1}{2} \max\_{\Omega} \left( \left| \left\| \boldsymbol{\varepsilon}\_{\mathcal{I}} \boldsymbol{\sigma}\_{1} \right| \right) \int\_{\Omega} q\_{1}^{2} d\Omega + \frac{1}{2} \max\_{\Omega} \left( \left| \left\boldsymbol{\varepsilon}\_{\mathcal{Y}} \boldsymbol{\sigma}\_{2} \right| \right) \int\_{\Omega} q\_{2}^{2} d\Omega + \max\_{\Omega} \left( \left| \boldsymbol{\sigma}\_{1} \right| \right) \int\_{\Omega} \left| q\_{1} \right| \boldsymbol{\varepsilon}\_{\mathcal{Y}} q\_{2} \right| d\Omega \\ &+ \max\_{\Omega} \left( \left| \left\boldsymbol{\sigma}\_{2} \right| \right) \int\_{\Omega} \left| q\_{2} \right| \boldsymbol{\varepsilon}\_{\mathcal{Y}} q\_{1} \left| d\Omega - \int\_{\partial\Omega} \frac{\overline{\boldsymbol{\sigma}}\_{1}}{2} q\_{1}^{2} n\_{\mathbf{x}} d\sigma - \int\_{\partial\Omega} \frac{\overline{\boldsymbol{\sigma}}\_{2}}{2} q\_{2}^{2} n\_{\mathbf{y}} d\sigma, \end{split} \tag{14}$$

$$\begin{split} I\_{3} &\leqslant \quad \frac{1}{2} \max\_{\Omega} \left( |\mathring{c}\_{3} \boldsymbol{\uppi}\_{1}| \right) \int\_{\Omega} q\_{1}^{2} d\Omega + \max\_{\Omega} (|\mathring{c}\_{3} \boldsymbol{\uppi}\_{2}|) \int\_{\Omega} |q\_{1} q\_{2}| d\Omega + \max\_{\Omega} (|\mathring{c}\_{2}|) \int\_{\Omega} |q\_{1} \mathring{c}\_{3} q\_{2}| d\Omega \\ &\quad + \max\_{\Omega} (|\mathring{c}\_{3} \boldsymbol{\uppi}\_{2}|) \int\_{\Omega} |q\_{1} q\_{2}| d\Omega + \max\_{\Omega} (|\mathring{c}\_{1}|) \int\_{\Omega} |q\_{2} \mathring{c}\_{3} q\_{1}| d\Omega + \frac{1}{2} \max\_{\Omega} (|\mathring{c}\_{3} \boldsymbol{\uppi}\_{2}|) \int\_{\Omega} q\_{2}^{2} d\Omega \\ &\quad - \int\_{\partial\Omega} \frac{\overline{v}\_{1}}{2} q\_{1}^{2} n\_{1} d\sigma - \int\_{\partial\Omega} \frac{\overline{v}\_{2}}{2} q\_{2}^{2} n\_{3} d\sigma, \\ &\quad \dots \end{split} \tag{15}$$

and

$$\begin{split} I\_4 &\ll \max\_{\Omega} \left( \left| \mathcal{J}\_0^1 \right| \right) \int\_{\Omega} q\_1^2 d\Omega + \max\_{\Omega} \left( \left| \gamma\_0^1 + \beta\_0^2 \right| \right) \int\_{\Omega} |q\_1 q\_2| d\Omega + \max\_{\Omega} \left( \left| \gamma\_0^2 \right| \right) \int\_{\Omega} q\_2^2 d\Omega \\ &+ \max\_{\Omega} \left( \frac{|a\_0^1|}{\sqrt{\tilde{h}}} \right) \int\_{\Omega} \sqrt{\tilde{h}} |h q\_1| d\Omega + \max\_{\Omega} \left( \frac{|a\_0^2|}{\sqrt{\tilde{h}}} \right) \int\_{\Omega} \sqrt{\tilde{h}} |h q\_2| d\Omega. \end{split} \tag{16}$$

We now apply the Young inequality to separate the nonlinear terms (in the perturbation state) for the upper bound of each of the inequalities (13)–(16). That implies the existence of *ε<sup>i</sup>* for p*i* " 1, ¨¨¨ , 18q, such that

$$\begin{split} \frac{1}{2}\partial\_{\tilde{t}}\mathcal{E}^{1}(t) + \mu \int\_{\Omega} \|\nabla q\|\_{2}^{2} d\Omega &\quad \leqslant \quad T\_{\text{bas}}(t) + \mathbb{C}\_{h} \int\_{\Omega} \bar{h}h^{2} d\Omega + \mathbb{C}\_{q\_{1}} \int\_{\Omega} q\_{1}^{2} d\Omega + \mathbb{C}\_{q\_{2}} \int\_{\Omega} q\_{2}^{2} d\Omega \\ &\quad + \left(\varepsilon\_{2} \max\_{\Omega} \left(\frac{\tilde{\sigma}\_{1}^{2}}{\sqrt{h}}\right) + \varepsilon\_{11} \max\_{\Omega} (|\bar{v}\_{2}|) \right) \int\_{\Omega} \left(\partial\_{x}q\_{1}\right)^{2} d\Omega \\ &\quad + \left(\varepsilon\_{8} \max\_{\Omega} \left(\frac{|\bar{v}\_{1}\tilde{v}\_{2}|}{\sqrt{h}}\right) + \varepsilon\_{15} \max\_{\Omega} (|\bar{v}\_{1}|) \right) \int\_{\Omega} \left(\partial\_{y}q\_{1}\right)^{2} d\Omega \\ &\quad + \left(\varepsilon\_{6} \max\_{\Omega} \left(\frac{|\bar{v}\_{1}\tilde{v}\_{2}|}{\sqrt{h}}\right) + \varepsilon\_{13} \max\_{\Omega} (|\bar{v}\_{2}|) \right) \int\_{\Omega} \left(\partial\_{x}q\_{2}\right)^{2} d\Omega \\ &\quad + \left(\varepsilon\_{4} \max\_{\Omega} \left(\frac{\sigma\_{2}^{2}}{\sqrt{h}}\right) + \varepsilon\_{10} \max\_{\Omega} (|\bar{v}\_{1}|) \right) \int\_{\Omega} \left(\partial\_{y}q\_{2}\right)^{2} d\Omega, \end{split}$$

where

*Ch* " <sup>1</sup> *ε*1 max Ω ´ *g* › ›∇¯ *h* › › ?¯ *h* ¯ ` 1 *ε*2 max Ω ´ *<sup>v</sup>*¯<sup>2</sup> ?1 ¯ *h* ¯ ` 1 *ε*3 max Ω ´ˇ ˇB*xv*¯<sup>2</sup> 1 ˇ ˇ ?¯ *h* ¯ ` 1 *ε*4 max Ω ´ *<sup>v</sup>*¯<sup>2</sup> ?2 ¯ *h* ¯ ; ` 1 *ε*5 max Ω ´ˇ ˇB*yv*¯<sup>2</sup> 2 ˇ ˇ ?¯ *h* ¯ ` 1 *ε*6 max Ω ´ˇ <sup>ˇ</sup>*v*¯1 *<sup>v</sup>*¯2 ˇ ˇ ?¯ *h* ¯ ` 1 *ε*7 max Ω ´ˇ <sup>ˇ</sup>B*x*p*v*¯1 *<sup>v</sup>*¯2 <sup>q</sup> ˇ ˇ ?¯ *h* ¯ ` 1 *ε*8 max Ω ´ˇ <sup>ˇ</sup>*v*¯1 *<sup>v</sup>*¯2 ˇ ˇ ?¯ *h* ¯ ` 1 *ε*9 max Ω ´ˇ <sup>ˇ</sup>B*y*p*v*¯1 *<sup>v</sup>*¯2 <sup>q</sup> ˇ ˇ ?¯ *h* ¯ ` 1 *ε*<sup>17</sup> max Ω ˜ˇ ˇ*α*1 0 ˇ ˇ ?¯ *h* ¸ ` 1 *ε*<sup>18</sup> max Ω ˜ˇ ˇ*α*2 0 ˇ ˇ ?¯ *h* ¸ ,

$$\begin{split} \mathbb{C}\_{q\_{1}} &= \ & \varepsilon\_{1} \max\_{\Omega} \left( \frac{\operatorname{g} \| \nabla \bar{h} \|}{\sqrt{\eta}} \right) + \varepsilon\_{3} \max\_{\Omega} \left( \frac{\operatorname{\boldsymbol{\varepsilon}}\_{\text{2}} \boldsymbol{\sigma}\_{1}^{2}}{\sqrt{\eta}} \right) + \varepsilon\_{9} \max\_{\Omega} \left( \frac{\operatorname{\boldsymbol{\varepsilon}}\_{\text{y}} (\boldsymbol{\overline{\boldsymbol{\varepsilon}}\_{1} \boldsymbol{\sigma}\_{2})}{\sqrt{\eta}} \right) + \max\_{\Omega} \left( \left| \boldsymbol{\varepsilon}\_{\text{2}} \boldsymbol{\overline{\boldsymbol{\varepsilon}}\_{1}} \right| \right) \\ &+ \frac{1}{\varepsilon\_{10}} \max\_{\Omega} (|\boldsymbol{\varepsilon}\_{1}|) + \frac{1}{\varepsilon\_{12}} \max\_{\Omega} (|\boldsymbol{\varepsilon}\_{\text{2}} \boldsymbol{\varepsilon}\_{1}|) + \frac{1}{\varepsilon\_{13}} \max\_{\Omega} (|\boldsymbol{\overline{\boldsymbol{\varepsilon}}\_{2}}|) + \frac{1}{\varepsilon\_{14}} \max\_{\Omega} (|\boldsymbol{\varepsilon}\_{\text{2}} \boldsymbol{\varepsilon}\_{1}|) + \max\_{\Omega} \left( |\boldsymbol{\varepsilon}\_{\text{y}} \boldsymbol{\varepsilon}\_{1}| \right) \\ &+ \frac{1}{\varepsilon\_{16}} \max\_{\Omega} \left( \left| \boldsymbol{\gamma}\_{\text{0}}^{1} + \boldsymbol{\beta}\_{\text{0}}^{2} \right| \right) + \varepsilon\_{17} \max\_{\Omega} \left( \frac{|\boldsymbol{\alpha}\_{\text{0}}^{1}|}{\sqrt{\boldsymbol{\varepsilon}}} \right), \end{split}$$

$$\begin{split} \mathbb{C}\_{q\_{2}} &= \ \varepsilon\_{1} \max\_{\Omega} \left( \frac{g \| \nabla \tilde{h} \|}{\sqrt{\mathfrak{f}}} \right) + \varepsilon\_{5} \max\_{\Omega} \left( \frac{\left| \mathbb{\tilde{c}}\_{y} \boldsymbol{\sigma}\_{2}^{2} \right|}{\sqrt{\mathfrak{f}}} \right) + \varepsilon\_{7} \max\_{\Omega} \left( \frac{\left| \mathbb{\tilde{c}}\_{x} (\mathbb{\tilde{v}}\_{1} \boldsymbol{\sigma}\_{2}) \right|}{\sqrt{\mathfrak{f}}} \right) + \max\_{\Omega} \left( \left| \mathbb{\tilde{c}}\_{y} \boldsymbol{\sigma}\_{2} \right| \right) \\ &+ \frac{1}{\varepsilon\_{11}} \max\_{\Omega} (|\boldsymbol{\sigma}\_{2}|) + \varepsilon\_{12} \max\_{\Omega} (|\boldsymbol{\biglangle} \boldsymbol{\sigma}\_{2} \boldsymbol{\bigrangle}|) + \varepsilon\_{14} \max\_{\Omega} \left( |\boldsymbol{\biglangle} \boldsymbol{\sigma}\_{y} \boldsymbol{\sigma}\_{1}| \right) + \frac{1}{\varepsilon\_{15}} \max\_{\Omega} (|\boldsymbol{\sigma}\_{1}|) + \max\_{\Omega} \left( |\boldsymbol{\sigma}\_{1}| \right) + \max\_{\Omega} \left( \left| \boldsymbol{\gamma}\_{0}^{2} \right| \right) \\ &+ \varepsilon\_{16} \max\_{\Omega} \left( \left| \boldsymbol{\gamma}\_{0}^{1} + \boldsymbol{\beta}\_{0}^{2} \right| \right) + \varepsilon\_{18} \max\_{\Omega} \left( \frac{|\boldsymbol{\alpha}\_{0}^{2}|}{\sqrt{\mathfrak{h}}} \right), \end{split}$$

and

$$\begin{aligned} T\_{\text{bond}}(t) &= \begin{array}{c} \ \ \ \end{array} \left( \bar{\boldsymbol{v}}\_{1} \boldsymbol{q}\_{1}^{2} \boldsymbol{n}\_{\boldsymbol{x}} + \bar{\boldsymbol{v}}\_{2} \boldsymbol{q}\_{2}^{2} \boldsymbol{n}\_{\boldsymbol{y}} \right) d\boldsymbol{\sigma} - \int\_{\Gamma\_{1}} (\boldsymbol{q}\_{1}^{2} + \boldsymbol{q}\_{2}^{2}) \bar{\boldsymbol{\sigma}} \cdot \mathbf{n} \, d\boldsymbol{\sigma} \\ &+ \mu \int\_{\Gamma\_{1}} (\boldsymbol{q}\_{1} \nabla \boldsymbol{q}\_{1} + \boldsymbol{q}\_{2} \nabla \boldsymbol{q}\_{2}) \cdot \mathbf{n} \, d\boldsymbol{\sigma} - \int\_{\Gamma\_{1}} \boldsymbol{h} (\boldsymbol{\mathcal{B}} \cdot \boldsymbol{q}) \cdot \mathbf{n} \, d\boldsymbol{\sigma} .\end{aligned}$$

Note that *Ch* , *Cq*<sup>1</sup> and *Cq*<sup>2</sup> are constant while *T*bord is time variable. Let us denote

$$\mathcal{C}\_{\mathsf{m}} = \max \left( \frac{\mathcal{C}\_{\mathsf{h}}}{\mathcal{S}}, \mathcal{C}\_{q\_{1}}, \mathcal{C}\_{q\_{2}} \right),$$

and

$$\mathbb{C}\_{v} = \max\left( \begin{array}{c} \varepsilon\_{2}\max\left(\frac{\tilde{\boldsymbol{\sigma}}\_{1}^{2}}{\sqrt{\tilde{\boldsymbol{h}}}}\right) + \varepsilon\_{11}\max\left(|\boldsymbol{\vartheta}\_{2}|\right), \ \varepsilon\_{8}\max\left(\frac{|\boldsymbol{\vartheta}\_{1}\boldsymbol{\vartheta}\_{2}|}{\sqrt{\tilde{\boldsymbol{h}}}}\right) + \frac{1}{\varepsilon\_{15}}\max\left(|\boldsymbol{\vartheta}\_{1}|\right), \\\\ \varepsilon\_{6}\max\left(\frac{|\boldsymbol{\vartheta}\_{1}\boldsymbol{\vartheta}\_{2}|}{\sqrt{\tilde{\boldsymbol{h}}}}\right) + \varepsilon\_{13}\max\left(|\boldsymbol{\vartheta}\_{2}|\right), \ \varepsilon\_{4}\max\left(\frac{\tilde{\boldsymbol{v}}\_{2}^{2}}{\sqrt{\tilde{\boldsymbol{h}}}}\right) + \varepsilon\_{10}\max\left(|\boldsymbol{\vartheta}\_{1}|\right). \end{array} \right).$$

Since the *ε<sup>i</sup>* can be chosen arbitrarily, we then take *ε*2, *ε*4, *ε*6, *ε*8, *ε*10, *ε*11, *ε*<sup>13</sup> and *ε*<sup>15</sup> such that *<sup>μ</sup>* <sup>2</sup> ě *Cv*. Therefore, it comes that

$$\begin{aligned} \left\| \left. \left\| \left. \left\| E^1(t) + \frac{\mu}{2} \int\_{\Omega} \| \nabla q \| \right\| \_2^2 d\Omega \right\| \right. \end{aligned} \right\|\_{\mathrm{mod}} & \leqslant \left. T\_{\mathrm{mod}}(t) + \mathsf{C}\_{\mathrm{m}} \mathrm{E}^1(t) \right. \\ \left\| \left. \left\| \left. \left\| \nabla q \right\| \right\| \_2^2 d\Omega \right\vert \right. \end{aligned} \right\|\_{\mathrm{mod}} & \leqslant \left. T\_{\mathrm{mod}}(t) + \mathsf{C}\_{\mathrm{m}} \mathrm{E}^1(t) + \mathsf{C}\_{\mathrm{m}} \frac{\mu}{2} \int\_0^t \int\_{\Omega} \left\| \nabla q \right\|\_2^2 d\Omega dt. \end{aligned}$$

Finally, with (12), the energy *E* of the stabilization problem (6) satisfies

$$
\partial\_t E(t) - \mathbb{C}\_m E(t) \preccurlyeq\_\varepsilon T\_{\text{bond}}(t). \tag{17}
$$

*4.2. Short-Time Control Building Process*

In this section, we address the existence and the design of the boundary feedback control law in the time interval r0, *T*1s, which stabilizes the perturbation state in the sense that the energy decreases.

**Lemma 2.** *Let r be a continuous time function for which the integral diverges when t tends to* `8*; there exists a nonlinear control law V*<sup>1</sup> " p*u*<sup>11</sup> , *u*<sup>12</sup> q *to set as the boundary condition on* Γ<sup>1</sup> *such that, for all t* P r0, *T*1s*, the energy E satisfies the following estimate:*

$$E^1(t) + \int\_0^t \left\| \nabla q \right\|\_{L^2(\Omega)}^2 ds \prec \mathcal{E}^1(0) \exp\left( \int\_0^t -r(s) ds \right) \quad \forall t \in [0, T\_1]. \tag{18}$$

This lemma is an intermediate result that proves the existence of a boundary control law in the time interval r0, *T*1s.

**Proof.** We have shown the existence of a solution in the time interval r0, *T*1s in Lemma 1. Let us now consider the energy estimate (17) with *T*bord p*t*q expressed in terms of the control commands

$$T\_{\rm bod}(t) = a\_1 \mu\_{1\_1}^2(t) + b\_1(h)\mu\_{1\_1}(t) + a\_2 \mu\_{2\_1}^2(t) + b\_2(h)\mu\_{2\_1}(t),$$

where

*<sup>a</sup>*<sup>1</sup> " ´ <sup>ż</sup> Γ1 ` 2*v*¯1*nx* ` *v*¯2*ny* ˘ *<sup>d</sup>σ*, *<sup>a</sup>*<sup>2</sup> " ´ <sup>ż</sup> Γ1 ` *v*¯1*nx* ` 2*v*¯2*ny* ˘ *dσ*, *<sup>b</sup>*<sup>1</sup> <sup>p</sup>*h*q"´ <sup>ż</sup> Γ1 ´ <sup>p</sup>*c*<sup>2</sup> ´ *<sup>v</sup>*¯ 2 <sup>1</sup> q*nx* ´ *v*¯1 *v*¯2*ny* ¯ *h dσ* ` *μ* ż Γ1 ∇*q*<sup>1</sup> ¨ **n** *dσ*, *<sup>b</sup>*<sup>2</sup> <sup>p</sup>*h*q"´ <sup>ż</sup> Γ1 ´ <sup>p</sup>*c*<sup>2</sup> ´ *<sup>v</sup>*¯ 2 <sup>1</sup> q*ny* ´ *v*¯1 *v*¯2*nx* ¯ *h dσ* ` *μ* ż Γ1 ∇*q*<sup>2</sup> ¨ **n** *dσ*.

The energy estimate can then be written in terms of the control command as follows

$$
\frac{1}{2}\hat{c}\_1 E(t) - \mathbb{C}\_m E(t) \lessapprox a\_1 u\_{1\_1}^2(t) + b\_1(h)u\_{1\_1}(t) + a\_2 u\_{2\_1}^2(t) + b\_2(h)u\_{2\_1}(t).
$$

Now we introduce the positive and continuous function *r* which stands for the stabilization rate. We also denote by *F*<sup>0</sup> the positive function given by

$$F\_{\mathbb{O}}(t) = E(0) \exp\left(-\int\_{\mathbb{O}}^{t} r(s)ds\right).$$

such that

$$a\_1 u\_{1\_1}^2(t) + b\_1(h)u\_{1\_1}(t) + a\_2 u\_{2\_1}^2(t) + b\_2(h)u\_{2\_1}(t) \prec \frac{1}{2} \frac{\partial F\_0}{\partial t} - C\_m F\_0. \tag{19}$$

Furthermore, we denote by *<sup>G</sup>*<sup>0</sup> " <sup>1</sup> <sup>2</sup> B*tF*<sup>0</sup> ´ *Cm F*<sup>0</sup> , and we set the following two inequalities

$$a\_1 u\_{1\_1}^2(t) + b\_1(h)u\_{1\_1}(t) - \frac{1}{2} \mathcal{G}\_0 \leqslant 0 \quad \text{and} \quad a\_2 u\_{2\_1}^2(t) + b\_2(h)u\_{2\_1}(t) - \frac{1}{2} \mathcal{G}\_0 \leqslant 0,$$

so that the inequality (19) holds. The solutions of the associated second-order polynomials are, respectively,

$$\begin{aligned} \tilde{\xi}\_{1\_1} &= \frac{-b\_1 - \sqrt{b\_1^2 + 2a\_1 G\_0}}{2a\_1}, & \tilde{\xi}\_{1\_2} &= \frac{-b\_1 + \sqrt{b\_1^2 + 2a\_1 G\_0}}{2a\_1}, \\ \tilde{\xi}\_{2\_1} &= \frac{-b\_2 - \sqrt{b\_2^2 + 2a\_2 G\_0}}{2a\_2}, & \tilde{\xi}\_{2\_2} &= \frac{-b\_2 + \sqrt{b\_2^2 + 2a\_2 G\_0}}{2a\_2}. \end{aligned}$$

because *a*<sup>1</sup> and *a*<sup>2</sup> are negative by construction (see statement (S1)). The coefficients *b*<sup>1</sup> and *b*<sup>2</sup> depend on the perturbation height *h* and on the limit, towards the boundary, of the *L*<sup>2</sup> norm of the gradient of the perturbation flow; therefore, to guarantee the boundedness of the control command, we define *ui*<sup>1</sup> (for i=1;2) using the following combination

$$u\_{i1} = \max\{-sign(b\_i), 0\} \mathfrak{F}\_{i1} + \max\{sign(b\_i), 0\} \mathfrak{F}\_{i2}.\tag{20}$$

The function *sign*p*x*q returns the sign of the real *x*. The control laws defined at (20) guarantee that the boundary condition *V*<sup>1</sup> " p*u*<sup>11</sup> , *u*<sup>21</sup> q is bounded and decreases the energy of the perturbation system thanks to

$$E^1(t) + \int\_0^t \|\nabla q\|\_{L^2(\Omega)}^2 ds \le E^1(0) \exp\left(\int\_0^t -r(s)ds\right) \quad \forall t \in [0, T\_1].$$

It is important to note that the control *V*<sup>1</sup> does not act on the system after the energy reaches *E*p*T*<sup>1</sup> q.

#### **5. Stabilization Result**

In this section, we establish the existence and uniqueness of the weak solution of the linearized system (6) equipped with the feedback control law, which is devised by cascade and achieves an exponential convergence of the state variables <sup>p</sup>*h*, *q*<sup>q</sup> towards <sup>p</sup>0, 0, 0q. We start by the existence of a sequence of intervals by replicating Lemma 1. For that, we adapt the energy definition for all time *t* ą 0.

**Definition 2** (Energy)**.** *We consider the following definition of the energy:*

$$\begin{aligned} E(t) &= \left\| \sqrt{g\hbar}h \right\|\_{L^2(\Omega)}^2 + \left\| q\_1 \right\|\_{L^2(\Omega)}^2 + \left\| q\_2 \right\|\_{L^2(\Omega)}^2 + \mu \int\_{T\_k}^t \left\| \nabla q \right\|\_{L^2(\Omega)}^2 d\sigma \\ &= \left. E^1(t) + \mu \int\_{T\_k}^t \left\| \nabla q \right\|\_{L^2(\Omega)}^2 d\sigma\_\prime \end{aligned} \tag{21}$$

*where <sup>E</sup>*1p*t*<sup>q</sup> *is the non-viscous energy and Tk is the lower bound of the time interval* <sup>r</sup>*Tk* , *Tk*`<sup>1</sup> <sup>s</sup>*, which is defined in the next lemma.*

**Lemma 3.** *There exists a sequence of intervals* `"*Tk* , *Tk*`<sup>1</sup> ‰˘ *<sup>k</sup>*ě<sup>0</sup> *such that*


$$E(t) \ll E^1(0) \exp\left(-\int\_0^{T\_{k+1}} r(s)ds\right). \tag{22}$$

**Proof.** for the sake of clarity, we proceed by induction to prove Lemma 3.


$$\begin{cases} \ y'(t) = f\_k(t, y) \\ \ y(t = T\_k) = y\_k. \end{cases} \tag{23}$$

Applying Lemma 1, it comes that › › *fk* › › <sup>2</sup> ď *Kk*`<sup>1</sup> and *Tk*`<sup>1</sup> " *E*p*Tk* q{*Kk* ` *Tk* such that (23) has a solution in " *Tk* , *Tk*`<sup>1</sup> ‰ . Thanks to Lemma 2, it exists a stabilizing control command *V<sup>k</sup>*`<sup>1</sup> , and for all *t* P " *Tk* , *Tk*`<sup>1</sup> ‰ , we have

$$\begin{aligned} E(t) &\quad \leqslant \quad E^1(T\_k) \exp\left(-\int\_{T\_k}^t r(s)ds\right), \\ &\leqslant \quad E(T\_k) \exp\left(-\int\_{T\_k}^t r(s)ds\right). \end{aligned}$$

Since *Tk* P " *Tk*´<sup>1</sup> , *Tk* ‰ , we have

$$E(T\_k) \ll E^1(T\_{k-1}) \exp\left(-\int\_{T\_{k-1}}^{T\_k} r(s)ds\right),$$

which implies that

$$\begin{aligned} E(t) &\quad \leqslant \quad E^1(T\_{k-1}) \exp\left(-\int\_{T\_{k-1}}^{T\_k} r(s)ds\right) \exp\left(-\int\_{T\_k}^t r(s)ds\right) \\ &\quad \leqslant \quad E^1(T\_{k-1}) \exp\left(-\int\_{T\_{k-1}}^t r(s)ds\right) \\ &\vdots \\ &\quad \gets \quad E^1(0) \exp\left(-\int\_0^t r(s)ds\right). \end{aligned}$$

The statement is then true at rank *k* ` 1, and that proves the Lemma 3.

We have shown the existence of the sequence of interval `"*Tk* , *Tk*`<sup>1</sup> ‰˘ *k*ě1 and the existence of a boundary control command *V<sup>k</sup>* in each interval " *Tk* , *Tk*`<sup>1</sup> ‰ . We can now design the feedback control law for all *t* ě 0.

**Definition 3** (Control law)**.** *The boundary control law V for the stabilization problem* (6) *is given by*

$$\mathcal{V}(t) = \sum\_{k=0}^{\infty} \mathcal{V}\_k(t) \mathbb{1}\_{[T\_{k'}T\_{k+1}]}(t),\tag{24}$$

*where the local control command V<sup>k</sup>* " p*u*1*<sup>k</sup>* , *u*2*<sup>k</sup>* q *is defined in* r*Tk* , *Tk*`<sup>1</sup> s *by*

$$\mu\_{i\_k} = \max(-\operatorname{sign}(b\_i), 0)\tilde{\varsigma}\_{i\_1}^k + \max(\operatorname{sign}(b\_i), 0)\tilde{\varsigma}\_{i\_2}^k \text{ for } i = 1; 2.1$$

*The quantities ξ<sup>k</sup> <sup>i</sup>*<sup>1</sup> *and <sup>ξ</sup><sup>k</sup> <sup>i</sup>*<sup>2</sup> *are solutions of second-order polynomials and are written as*

$$\begin{aligned} \mathfrak{f}\_{1\_1}^k &= \frac{-b\_1 - \sqrt{b\_1^2 + 2a\_1 G\_k}}{2a\_1}, & \mathfrak{f}\_{1\_2}^k &= \frac{-b\_1 + \sqrt{b\_1^2 + 2a\_1 G\_k}}{2a\_1}, \\ \mathfrak{f}\_{2\_1}^k &= \frac{-b\_2 - \sqrt{b\_2^2 + 2a\_2 G\_k}}{2a\_2}, & \mathfrak{f}\_{2\_2}^k &= \frac{-b\_2 + \sqrt{b\_2^2 + 2a\_2 G\_k}}{2a\_2}. \end{aligned}$$

*where the coefficients ai and bi are given in Section 4.2 where the function Gk is defined in the time interval* r*Tk* , *Tk*`<sup>1</sup> s *as follows:*

$$G\_k(t) = \frac{1}{2}\hat{c}\_t F\_k(t) - \mathbb{C}\_m F\_k(t), \quad \text{with} \quad F\_k(t) = E(T\_k) \exp\left(-\int\_{T\_k}^t r(s)ds\right). \tag{25}$$

*The time function r represents the stabilization rate, and Cm is a constant depending on the steady state.*

We can now state our main result.

**Theorem 1** (main result)**.** *Let r be a continuous time function for which the integral over the interval* r0, *t*s *diverges when t tends to* `8*. Then, there exists a sequence of intervals* ` r*Tk* , *Tk*`<sup>1</sup> s ˘ *k*ě0 *such that*

$$\bigcup\_{k=0}^{+\infty} [T\_{k'}T\_{k+1}] = [0, +\infty]\_{\prime}$$

*and the stabilization problem* (6) *with the boundary conditions* (24) *admits a unique solution* <sup>p</sup>*h*, *q*<sup>q</sup> *for which the energy E decreases according to the following estimate:*

$$E(t) \ll E^1(0) \exp\left(\int\_0^t -r(s)ds\right). \tag{26}$$

The continuity of the functions *<sup>x</sup>* ÞÑ ?*<sup>x</sup>* and *<sup>x</sup>* ÞÑ *<sup>x</sup>*<sup>2</sup> and of the hydrodynamic water level *h* imply that the control law *V* built by concatenating the stabilizing control commands *V<sup>k</sup>* is continuous. It is also worth noticing that the control vanishes when the energy reaches zero, and that the sequence of the time intervals is well-defined, i.e.,

$$\bigcup\_{k=0}^{+\infty} [\![T\_{k'} \; T\_{k+1} \; ] = ]0, +\infty[\![.]$$

**Proof.** The proof is performed using the Faedo–Galerkin method. As outlined at the beginning of the proof of Lemma 1, we consider again t*ai*u1ď*i*ď*<sup>n</sup>* (respectively, by t*ei*u1ď*i*ď*n*) by a finite Hilbertian basis of the space *H*1pΩq (respectively *H*<sup>1</sup> Γ1 pΩq). Let *Wn* " *Vect*- *a*<sup>1</sup> , ¨¨¨ , *an* ( ˆ *Vect*- *e*<sup>1</sup> , ¨¨¨ ,*en* ( ˆ *Vect*- *e*<sup>1</sup> , ¨¨¨ ,*en* ( be the vector space of finite dimension generated by - *ai* ( ˆ - *e*1*i* ( ˆ - *ei* ( . Let p*h*<sup>0</sup> *<sup>n</sup>*, *q*<sup>0</sup> *<sup>n</sup>*<sup>q</sup> be a sequence in *Wn*pΩ<sup>q</sup> converging to <sup>p</sup>*h*0, *q*0<sup>q</sup> in *<sup>L</sup>*2pΩq. The weak form associated with the problem (6) in *Wn*pΩq is given by:

$$
\int\_{\Omega} \oint\_{\Omega} \boldsymbol{q} \cdot \boldsymbol{\hat{n}} \, d\boldsymbol{\Omega} - \int\_{\Omega} \boldsymbol{q}\_n \nabla \boldsymbol{q} d\boldsymbol{\Omega} = - \int\_{\Gamma\_1} \boldsymbol{q} \boldsymbol{q}\_n \cdot \mathbf{n} d\boldsymbol{\sigma},\tag{27}
$$

$$\begin{aligned} \int\_{\Omega} \hat{\imath}\_{l} q\_{l\_{n}} \phi \, d\Omega &+ \int\_{\Omega} \left( \text{div} q\_{n} \mathbb{G}\_{1} \right) \phi \, d\Omega - \int\_{\Omega} q\_{l\_{n}} \text{div} (\phi \vec{\nu}) \, d\Omega + \mu \int\_{\Omega} \nabla q\_{l\_{n}} \nabla \phi \, d\Omega \\ &- \int\_{\Omega} h\_{n} \text{div} (\phi \mathcal{B}\_{1}) \, d\Omega + \int\_{\Omega} \beta\_{0}^{1} q\_{l\_{n}} \phi \, d\Omega + \int\_{\Omega} \gamma\_{0}^{1} q\_{l\_{n}} \phi \, d\Omega + \int\_{\Omega} h\_{n} a\_{0}^{1} \phi \, d\Omega \\ &= - \int\_{\Gamma\_{1}} (q\_{l\_{n}} \phi) \vec{\nu} \cdot \mathbf{n} \, d\sigma + \mu \int\_{\Gamma\_{1}} (\nabla q\_{l\_{n}} \cdot \mathbf{n}) \phi \, d\sigma - \int\_{\Gamma\_{1}} h\_{n} \phi (\mathcal{B}\_{1} \cdot \mathbf{n}) \, d\sigma, \\ \int\_{\Omega} \vec{\nu} q\_{l\_{2}} \psi \, d\Omega &+ \int\_{\Omega} (\text{div} q\_{n} \mathbb{G}\_{2}) \psi \, d\Omega - \int\_{\Omega} q\_{l\_{n}} \text{div} (\psi \vec{\nu}) \, d\Omega + \mu \int\_{\Omega} \nabla q\_{l\_{n}} \nabla \psi \, d\Omega \end{aligned} \tag{28}$$

$$\begin{split} & -\int\_{\Omega} h\_n \text{div}(\boldsymbol{\Psi} \mathbf{B}\_2) \, d\Omega + \int\_{\Omega} \boldsymbol{\beta}\_0^2 q\_{l\_n} \boldsymbol{\Psi} \, d\Omega + \int\_{\Omega} \gamma\_0^2 q\_{l\_n} \boldsymbol{\Psi} \, d\Omega + \int\_{\Omega} h\_n a\_0^2 \boldsymbol{\Psi} \, d\Omega \\ &= \quad - \int\_{\Gamma\_1} (q\_{l\_n} \boldsymbol{\Psi}) \boldsymbol{\vec{\sigma}} \cdot \mathbf{n} \, d\boldsymbol{\sigma} + \mu \int\_{\Gamma\_1} (\nabla q\_{l\_n} \cdot \mathbf{n}) \boldsymbol{\Psi} \, d\sigma - \int\_{\Gamma\_1} h\_n \boldsymbol{\Psi}(\mathbf{B}\_2 \cdot \mathbf{n}) \, d\sigma. \end{split} \tag{29}$$

Referring to the energy estimate, it comes that

$$E\_n^1(t) \prec E\_n(t) \prec E\_n^1(0) \exp\left(\int\_0^t -r(s)ds\right) \quad \forall t \in [0, T],\tag{30}$$

with

$$E\_n^1(t) = \left\|\sqrt{g\bar{h}}h\_n\right\|\_{L^2(\Omega)}^2 + \left\|q\_{1\_n}\right\|\_{L^2(\Omega)}^2 + \left\|q\_{2\_n}\right\|\_{L^2(\Omega)}^2.$$

For *<sup>T</sup>* <sup>ě</sup> 0 to be sufficiently large, and a steady state *<sup>U</sup>*¯ sufficiently regular, the integration over the time interval r0, *T*s of (30) gives us the existence of a positive constant *C* ě 0, such that

$$\left\| h\_n \right\|\_{L^2(0,T,\Omega)}^2 + \left\| \boldsymbol{q}\_n \right\|\_{L^2(0,T,\Omega)}^2 \leqslant \mathsf{C}.$$

This latter inequality implies that <sup>p</sup>*hn* , *<sup>q</sup><sup>n</sup>* <sup>q</sup> is bounded in *<sup>L</sup>*<sup>2</sup> ` 0, *T*, *L*2pΩq<sup>3</sup> ˘ , which is a Hilbert space. Therefore, we can extract a sub-sequence, denoted also by <sup>p</sup>*hn* , *<sup>q</sup><sup>n</sup>* <sup>q</sup>, converging weakly to the limit <sup>p</sup>*h*, *<sup>q</sup>*<sup>q</sup> in *<sup>L</sup>*<sup>2</sup> ` 0, *T*, *L*2pΩq<sup>3</sup> ˘ . Let us now introduce the following spaces *H*<sup>1</sup> *<sup>T</sup>*, *St*p*T*q and *Sl*p*T*q

$$\begin{aligned} &H\_T^1 &= \left\{ \mathcal{g} \in H^1(0, T), \text{ such that } \mathcal{g}(T) = 0 \right\}, \\ &S\_l(T) &= \left\{ \mathcal{g} : \left.\boldsymbol{\varrho}(t, \mathbf{x}, \mathbf{y}) = \mathcal{g}\_1(t) \sum\_{i=1}^{n\_0} a\_i \boldsymbol{e}\_i(\mathbf{x}, \mathbf{y}), \text{ such that } \mathcal{g}\_1 \in H^1\_T \text{ and } a\_i \in \mathbb{R} \right\}, \\ &S\_l(T) &= \left\{ f \in H^1\_T \times H^1(\Omega), \text{ such that } \boldsymbol{f}(T, \mathbf{x}, \mathbf{y}) = 0 \right\}. \end{aligned}$$

For the mass equation: the integration over the time interval r0, *T*s of (27) results in,

$$-\int\_{0}^{T} (\boldsymbol{h}\_{n}, \boldsymbol{\mathcal{E}}\boldsymbol{\varrho})\_{\Omega} dt \quad + \quad (\boldsymbol{h}\_{n}(0, \boldsymbol{x}, \boldsymbol{y}), \boldsymbol{\varrho}(0, \boldsymbol{x}, \boldsymbol{y}))\_{\Omega} - (\boldsymbol{h}\_{n}(T, \boldsymbol{x}, \boldsymbol{y}), \boldsymbol{\varrho}(T, \boldsymbol{x}, \boldsymbol{y}))\_{\Omega}$$

$$-\quad \int\_{0}^{T} (\boldsymbol{q}\_{n}, \nabla \boldsymbol{\varrho})\_{\Omega} dt = -\int\_{0}^{T} \int\_{\hat{c}\Omega} \boldsymbol{\varrho} \boldsymbol{q}\_{n} \cdot \mathbf{n} d\boldsymbol{\sigma} dt,\tag{31}$$

where *ϕ* P *H*1pΩq. Taking *ϕ* P *St*p*T*q ˆ *H*1pΩq, that is *ϕ* " *g*1p*t*q ř*n*<sup>0</sup> *<sup>i</sup>*"<sup>1</sup> *aiei*p*x*, *<sup>y</sup>*q " *<sup>a</sup>*1*e*1p*x*, *<sup>y</sup>*q, the equality (31) becomes

$$\begin{split} -\int\_{0}^{T} (h\_{\boldsymbol{\nu}}, e\_{1})\_{\Omega} \hat{e}\_{1} \varrho\_{1}(t) dt &- \left. (h\_{\boldsymbol{\nu}}(0, \mathbf{x}, y), e\_{1}(0, \mathbf{x}, y))\_{\Omega} \varrho\_{1}(0) - \int\_{0}^{T} (q\_{\boldsymbol{\nu}}, \nabla e\_{1})\_{\Omega} \varrho\_{1}(t) dt \right| \\ &= \left. - \int\_{0}^{T} \left( \varrho\_{1}(t) \boldsymbol{\nu}\_{1}(t) \int\_{\Gamma\_{1}} e\_{1} n\_{\boldsymbol{x}} d\sigma \right) dt - \int\_{0}^{T} \left( \varrho\_{1}(t) \boldsymbol{\nu}\_{2}(t) \int\_{\Gamma\_{1}} e\_{1} n\_{\boldsymbol{y}} d\sigma \right) dt \right| \\ &= \left. - \int\_{\Gamma\_{1}} e\_{1} n\_{\boldsymbol{x}} d\sigma \int\_{0}^{T} \varrho\_{1}(t) \boldsymbol{\nu}\_{1}(t) dt - \int\_{\Gamma\_{1}} e\_{1} n\_{\boldsymbol{y}} d\sigma \int\_{0}^{T} \varrho\_{1}(t) \boldsymbol{\nu}\_{2}(t) dt. \end{split}$$

Since *St*p*T*q is dense in *Sl*p*T*q, taking the limit when *n* tends to `8, we obtain by compactness:

$$\begin{aligned} -\int\_0^T \left(\tilde{h}\_\tau e\_1\right)\_\Omega \hat{c}\_l \varrho\_1(t) dt &- \quad \left(\tilde{h}(0, \mathbf{x}, y), e\_1(0, \mathbf{x}, y)\right)\_\Omega \varrho\_1(0) - \int\_0^T \left(\tilde{q}\_\tau \nabla e\_1\right)\_\Omega \varrho\_1(t) dt \\ &= \quad - \quad \int\_{\Gamma\_1} \varepsilon\_1 n\_\Gamma d\sigma \int\_0^T \varrho\_1(t) v\_1(t) dt - \int\_{\Gamma\_1} \varepsilon\_1 n\_\mathcal{Y} d\sigma \int\_0^T \varrho\_1(t) v\_2(t) dt, \end{aligned}$$

where *e*<sup>1</sup> is taken in *H*1pΩq and *g* in *H*<sup>1</sup> *<sup>T</sup>*, respectively.

Since *D*p0, *T*q ˆ *D*pΩq Ă *H*1p0, *T*q ˆ *H*1pΩq and *L*2p*Q*q Ă *D*<sup>1</sup> p*Q*q, we consider from now on the test function *ϕ* P *D*p*Q*q ˆ *D*pΩq. That allows us to drop the second member in the equation above because if a function belongs to *D*p*Q*q, it vanishes at the boundary. Subsequently, we can write

$$-\left(\hbar\_{\prime}\hat{c}\_{\mathrm{I}}\boldsymbol{\varrho}\right)\_{\left(D(0,T)\times D(\Omega)\right)',\left(D(0,T)\times D(\Omega)\right)} - \left(\mathfrak{q}\_{\prime}\nabla\boldsymbol{\varrho}\right)\_{\left(D(0,T)\times D(\Omega)\right)',\left(D(0,T)\times D(\Omega)\right)} = 0.$$

Finally, we conclude that , in the distributions sense,

$$
\partial\_t h + d \dot{v} \eta = 0.
$$

For the first equation of the momentum, we have

$$\begin{split} -\left(\hat{\boldsymbol{\sigma}}\boldsymbol{q}\_{1\boldsymbol{k}},\boldsymbol{\Psi}\right)\_{\Omega} &- \quad \left(q\_{1\boldsymbol{k}}(0,\boldsymbol{x},\boldsymbol{y}),\boldsymbol{\Psi}(0,\boldsymbol{x},\boldsymbol{y})\right)\_{\Omega} + \left(q\_{1\boldsymbol{k}}(\boldsymbol{T},\boldsymbol{x},\boldsymbol{y}),\boldsymbol{\Psi}(\boldsymbol{T},\boldsymbol{x},\boldsymbol{y})\right)\_{\Omega} + \int\_{0}^{T} \left(\hat{\boldsymbol{\mu}}\mathrm{div}\boldsymbol{q}\_{\boldsymbol{k}},\boldsymbol{\Psi}\right)\_{\Omega} dt \\ &- \quad \int\_{0}^{T} \left(q\_{1\boldsymbol{k}},\mathrm{div}\left(\boldsymbol{\psi}\boldsymbol{\bar{\sigma}}\right)\right)\_{\Omega} dt + \mu \int\_{0}^{T} \left(\nabla q\_{1\boldsymbol{k}},\nabla\boldsymbol{\Psi}\right)\_{\Omega} dt - \int\_{0}^{T} \left(h\_{1\boldsymbol{k}}\,\mathrm{div}\left(\boldsymbol{\psi}\boldsymbol{\mathcal{B}}\_{1}\right)\right)\_{\Omega} dt \\ &+ \quad \int\_{0}^{T} \left(\beta\_{0}^{1}q\_{1\boldsymbol{k}},\boldsymbol{\Psi}\right)\_{\Omega} dt + \int\_{0}^{T} \left(\gamma\_{0}^{0}q\_{2\boldsymbol{k}},\boldsymbol{\Psi}\right)\_{\Omega} dt + \int\_{0}^{T} \left(a\_{0}^{1}h\_{2\boldsymbol{k}},\boldsymbol{\Psi}\right)\_{\Omega} dt \\ &= \quad \quad \int\_{0}^{T} \int\_{\Gamma\_{1}} \left(v\_{1}\boldsymbol{\Psi}\right)\boldsymbol{\vec{\sigma}}\cdot\mathbf{n} d\boldsymbol{\sigma} dt + \mu \int\_{0}^{T} \int\_{\Gamma\_{1}} \left(\nabla q\_{1\boldsymbol{k}}\cdot\mathbf{n}\right)\boldsymbol{\Psi}\boldsymbol{\$$

Taking *ψ* P *St*p*T*q, *ψ* " *g*2p*t*q ř*n*<sup>0</sup> *<sup>i</sup>*"<sup>1</sup> *<sup>b</sup>*<sup>1</sup> *<sup>i</sup> ei*p*x*, *y*q " *g*2p*t*q*e*2p*x*, *y*q, the relation (32) becomes

´ ` B*tq*1*<sup>k</sup>* , *ψ* ˘ <sup>Ω</sup> ´ ` *q*1*<sup>k</sup>* p0, *x*, *y*q,*e*2p0, *x*, *y*q ˘ <sup>Ω</sup>*g*2p0q ` <sup>ż</sup> *<sup>T</sup>* 0 ` *<sup>v</sup>*¯1 *divqk*,*e*2<sup>q</sup> ˘ <sup>Ω</sup>*g*2p*t*q*dt* ´ ż *T* 0 ` *<sup>q</sup>*1*<sup>k</sup>* , *div*p*e*2*v*¯<sup>q</sup> ˘ <sup>Ω</sup>*g*2p*t*q*dt* ` *μ* ż *T* 0 ` ∇*q*1*<sup>k</sup>* , ∇*e*<sup>2</sup> ˘ <sup>Ω</sup>*g*2p*t*q*dt* ´ ż *T* 0 ` *hk* , divp*e*2*B*<sup>1</sup>¨<sup>q</sup> ˘ <sup>Ω</sup>*g*2p*t*q*dt* ` ż *T* 0 ´ *β*1 <sup>0</sup>*q*1*<sup>k</sup>* ,*e*<sup>2</sup> ¯ Ω*g*2p*t*q*dt* ` ż *T* 0 ´ *γ*1 <sup>0</sup>*q*2*<sup>k</sup>* ,*e*<sup>2</sup> ¯ Ω*g*2p*t*q*dt* ` ż *T* 0 ´ *α*1 <sup>0</sup>*hk* ,*e*<sup>2</sup> ¯ Ω*g*2p*t*q*dt* " ż *T* 0 *<sup>v</sup>*1p*t*q*g*2p*t*q*dt* <sup>ż</sup> Γ1 *<sup>e</sup>*2*v***¯** ¨ **<sup>n</sup>***d<sup>σ</sup>* ` *<sup>μ</sup>* ż *T* 0 *g*2p*t*q ż Γ1 ` ∇*q*1*<sup>k</sup>* ¨ **n** ˘ *e*<sup>2</sup> *dσdt* ´ ż *T* 0 *g*2p*t*q ż Γ1 *hk <sup>e</sup>*2p*B*<sup>1</sup>¨ ¨ **<sup>n</sup>**<sup>q</sup> *<sup>d</sup>σdt*.

Since the test function is in the space *St*p*T*q that is dense in *Sl*p*T*q, by taking the limit, it follows that

´ ` B*tq*<sup>1</sup> , *ψ* ˘ <sup>Ω</sup> ´ ` *q*<sup>1</sup> p0, *x*, *y*q,*e*2p0, *x*, *y*q ˘ <sup>Ω</sup>*g*2p0q ` <sup>ż</sup> *<sup>T</sup>* 0 ` *<sup>v</sup>*¯1div*q*,*e*2<sup>q</sup> ˘ <sup>Ω</sup>*g*2p*t*q*dt* ´ ż *T* 0 ` *<sup>q</sup>*<sup>1</sup> , divp*e*2*v*¯<sup>q</sup> ˘ <sup>Ω</sup>*g*2p*t*q*dt* ` *μ* ż *T* 0 ` ∇*q*<sup>1</sup> , ∇*e*<sup>2</sup> ˘ <sup>Ω</sup>*g*2p*t*q*dt* ´ ż *T* 0 ` *<sup>h</sup>*, divp*e*2*B*<sup>1</sup>¨<sup>q</sup> ˘ <sup>Ω</sup>*g*2p*t*q*dt* ` ż *T* 0 ´ *β*1 <sup>0</sup>*q*<sup>1</sup> ,*e*<sup>2</sup> ¯ Ω*g*2p*t*q*dt* ` ż *T* 0 ´ *γ*1 <sup>0</sup>*q*<sup>2</sup> ,*e*<sup>2</sup> ¯ Ω*g*2p*t*q*dt* ` ż *T* 0 ´ *α*1 <sup>0</sup>*h*,*e*<sup>2</sup> ¯ Ω*g*2p*t*q*dt* " ż *T* 0 *<sup>v</sup>*1p*t*q*g*2p*t*q*dt* <sup>ż</sup> Γ1 *<sup>e</sup>*2*v*¯**n***d<sup>σ</sup>* ` *<sup>μ</sup>* ż *T* 0 *g*2p*t*q ż Γ1 ` ∇*q*1**n** ˘ *e*<sup>2</sup> *dσdt* ´ ż *T* 0 *g*2p*t*q ż Γ1 *he*2p*B*<sup>1</sup>¨**n**<sup>q</sup> *<sup>d</sup>σdt*,

with *e*<sup>2</sup> P *H*1pΩq and *g*<sup>2</sup> P *H*<sup>1</sup> *T*.

As previously for the mass conservation equation, we obtain in the distributions sense that

$$
\hat{\sigma}\_t q\_1 + \vec{\sigma}\_1 \text{div} \boldsymbol{q} + \nabla q\_1 \cdot \vec{\sigma} - \mu \Delta \boldsymbol{q} + \mathbf{B}\_{1\cdot} \nabla h + \beta\_0^1 q\_1 + \gamma\_0^1 q\_2 + a\_0^1 h = 0.
$$

Following the same process, we establish the existence result for the second equation of the momentum conservation. Since *E*p*t*q ď lim inf*k*Ñ8 *Ek* p*t*q, (26) follows from (30).

#### **6. Conclusions**

We presented a methodology for building a local boundary control law for the exponential stabilization of two-dimensional shallow viscous water flow. The control law acts only on the volumetric flow parameter in a portion of the boundary and is built by cascade over a sequence of intervals that are given by the existence of weak solutions of the perturbation state. The latter state is obtained by neglecting higher orders terms in the linearization. Nevertheless, it is desirable to address the construction of the control law using the nonlinear model directly. A prospective direction toward improving the presented approach, to address in future, is to consider higher order terms in the approximation in the reformulation of the governing equations of around the equilibrium.

**Author Contributions:** Conceptualization, B.M.D. and O.D.; Formal analysis, B.M.D. and M.S.G.; Funding acquisition, B.M.D.; Methodology, M.S.G. and O.D.; Supervision, O.D.; Writing—original draft, B.M.D.; Writing—review & editing, M.S.G. and O.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the CIPR (Center for Integrative Petroleum Research), College of Petroleum Engineering and Geosciences at King Fahd University of Petroleum and Minerals, Startup funds.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

