*Case ii:*

Following a similar approach, as discussed earlier, Equation (3) gives

$$A\_{\delta}w^{j+1} - \theta \left[ w\_{xx}^{j+1} + w\_{yy}^{j+1} \right] = (1 - \theta) \left[ w\_{xx}^{j} + w\_{yy}^{j} \right] + \mathcal{B}(\mathbf{x}, y, t^{j+1}) + 2A\_{\delta}w^{j} - A\_{\delta}w^{j-1}$$

$$-A\_{\delta} \sum\_{k=1}^{j} \left[ w^{j-k+1} - 2w^{j-k} + w^{j-k-1} \right] \mathcal{B}(k). \tag{20}$$

Now we approximate *wj*+<sup>1</sup> *xxyy*(*<sup>x</sup>*, *y*) with a two dimensional truncated Haar wavelets series as:

$$w\_{xxyy}^{j+1}(\mathbf{x}, y) = \sum\_{i=1}^{2M} \sum\_{l=1}^{2M} a\_{i,l}^{j+1} \mathcal{H}\_i(\mathbf{x}) \mathcal{H}\_i(y), \tag{21}$$

where *aj*+<sup>1</sup> *i*,*l* are unknowns to be determined. Integration of Equation (21) w.r.t. to *y*, between 0 and *y*, gives

$$w\_{\rm xxy}^{j+1}(\mathbf{x}, y) = \sum\_{i=1}^{2M} \sum\_{l=1}^{2M} a\_{i,l}^{j+1} \mathcal{H}\_i(\mathbf{x}) \mathcal{P}\_{l,1}(y) + w\_{\rm xxy}^{j+1}(\mathbf{x}, 0). \tag{22}$$

Integrating Equation (22) w.r.t *y* from 0 to 1, the unknown term *wj*+<sup>1</sup> *xxy* (*<sup>x</sup>*, 0) is given by

$$w\_{\rm xxy}^{j+1}(\mathbf{x},0) = w\_{\rm xx}^{j+1}(\mathbf{x},1) - w\_{\rm xx}^{j+1}(\mathbf{x},0) - \sum\_{i=1}^{2M} \sum\_{l=1}^{2M} a\_{i,l}^{j+1} \mathcal{H}\_{i}(\mathbf{x}) \mathcal{P}\_{l,2}(1). \tag{23}$$

Substituting Equation (23) in Equation (22), the obtained result is

$$w\_{\mathbf{x}xy}^{j+1}(\mathbf{x},y) = \sum\_{i=1}^{2M} \sum\_{l=1}^{2M} a\_{i,l}^{j+1} \mathcal{H}\_i(\mathbf{x}) \left[ \mathcal{P}\_{l,1}(y) - \mathcal{P}\_{l,2}(1) \right] + w\_{\mathbf{x}x}^{j+1}(\mathbf{x},1) - w\_{\mathbf{x}x}^{j+1}(\mathbf{x},0). \tag{24}$$

Integrating Equation (24) from 0 to *y*, we ge<sup>t</sup>

$$w\_{\text{xx}}^{j+1}(\mathbf{x}, y) = \sum\_{i=1}^{2M} \sum\_{l=1}^{2M} a\_{i,l}^{j+1} \mathcal{H}\_i(\mathbf{x}) \left[ \mathcal{P}\_{l,2}(y) - y \mathcal{P}\_{l,2}(1) \right] + y w\_{\text{xx}}^{l+1}(\mathbf{x}, 1) + (1 - y) w\_{\text{xx}}^{l+1}(\mathbf{x}, 0). \tag{25}$$

Repeating the same procedure one can easily derive the subsequent expressions

$$\left(w\_{yy}^{l+1}(\mathbf{x}, y) = \sum\_{i=1}^{2M} \sum\_{l=1}^{2M} a\_{i,l}^{j+1} \left[\mathcal{P}\_{i,2}(\mathbf{x}) - \mathbf{x} \mathcal{P}\_{i,2}(1)\right] \mathcal{H}\_l(y) + \mathbf{x} w\_{yy}^{l+1}(1, y) + (1 - \mathbf{x}) w\_{yy}^{l+1}(0, y). \tag{26}$$

$$\begin{split} w\_{\mathbf{x}}^{l+1}(\mathbf{x},\mathbf{y}) &= \sum\_{i=1}^{2M} \sum\_{l=1}^{2M} a\_{i,l}^{l+1} \left[ \mathcal{P}\_{i,1}(\mathbf{x}) - \mathcal{P}\_{i,2}(1) \right] \left[ \mathcal{P}\_{l,2}(\mathbf{y}) - y \mathcal{P}\_{l,2}(1) \right] + yw\_{\mathbf{x}}^{l+1}(\mathbf{x},1) \\ &+ (1-y)w\_{\mathbf{x}}^{l+1}(\mathbf{x},0) + w^{l+1}(1,y) - w^{l+1}(0,y) - yw^{l+1}(1,1) + yw^{l+1}(0,1) \\ &+ (y-1)w^{l+1}(1,0) + (1-y)w^{l+1}(0,0). \end{split} \tag{27}$$

$$\begin{split} w\_{y}^{l+1}(\mathbf{x},y) &= \sum\_{i=1}^{2M} \sum\_{l=1}^{2M} a\_{i,l}^{l+1} \left[ \mathcal{P}\_{i,2}(\mathbf{x}) - \mathbf{x} \mathcal{P}\_{i,2}(1) \right] \left[ \mathcal{P}\_{l,1}(y) - \mathcal{P}\_{l,2}(1) \right] + \mathbf{x} w\_{y}^{l+1}(1,y) \\ &+ (1-\mathbf{x}) w\_{y}^{l+1}(0,y) + w^{l+1}(\mathbf{x},1) - w^{l+1}(\mathbf{x},0) - \mathbf{x} w^{l}(1,1) + \mathbf{x} w^{l+1}(1,0) \\ &+ (\mathbf{x}-1) w^{l+1}(0,1) + (1-\mathbf{x}) w^{l+1}(0,0). \end{split} \tag{28}$$

$$\begin{split} wv^{l+1}(\mathbf{x},\mathbf{y}) &= \sum\_{i=1}^{2M} \sum\_{l=1}^{2M} d\_{i,l}^{j+1} \left[ \mathcal{P}\_{i,2}(\mathbf{x}) - \mathbf{x} \mathcal{P}\_{i,2}(1) \right] \left[ \mathcal{P}\_{l,2}(\mathbf{y}) - \mathcal{Y} \mathcal{P}\_{l,2}(1) \right] + yw^{l+1}(\mathbf{x},1) \\ &- yw^{j+1}(0,1) + (1-y) \left[ w^{j+1}(\mathbf{x},0) - w^{j+1}(0,0) \right] + \mathbf{x}w^{j+1}(1,\mathbf{y}) \\ &- \mathbf{x}w^{j+1}(0,y) - \mathbf{x}y \left[ w^{j+1}(1,1) - w^{j+1}(0,1) \right] + \mathbf{x} \left( y-1 \right) w^{j+1}(1,0) \\ &+ \mathbf{x}(1-y)w^{j+1}(0,0) + w^{j+1}(0,y). \end{split} \tag{29}$$

Substitution of Equations (25), (26) and (29) in Equation (20) and using the collocation points, *xm* = *<sup>m</sup>*−0.5 2*M* , *yn* = *<sup>n</sup>*−0.5 2*M* , *m*, *n* = 1, 2, . . . 2*M*, produces the following system of equations

$$\sum\_{i=1}^{2M} \sum\_{l=1}^{2M} a\_{i,l}^{j+1} \left[ A\_\delta \mathcal{D}(\mathbf{i}, l, m, n) - \theta \mathcal{E}(\mathbf{i}, l, m, n) - \theta \mathcal{F}(\mathbf{i}, l, m, n) \right] = \mathcal{L}(m, n) + \mathcal{M}(m, n), \tag{30}$$

where

<sup>D</sup>(*<sup>i</sup>*, *l*, *m*, *n*) = [P*i*,<sup>2</sup>(*xm*) − *xm*P*i*,<sup>2</sup>(1)] [P*l*,<sup>2</sup>(*yn*) − *yn*P*l*,<sup>2</sup>(1)] , E(*<sup>i</sup>*, *l*, *m*, *n*) = <sup>H</sup>*i*(*xm*)[P*l*,<sup>2</sup>(*yn*) − *yn*P*l*,<sup>2</sup>(1)] , F(*<sup>i</sup>*, *l*, *m*, *n*) = [P*i*,<sup>2</sup>(*xm*) − *xm*P*i*,<sup>2</sup>(1)] H*l*(*yn*), L(*<sup>m</sup>*, *n*)=(<sup>1</sup> − *θ*) *wjxx* + *wjyy* , <sup>+</sup>B(*xm*, *yn*, *tj*+<sup>1</sup>) + <sup>2</sup>*Aδ<sup>w</sup><sup>j</sup>* − *<sup>A</sup>δ<sup>w</sup>j*−<sup>1</sup> − *Aδ j* ∑ *k*=1 *wj*−*k*+<sup>1</sup> − 2*wj*−*<sup>k</sup>* + *wj*−*k*−<sup>1</sup> *<sup>B</sup>*(*k*), M(*<sup>m</sup>*, *n*) = −*Aδynwj*+<sup>1</sup> *x* (*xm*, 1) − *ynwj*+<sup>1</sup>(0, 1)+(1 − *yn*)1*wj*+<sup>1</sup>(*xm*, 0) − *wj*+<sup>1</sup>(0, 0)2 + *xmwj*+<sup>1</sup>(1, *yn*) − *xmwj*+<sup>1</sup>(0, *yn*) − *xmyn*1*wj*+<sup>1</sup>(1, 1) − *wj*+<sup>1</sup>(0, 1)2 + *xm* (*yn* − 1) *wj*+<sup>1</sup>(1, 0) + *xm*(<sup>1</sup> − *yn*)*wj*+<sup>1</sup>(0, 0) + *wj*+<sup>1</sup>(0, *yn*) + *<sup>θ</sup>ynwj*+<sup>1</sup> *xx* (*xm*, 1) + (1 − *yn*)*wj*+<sup>1</sup> *xx* (*xm*, 0) + *xmwj*+<sup>1</sup> *yy* (1, *yn*) + (1 − *xm*) *wj*+<sup>1</sup> *yy* (0, *yn*).

Equation (30) represents 2*M* × 2*M* equations in so many unknowns which can be solved easily. After calculation of these unknowns, an approximate solution can be obtained from Equation (29).

#### **4. Stability Analysis**

Here we present the stability analysis of the proposed scheme for (1 + 2)-dimensional problems; a similar result can be proved for (1 + 1)-dimensional problems. In matrix form Equations (25), (26) and (29) can be written as

$$w\_{xx}^{j+1} = \mathcal{U}\mathfrak{a}^{j+1} + \mathcal{U}^{j+1},\tag{31}$$

$$w\_{yy}^{j+1} = \mathcal{V}a^{j+1} + \mathcal{V}^{j+1},\tag{32}$$

$$w^{j+1} = \mathcal{Z}u^{j+1} + \mathcal{Z}^{j+1},\tag{33}$$

where *αj*+<sup>1</sup> = *<sup>α</sup>j*+<sup>1</sup>(*<sup>i</sup>*, *l*), U, V, Z and U˜ *j*+1, V˜ *j*+1, Z˜*j*+<sup>1</sup> are interpolation matrices of *wj*+<sup>1</sup> *xx* , *wj*+<sup>1</sup> *yy* , *wj*+<sup>1</sup> at collocation points and boundary terms, respectively. Now using Equations (31), (32) and (33) in Equation (20), we ge<sup>t</sup>

$$
\left[A\_{\delta}Z-\theta\left(\mathcal{U}+\mathcal{V}\right)\right]a^{j+1}=\left[2A\_{\delta}Z+(1-\theta)\left(\mathcal{U}+\mathcal{V}\right)\right]a^{j}+\mathcal{G}^{j+1},\tag{34}
$$

where G*j*+<sup>1</sup> = −*Aδ*Z˜*j*+<sup>1</sup> + *θ*-U˜ *j*+1 + V˜ *<sup>j</sup>*+<sup>1</sup> + 2*Aδ* ˜ Z*j* + (1 − *θ*)( ˜ U*j* + ˜ V*j*) + **B***j*+<sup>1</sup> − *<sup>A</sup>δ<sup>w</sup>j*−<sup>1</sup> − *Aδ* <sup>∑</sup>*jk*=<sup>1</sup> *wj*−*k*+<sup>1</sup> − 2*wj*−*<sup>k</sup>* + *<sup>w</sup>j*−*k*−<sup>1</sup>*B*(*k*).

Now From Equation (34) one can write

$$a^{j+1} = \mathcal{C}^{-1} \mathcal{T} a^j + \mathcal{C}^{-1} \mathcal{G}^{j+1},\tag{35}$$

where C = -*<sup>A</sup>δ*<sup>Z</sup> − *θ*-U + <sup>V</sup>., T = 2*Aδ*Z + (1 − *θ*)-U + <sup>V</sup>. Putting Equation (35) in Equation (33) we ge<sup>t</sup>

$$w^{j+1} = \mathcal{Z}\mathcal{C}^{-1}\mathcal{T}a^j + \mathcal{Z}\mathcal{C}^{-1}\mathcal{G}^{j+1} + \mathcal{Z}^{j+1}.\tag{36}$$

Using Equation (33) in Equation (36) we have

$$w^{j+1} = \mathcal{ZC}^{-1} \mathcal{T} \mathcal{Z}^{-1} w^j - \mathcal{ZC}^{-1} \mathcal{T} \mathcal{Z}^{-1} \tilde{\mathcal{Z}}^j + \mathcal{ZC}^{-1} \mathcal{G}^{j+1} + \tilde{\mathcal{Z}}^{j+1}.\tag{37}$$

The above equation shows a recurrence relation of a full discretization scheme which allow us refinement in time. If *w*˜ *j*+1 is numerical solution then

$$
\hat{w}^{j+1} = \mathcal{Z}\mathcal{C}^{-1}\mathcal{T}\mathcal{Z}^{-1}\tilde{w}^{j} - \mathcal{Z}\mathcal{C}^{-1}\mathcal{T}\mathcal{C}^{-1}\mathcal{Z}^{j} + \mathcal{Z}\mathcal{C}^{-1}\mathcal{G}^{j+1} + \mathcal{Z}^{j+1}.\tag{38}
$$

Let *ej*+<sup>1</sup> = *wj*+<sup>1</sup> − *w*˜ *j*+1 be the error at (*j* + 1)th time level. Subtracting Equation (37) from Equation (38) then

$$e^{j+1} = \Lambda e^j,$$

where Λ = ZC−<sup>1</sup>T Z−<sup>1</sup> is the amplification matrix. According to Lax-Richtmyer criterion, the scheme will be stable if Λ ≤ 1. It has been verified computationally that Λ ≤ 1. For *J* = 1 the spectral radius is 0.01025 which lies in the stability domain.

#### **5. Convergence Analysis**

The convergence analysis of scheme (18) and (29) is similar to the following theorems, therefore the proofs are omitted.

**Lemma 1** (see [24])**.** *If w*(*x*) ∈ *L*<sup>2</sup>(*R*) *with w*(*x*) ≤ *ρ, for all x* ∈ (0, <sup>1</sup>)*, ρ* > 0 *and w*(*x*) = ∑∞*<sup>i</sup>*=<sup>0</sup> *ai*H*i*(*x*) *then* | *ai* |≤ *ρ* 2*j*+<sup>1</sup> .

**Lemma 2** (see [25])**.** *If f*(*<sup>x</sup>*, *y*) *satisfies a Lipschitz condition on* [0, 1] × [0, 1]*, that is, there exists a positive L such that for all* (*<sup>x</sup>*1, *y*),(*<sup>x</sup>*2, *y*) ∈ [0, 1] × [0, 1] *we have* | *f*(*<sup>x</sup>*1, *y*) − *f*(*<sup>x</sup>*2, *y*) |≤ *L* | *x*1 − *x*2 | *then*

$$a\_{i,l}^2 \le \frac{L^2}{2^{4j+4}m^2}$$

**Theorem 1.** *If w*(*x*) *and <sup>w</sup>*2*M*(*x*) *are the exact and approximate solution of Equation (1), then the error norm EJ at Jth resolution level is*

$$\|\|E\_{\mathcal{I}}\|\| \le \frac{4\rho}{3} \left(\frac{1}{2^{\mathcal{I}+1}}\right)^2. \tag{39}$$

**Proof.** See [26].

**Theorem 2.** *Assume <sup>w</sup>*(*<sup>x</sup>*, *y*) *and <sup>w</sup>*2*<sup>M</sup>*(*<sup>x</sup>*, *y*) *be the exact and approximate solution of Equation (3), then*

$$\|\|\|\|\|\|\_{I} \|\| \leq \frac{L}{4\sqrt{255}} \frac{1}{2^{4/\delta}}.\tag{40}$$

**Proof.** See [27].

#### **6. Illustrative Test Problems**

In this part, we chose some test problem to confirm the reliability and efficiency of the present scheme. For validation of our results *L*∞ and *L*2 error norm are figured out which are defined as follows:

$$L\_2 = \sqrt{\sum\_{i=1}^{2M} \left( w^{ext} - w^{app} \right)^2}, \quad L\_\infty = \max\_{1 \le i \le 2M} \left| w^{ext} - w^{app} \right|, \tag{41}$$

where *wapp* and *wext* are respectively approximate and exact solutions. *Problem 5.1*

> Let us take the following (1 + 1)-dimensional TFDWE with damping

$$\mathbf{u}^{\varepsilon}D\_{t}^{\delta}w(\mathbf{x},t) \quad = -w\_{l}(\mathbf{x},t) + w\_{\text{xx}}(\mathbf{x},t) + \mathcal{A}(\mathbf{x},t), \quad \mathbf{x} \in [0,1], \quad t \in [0,1], \quad 1 < \delta \le 2,\tag{42}$$

with A(*<sup>x</sup>*, *t*) = <sup>2</sup>*x*(<sup>1</sup>−*<sup>x</sup>*)*<sup>t</sup>* 2−*δ* <sup>Γ</sup>(<sup>3</sup>−*<sup>δ</sup>*) + 2*tx*(1 − *x*) + 2*t* 2. Initial and boundary conditions are derived from the exact solution *<sup>w</sup>*(*<sup>x</sup>*, *t*) = *t* <sup>2</sup>*x*(<sup>1</sup> − *<sup>x</sup>*). This problem has been solved for parameters *J* = 4, *t* = 0.01, 0.1, 1, *δ* = 1.1, 1.3, 1.5, 1.7, 1.9. The obtained error norms are shown in Table 1. From table it is obvious that results of the present scheme match well with exact solution. Also in Table 2 it has been observed that accuracy increases with increasing resolution level which shows the convergence in the spatial direction. In the same table, the results have been matched with existing results in the literature which clarify that computed solutions are in good agreemen<sup>t</sup> with the work of Chen et al. [28]. Table 3 shows convergence in time for fixed *dx* = 1/32. The convergence rate of the proposed scheme has been addressed in Table 4. the graphical solution and error plot are given in Figure 1. From this Figure it is clear that approximate solutions are matchable with exact.

**Table 1.** Error norms of problem 5.1 for at *J* = 4.


**Table 2.** Comparison of maximum error of problem 5.1 with previous work at *t* = 1 and *δ* = 1.7.


**Table 3.** Error norms of problem 5.1 for different values of *τ* and *δ*.



**Table 4.** Convergence rate of maximum error of problem 5.1 at *t* = 1 and *δ* = 1.7.

**Figure 1.** Graphical behaviour of problem 5.1 when *t* = 1, *δ* = 1.5.

*Problem 5.2:*

> Consider the following TFDWE with damping

$$\mathbf{u}^{\varepsilon}D\_{t}^{\delta}w(\mathbf{x},t) \ = -w\_{l}(\mathbf{x},t) + w\_{\text{xx}}(\mathbf{x},t) + \mathcal{A}(\mathbf{x},t), \quad \mathbf{x} \in [0,1], \quad t \in [0,1], \quad 1 < \delta \le 2,\tag{43}$$

coupled with initial and boundary conditions

$$\begin{cases} w(\mathbf{x},0) = 0, \quad w\_l(\mathbf{x},0) = 0 \quad \mathbf{x} \in (0,1) \\ w(0,t) = t^3, \quad w(1,t) = et^3, \quad t \in [0,1]. \end{cases} \tag{44}$$

The exact solution and source term are given by *<sup>w</sup>*(*<sup>x</sup>*, *t*) = *ext*<sup>3</sup> and A(*<sup>x</sup>*, *t*) = 6*t*3−*δe<sup>x</sup>* <sup>Γ</sup>(<sup>4</sup>−*<sup>δ</sup>*) + 3*t*2*e<sup>x</sup>* − *t*3*ex*. In Table 5 the obtained error norms are shown for parameters *t* = 0.01, 0.1, *δ* = 1.1, 1.3, 1.5, 1.7, 1.9, *J* = 4. Table 5 shows that exact and approximate solutions agree with each other. The solution profile and absolute error are displayed Figure 2. From the Figure, the coincidence of both solutions are visible.

*δ t* **= 0.01,** *τ* **= 0.0001** *t* **= 0.1,** *τ* **= 0.001** *L***∞** *L***2** *L***∞** *L***2** 1.1 1.7079 × 10−<sup>7</sup> 6.8446 × 10−<sup>7</sup> 1.2504 × 10−<sup>4</sup> 5.3397 × 10−<sup>4</sup> 1.3 6.5331 × 10−<sup>7</sup> 2.5683 × 10−<sup>6</sup> 4.4278 × 10−<sup>4</sup> 1.8777 × 10−<sup>3</sup> 1.5 1.2494 × 10−<sup>6</sup> 4.7989 × 10−<sup>6</sup> 8.7071 × 10−<sup>4</sup> 3.6354 × 10−<sup>3</sup> 1.7 1.3386 × 10−<sup>6</sup> 5.0827 × 10−<sup>6</sup> 1.0489 × 10−<sup>3</sup> 4.2541 × 10−<sup>3</sup> 1.9 5.6739 × 10−<sup>7</sup> 2.1936 × 10−<sup>6</sup> 5.1085 × 10−<sup>4</sup> 2.0046 × 10−<sup>3</sup>

**Table 5.** Error norms of problem 5.2 at *J* = 4.

*Mathematics* **2019**, *7*, 923

**Figure 2.** Graphical behaviour of problem 5.2 at *t* = 0.3, *δ* = 1.1.

*Problem 5.3:*

> Now we consider (1+2)-dimensional TFDWE [29]

$$\Delta^{\mathbb{C}}D\_t^{\delta}w(\mathbf{x},y,t) = \Delta w(\mathbf{x},y,t) + \mathcal{B}(\mathbf{x},y,t), \quad (\mathbf{x},y) \in [0,1] \times [0,1], \quad t \in [0,1], \quad 1 < \delta \le 2,\tag{45}$$

with exact solution *<sup>w</sup>*(*<sup>x</sup>*, *y*, *t*) = *sin*(*πx*)*sin*(*πy*)*tδ*+3, and source term

$$\mathcal{B}(x, y, t) = \sin(\pi x) \sin(\pi y) \left[ \frac{\Gamma(\delta + 3)t^2}{2} - 2t^{\delta + 2} \right].$$

We solved this problem for resolution level *J* = 4 and the obtained results are recorded in Table 6 for different values of time and *τ*. From Table 6 it is clear that the proposed scheme works well for the solution of two dimensional problems. Table 7 shows the comparison of the computed results with the previous work of Zhang [29]. One can see that our results are matchable with existing results. The same table presents convergence in time for (1 + 2)-dimensional problems. The graphical solution and absolute error of the problem are shown in Figure 3. It is obvious from Figure 3 that the exact and approximate solutions have strong agreement.


**Table 6.** Comparison of problem 5.4 at *t* = 1 and *δ* with previous results.


**Table 7.** Error norms of problem 5.3 for different values of *τ* and *δ*.

**Figure 3.** Graphical behaviour of problem 5.3 when *t* = 0.5, *δ* = 1.9.

*Problem 5.4:*

> Consider the following TFDWE with reaction term [19]

$$\mathbf{u}^{\varepsilon}D\_t^{\delta}w(\mathbf{x},t) + w(\mathbf{x},t) \ = w\_{\text{xx}}(\mathbf{x},t) + \mathcal{A}(\mathbf{x},t), \quad \mathbf{x} \in [0,1], \quad t \in [0,1], \quad 1 < \delta \le 2,\tag{46}$$

coupled with initial and boundary conditions

$$\begin{cases} w(\mathbf{x},0) = 0, \quad w\_t(\mathbf{x},0) = 0 \quad \mathbf{x} \in (0,1) \\ w(0,t) = o, \quad w(1,t) = 0, \quad t \in [0,1] \end{cases} \tag{47}$$

where the forcing terms are A(*<sup>x</sup>*, *t*) = 2*t* <sup>2</sup>−*δx*(<sup>1</sup>−*<sup>x</sup>*) <sup>Γ</sup>(<sup>3</sup>−*<sup>δ</sup>*) + *t* <sup>2</sup>*x*(<sup>1</sup> − *x*) − 2*t* 2. This problem has been solved with the help of the proposed scheme. In Table 8 we presented the solutions at different points. Also the obtained results have been compared with the work presented in Reference [19]. It is clear from table that our results are more accurate. From the table it is also obvious that the exact and numerical solutions are in good agreement. Exact verses numerical solutions are plotted in Figure 4. Graphical solutions also indicate that the proposed scheme works in the case where the reaction term exists.

**Table 8.** Absolute error at different points of example 5.4 at *τ* = 0.001.


**Figure 4.** Graphical behaviour of problem 5.4 at *δ* = 1.1, *t* = 1.
