*6.2. Two-Dimensional Case*

For the numerical experiment, we consider the mixed boundary value problem

$$\begin{cases} \frac{\partial^2 u(t, y)}{\partial t^2} - \frac{\partial^2 u(t, x, y)}{\partial x^2} - \frac{\partial^2 u(t, y)}{\partial y^2} = 2e^{-t} \sin x \sin y + \cos \left( u \left( t, x, y \right) u \left( t - 1, x, y \right) \right) \\\\ - \cos \left( e^{-t} \sin x \sin y \left( t - 1, x, y \right) \right), \ 0 < t < \infty, \ 0 < x, y < \pi, \\\\ u \left( t, x, y \right) = e^{-t} \sin x \sin y, \ 0 \le x, y \le \pi, \ -1 \le t \le 0, \\\\ u \left( t, 0, y \right) = u \left( t, \pi, y \right) = 0, \ 0 \le y \le \pi, \ t \ge 0, \\\\ u \left( t, x, 0 \right) = u \left( t, x, \pi \right) = 0, \ 0 \le x \le \pi, \ t \ge 0 \end{cases} \tag{56}$$

for the semilinear two dimensional delay hyperbolic equation. The exact solution of problem (56) is *u* (*t*, *x*) = *e*−*<sup>t</sup>* sin *x* sin *y*. We will consider the following iterative difference scheme of first order of approximation in *t* for the numerical solution of the initial-boundary value problem (56)

⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *ju<sup>k</sup>*+<sup>1</sup> *<sup>n</sup>*,*<sup>i</sup>* <sup>−</sup>2(*ju<sup>k</sup> n*,*i*)+*ju<sup>k</sup>*−<sup>1</sup> *n*,*i <sup>τ</sup>*<sup>2</sup> <sup>−</sup> *ju<sup>k</sup>*+<sup>1</sup> *n*+1,*i* <sup>−</sup>2(*ju<sup>k</sup>*+<sup>1</sup> *<sup>n</sup>*,*<sup>i</sup>* )+*ju<sup>k</sup>*+<sup>1</sup> *n*−1,*i <sup>h</sup>*<sup>2</sup> <sup>−</sup> *ju<sup>k</sup>*+<sup>1</sup> *n*,*i*+1−2(*ju<sup>k</sup>*+<sup>1</sup> *<sup>n</sup>*,*<sup>i</sup>* )+*ju<sup>k</sup>*+<sup>1</sup> *n*,*i*−1 *h*2 <sup>=</sup> <sup>2</sup>*e*−*tk* sin *xn* sin *xi* <sup>+</sup> cos *<sup>j</sup>*−1*u<sup>k</sup> n*,*i uk*−*<sup>N</sup> n*,*i* <sup>−</sup> cos *e*−*tk* sin *xn* sin *xiu<sup>k</sup>*−*<sup>N</sup> n*,*i* , *tk* = *kτ*, *xn* = *nh*, 1 ≤ *k* < ∞, 1 ≤ *n*, *i* ≤ *M* − 1, *Nτ* = 1, *Mh* = *π*, *uk <sup>n</sup>*,*<sup>i</sup>* <sup>=</sup> *<sup>e</sup>*−*tk* sin *xn* sin *xi*, *tk* <sup>=</sup> *<sup>k</sup>τ*, *xn* <sup>=</sup> *nh*, 0, *<sup>i</sup>* <sup>≤</sup> *<sup>M</sup>*, <sup>−</sup>*<sup>N</sup>* <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> 0, *ju<sup>k</sup>*+<sup>1</sup> *<sup>n</sup>*,*<sup>i</sup>* <sup>−</sup>*ju<sup>k</sup> n*,*i <sup>τ</sup>* <sup>−</sup> *<sup>τ</sup> h*2 *ju<sup>k</sup>*+<sup>1</sup> *<sup>n</sup>*+1,*<sup>i</sup>* <sup>−</sup>*<sup>j</sup> <sup>u</sup><sup>k</sup> <sup>n</sup>*+1,*<sup>i</sup>* − 2 *ju<sup>k</sup>*+<sup>1</sup> *<sup>n</sup>*,*<sup>i</sup>* <sup>−</sup>*<sup>j</sup> <sup>u</sup><sup>k</sup> n*,*i* +*<sup>j</sup> uk*+<sup>1</sup> *<sup>n</sup>*,*i*−<sup>1</sup> <sup>−</sup>*<sup>j</sup> <sup>u</sup><sup>k</sup> n*,*i*−1 − *τ h*2 *ju<sup>k</sup>*+<sup>1</sup> *<sup>n</sup>*,*i*+<sup>1</sup> <sup>−</sup>*<sup>j</sup> <sup>u</sup><sup>k</sup> <sup>n</sup>*,*i*+<sup>1</sup> − 2 *ju<sup>k</sup>*+<sup>1</sup> *<sup>n</sup>*,*<sup>i</sup>* <sup>−</sup>*<sup>j</sup> <sup>u</sup><sup>k</sup> n*,*i* +*<sup>j</sup> u*<sup>1</sup> *<sup>n</sup>*−1,*<sup>i</sup>* <sup>−</sup>*<sup>j</sup> <sup>u</sup><sup>k</sup> n*−1,*i* <sup>=</sup> *<sup>u</sup><sup>k</sup> n*,*i* <sup>−</sup>*uk*−<sup>1</sup> *n*,*i <sup>τ</sup>* , *k* = *mN* + 1, *m* = 0, 1, ..., *k* ≥ 1, *juk* 0,*<sup>i</sup>* =*<sup>j</sup> <sup>u</sup><sup>k</sup> <sup>M</sup>*,*<sup>i</sup>* <sup>=</sup> 0, 0 <sup>≤</sup> *<sup>i</sup>* <sup>≤</sup> *<sup>M</sup>*,*<sup>j</sup> <sup>u</sup><sup>k</sup> <sup>n</sup>*,0 =*<sup>j</sup> <sup>u</sup><sup>k</sup> <sup>n</sup>*,*<sup>M</sup>* = 0, 0 ≤ *n* ≤ *M*, 0 ≤ *k* < ∞, *j* = 1, 2, ... (57)

for the semilinear delay hyperbolic equation. Here and in the future *j* denotes the iteration index and an initial guess <sup>0</sup>*u<sup>k</sup> n*,*i* , *k* ≥ 1, 0 ≤ *n*, *i* ≤ *M* is to be made. For solving difference scheme (57), the numerical steps are given below. For 0 ≤ *k* < *N*, 0 ≤ *n*, *i* ≤ *M* the algorithm is as follows :

$$1. \quad j=1.$$

$$\text{2. }\_{j-1}\mathfrak{u}\_{n,i}^k \text{ is known.}$$


We write Equation (57) in matrix form

$$A\left(j\mu^{k+1}\right) + B\left(j\mu^{k}\right) + \mathbb{C}\left(j\mu^{k-1}\right) = Rq\langle\_{j-1}\mu^{k}, \mu^{k-N}\rangle, \quad 1 \le k < \infty,$$

$$u^{k} = \left\{e^{-t\_{k}}\sin\mathbf{x}\_{n}\sin\mathbf{x}\_{i}\right\}\_{n,j=0}^{M}, \quad -N \le k \le 0,\tag{58}$$

$$\frac{\mu^{k+1} - j\mu^{k}}{\tau} - \left\{\frac{\tau}{h^{2}}\left(j\mu\_{n+1,i}^{k+1} - j\mu\_{n+1,i}^{k} - 2\left(j\mu\_{n,i}^{k+1} - j\mu\_{n,i}^{k}\right) + j\mu\_{n,i-1}^{k+1} - j\mu\_{n,i-1}^{k}\right)\right\}\_{n,j=1}^{M-1}$$

$$-\left\{\frac{\tau}{h^{2}}\left(j\mu\_{n,i+1}^{k+1} - j\mu\_{n,i+1}^{k} - 2\left(j\mu\_{n,i}^{k+1} - j\mu\_{n,i}^{k}\right) + j\mu\_{n-1,i}^{k+1} - j\mu\_{n-1,i}^{k}\right)\right\}\_{n,j=1}^{M-1}$$

$$=\frac{\mathfrak{u}^k - \mathfrak{u}^{k-1}}{\tau},\ k = mN + 1, m = 0, 1, \ldots, k \ge 1.$$

Here *<sup>R</sup>*, *<sup>A</sup>*, *<sup>B</sup>*, and *<sup>C</sup>* are (*<sup>M</sup>* <sup>+</sup> <sup>1</sup>) <sup>×</sup> (*<sup>M</sup>* <sup>+</sup> <sup>1</sup>) <sup>×</sup> (*<sup>M</sup>* <sup>+</sup> <sup>1</sup>) given matrices and *<sup>ϕ</sup>*(*j*−1*uk*, *<sup>u</sup>k*−*N*) and *jus* , *s* = *k*, *k* ± 1 are given (*M* + 1) × (*M* + 1) × 1 column vectors. Therefore, we will use the same algorithm as the one dimensional case.

So, we have the initial value problem for the second order difference equation with respect to *k* with matrix coefficients. From Equations (53) and (54) it follows that

$$\mu^{k+1} = -A^{-1} \left( B\_j \mu^k - \mathbb{C}\_j \mu^{k-1} + A^{-1} \mathbb{R} \varphi^k(\!\_{j=1} \mu^k, \! \mu^{k-N}) \right), \ 1 \le k < \infty,$$

$$\mu^k = \left\{ e^{-t\_k} \sin \text{x}\_{\text{lt}} \sin \text{x}\_{\text{l}} \right\}\_{\text{n}; i=0}^M, \ -N \le k \le 0,$$

$$\mu^{k+1} = \psi \left( \mu^k, \mu^{k-1} \right), \ k = mN + 1, m = 0, 1, \ldots, k \ge 1. \tag{59}$$

Here, *ψ uk*, *uk*−<sup>1</sup> is the given (*M* + 1) × (*M* + 1) × 1 column vector.

In computations the initial guess is chosen as <sup>0</sup>*u<sup>k</sup>* <sup>=</sup> {sin *xn* sin *xi*}*<sup>M</sup> <sup>n</sup>*,*i*=<sup>0</sup> and when the maximum errors between two consecutive results of iterative difference scheme (57) become less than 10−6, the iterative process is terminated. We present numerical results for different values of *N* and *M* and *uk <sup>n</sup>*,*<sup>i</sup>* represent the numerical solutions of this difference scheme at (*tk*, *xn*, *xi*). The table is constructed for *N* = *M* = 20, 40, 80 in *t* ∈ [0, 1] , *t* ∈ [1, 2] , *t* ∈ [2, 3], respectively and the errors are computed by the following formula

$$mE\_{M,M}^N = \max\_{mN+1 \le k \le (m+1)N, 0 \le n, i \le M} \left| \mu \left( t\_{k'} \ge\_{n'} \ge\_{j} \right| \cdot \iota\_{n,i}^k \right|.$$

As can be seen from table, these numerical experiments support the theoretical statements. The number of iterations and maximum errors are decreasing with the increase of grid points.

In Table 2, as we increase values of *M* and *N* each time starting from *M* = *N* = 30 by a factor of 2 the errors in the first order of accuracy difference scheme decrease approximately by a factor of 1/2. The errors presented in tables indicate the the time convergence order is one. This result fits with the theoretical results perfectly.


**Table 2.** The errors of difference scheme (57) (Number of the iteration = *j*).

#### *6.3. Conclusions and Our Future Plans*

1. In the present paper, the main theorem on the existence and uniqueness of a bounded solution of the initial value problem for a semilinear hyperbolic equation with time delay in a Hilbert space with the self adjoint positive definite operator is established. In applications, the existence and uniqueness of a bounded solution of four problems for semilinear hyperbolic equations with time delay are obtained. A first order of accuracy difference scheme for the numerical solution of the abstract problem is presented. The theorem on the existence and uniqueness of an uniformly bounded solution of this difference scheme with respect to *τ* is established. In applications, the existence and uniqueness of a uniformly bounded solutions with respect to time and space stepsizes of difference schemes for four semilinear partial differential equations with time delay are obtained. Numerical results for the solution of difference schemes for one and two dimensional semilinear delay hyperbolic equation are presented.

2. We are interested in studying uniformly boundedness of solutions of high order of accuracy difference schemes uniformly with respect to time stepsize of approximate solutions of this initial-value problem, in which bounded solutions were established without any assumptions in respect to the

grid steps *τ* and *h*. We have not been able to establish such type of results for the solution of the very well-known second order difference scheme

$$\begin{cases} \frac{\underline{u}^{k+1} - 2\underline{u}^{k} + \underline{u}^{k-1}}{\underline{\tau}^{2}} + Au^{k} = f(t\_{k}, u^{k}, \frac{\underline{u}^{k-N} - \underline{u}^{k-N-1}}{\underline{\tau}}, \,\mathfrak{u}^{k-N}), \\\\ t\_{k} = k\tau, \; 1 \le k < \infty, \; N\tau = \omega, \\\\ \left(I + \tau^{2}A\right) \frac{\underline{u}^{k+1} - \underline{u}^{k}}{\underline{\tau}} = \frac{2\underline{u}^{k} - 3\underline{u}^{k-1} + \underline{u}^{k-2}}{\underline{\tau}}, \; k = mN, \; m = 1, \dots, \\\\ \left.u^{k} = q(t\_{k}), \; t\_{k} = k\tau, \; -N \le k \le 0. \end{cases}$$

Note that absolute stable two-step difference schemes of the high order of approximation for hyperbolic partial differential equations were presented and investigated in papers [11,54]. Applying methods of the present paper and papers [11,54] we can establish the similar stability and convergence results of this paper for the solution of the absolute stable two-step difference schemes of high order of approximation for semilinear delay hyperbolic equations.

3. Investigate the uniform to-step difference schemes and asymptotic formulas for the solution of initial value perturbation problem

$$\begin{cases} \varepsilon^2 u''(t) + Au(t) = f(t, u(t), u\_t(t - w), u(t - w)), & t > 0, \\\\ u(t) = \varrho(t), \ -w \le t \le 0 \end{cases}$$

for a semilinear delay hyperbolic equation in a Hilbert space *H* with the self adjoint positive definite operator *A* and with *ε* ∈ (0, ∞) parameter multiplying the highest order derivative term.

In [31], the uniform difference schemes and asymptotic formulas for the solution of initial value perturbation problem for a linear hyperbolic equation in a Hilbert space with the self adjoint positive definite operator and with *ε* ∈ (0, ∞) parameter multiplying the highest order derivative term were presented and investigated.

4. Investigate the initial value problem

$$\begin{cases} \begin{aligned} \boldsymbol{u}\_{ll}(t)dt + \boldsymbol{A}\boldsymbol{u}(t)dt &= \boldsymbol{f}(t, \; \boldsymbol{u}(t), \; \boldsymbol{u}\_{l}(t-w), \; \boldsymbol{u}(t-\omega))d\boldsymbol{w}\_{l}, \\\\ \boldsymbol{w}\_{l} &= \sqrt{\boldsymbol{t}}^{\boldsymbol{x}}\_{\boldsymbol{\xi}}, \; \boldsymbol{\xi} \in \boldsymbol{N}(0,1), \; t > 0, \\\\ \boldsymbol{u}(t) &= 0, \; -\omega \le t \le 0 \end{aligned} \end{cases}$$

for a semilinear stochastic hyperbolic equation with time delay in a Hilbert space *H* with the self adjoint positive definite operator *A*. Here, *wt* is a standard Wiener process given on the probability space (*Q*, *F*, *P*).

Note that absolute stable difference schemes for stochastic linear hyperbolic equations in Hilbert spaces were presented and investigated in [30].

Finally, in paper [55], a Lie algebra approach is applied to solve an SIS model where infection rate and recovery rate are time-varying. The method presented here has been used widely in chemical and physical sciences but not in epidemic applications due to insufficient symmetries.

**Author Contributions:** Investigation, A.A. and D.A.

**Funding:** This research was funded by "Russian Foundation for Basic Research (RFBR) grant number 16–01–00450."

**Acknowledgments:** The publication has been prepared with the support of the "RUDN University Program 5–100" and dedicated in memory of Pavel Evseevich Sobolevskii. The authors are grateful to Francisco Rodríguez and reviewers of this paper for the useful comments and relevant references.

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