Bounded Solutions of Semilinear Time Delay Hyperbolic Differential and Difference Equations

Allaberen Ashyralyev,Deniz Agirseven

doi:10.3390/math7121163

Abstract

In this paper, we study the initial value problem for a semilinear delay hyperbolic equation in Hilbert spaces with a self-adjoint positive definite operator. The mean theorem on the existence and uniqueness of a bounded solution of this differential problem for a semilinear hyperbolic equation with unbounded time delay term is established. In applications, the existence and uniqueness of bounded solutions of four problems for semilinear hyperbolic equations with time delay in unbounded term are obtained. For the approximate solution of this abstract differential problem, the two-step difference scheme of a first order of accuracy is presented. The mean theorem on the existence and uniqueness of a uniformly bounded solution of this difference scheme with respect to time stepsize 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 in unbounded term are obtained. In general, it is not possible to get the exact solution of semilinear hyperbolic equations with unbounded time delay term. Therefore, numerical results for the solution of difference schemes for one and two dimensional semilinear hyperbolic equation with time delay are presented. Finally, some numerical examples are given to confirm the theoretical analysis.

Highlights

Delay differential equations are used to model biological, physical, system engineering, and sociological processes as well as naturally occurring oscillatory systems.It is known that in differential and difference equations, the involvement of the delay term causes deep difficulties in the analysis of these equations
The stability estimates in Hölder norms for the solutions of the initial-boundary value problem for delay parabolic equations were established
Our goal in the present paper is to investigate the boundedness solution of problems for semilinear hyperbolic equations with unbounded time delay term

Summary

Introduction

Delay differential equations are used to model biological, physical, system engineering, and sociological processes as well as naturally occurring oscillatory systems (see, for examples, [1,2,3,4,5,6,7,8,9]). Ashyralyev, Agirseven, and Ceylan [19] investigated finding sufficient conditions for the existence and uniqueness of a bounded solution of the initial value problem for the semilinear delay parabolic equation in a Banach space. The method of operators as a tool for the study of the stability of the solution of local and nonlocal problems to hyperbolic differential and difference equations in Hilbert and Banach spaces has been systematically developed by many authors (see, e.g., [27,28,29,30,31,32,35,36,37]). The existence and uniqueness of a bounded solution of four problems for semilinear hyperbolic equations with time delay are obtained. Numerical results for the solution of difference schemes for one and two dimensional nonlinear hyperbolic equation with time delay are presented

Main Existence and Uniqueness Theorem of the Differential Problem

Applications

Applications of Theorem 8

Numerical Experiments

One Dimensional Case

Conclusions and Our Future Plans