Bounded Solutions of Semilinear Time Delay Hyperbolic Differential and Difference Equations

Allaberen Ashyralyev and Deniz Agirseven
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Allaberen Ashyralyev: Department of Mathematics, Near East University, Mersin 10 99138, Turkey
Deniz Agirseven: Department of Mathematics, Trakya University, Edirne 22030, Turkey

Mathematics, 2019, vol. 7, issue 12, 1-38

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.

Keywords: semilinear problems with delay; hyperbolic equations; difference scheme; stability; Hilbert space (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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