Next Article in Journal
Approximate Solutions for Fractional Boundary Value Problems via Green-CAS Wavelet Method
Previous Article in Journal
Characterizations of Positive Operator-Monotone Functions and Monotone Riemannian Metrics via Borel Measures
Previous Article in Special Issue
Exact and Nonstandard Finite Difference Schemes for Coupled Linear Delay Differential Systems
Open AccessArticle

Bounded Solutions of Semilinear Time Delay Hyperbolic Differential and Difference Equations

1
Department of Mathematics, Near East University, Mersin 10 99138, Turkey
2
Department of Mathematics, Peoples’ Friendship University Russia, Moscow 117198, Russia
3
Institute of Mathematics and Mathematical Modeling, Almaty 050010, Kazakhstan
4
Department of Mathematics, Trakya University, Edirne 22030, Turkey
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(12), 1163; https://doi.org/10.3390/math7121163
Received: 28 October 2019 / Revised: 25 November 2019 / Accepted: 27 November 2019 / Published: 2 December 2019
(This article belongs to the Special Issue Models of Delay Differential Equations)
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. View Full-Text
Keywords: semilinear problems with delay; hyperbolic equations; difference scheme; stability; Hilbert space semilinear problems with delay; hyperbolic equations; difference scheme; stability; Hilbert space
MDPI and ACS Style

Ashyralyev, A.; Agirseven, D. Bounded Solutions of Semilinear Time Delay Hyperbolic Differential and Difference Equations. Mathematics 2019, 7, 1163.

Show more citation formats Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Back to TopTop