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Bounded Solutions of Semilinear Time Delay Hyperbolic Differential and Difference Equations

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  • Allaberen Ashyralyev

    (Department of Mathematics, Near East University, Mersin 10 99138, Turkey
    Department of Mathematics, Peoples’ Friendship University Russia, Moscow 117198, Russia
    Institute of Mathematics and Mathematical Modeling, Almaty 050010, Kazakhstan)

  • Deniz Agirseven

    (Department of Mathematics, Trakya University, Edirne 22030, Turkey)

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.

Suggested Citation

  • Allaberen Ashyralyev & Deniz Agirseven, 2019. "Bounded Solutions of Semilinear Time Delay Hyperbolic Differential and Difference Equations," Mathematics, MDPI, vol. 7(12), pages 1-38, December.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:12:p:1163-:d:293210
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    References listed on IDEAS

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    1. Liang, Hui, 2015. "Convergence and asymptotic stability of Galerkin methods for linear parabolic equations with delays," Applied Mathematics and Computation, Elsevier, vol. 264(C), pages 160-178.
    2. Allaberen Ashyralyev & Pavel E. Sobolevskii, 2005. "Two new approaches for construction of the high order of accuracy difference schemes for hyperbolic differential equations," Discrete Dynamics in Nature and Society, Hindawi, vol. 2005, pages 1-31, January.
    3. A. Ashyralyev & J. Pastor & S. Piskarev & H. A. Yurtsever, 2015. "Second Order Equations in Functional Spaces: Qualitative and Discrete Well-Posedness," Abstract and Applied Analysis, Hindawi, vol. 2015, pages 1-63, July.
    4. Hao, Zhaopeng & Fan, Kai & Cao, Wanrong & Sun, Zhizhong, 2016. "A finite difference scheme for semilinear space-fractional diffusion equations with time delay," Applied Mathematics and Computation, Elsevier, vol. 275(C), pages 238-254.
    5. Deniz Agirseven, 2012. "Approximate Solutions of Delay Parabolic Equations with the Dirichlet Condition," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-31, June.
    6. H. Chi & H. Poorkarimi & J. Wiener & S. M. Shah, 1989. "On the exponential growth of solutions to non-linear hyperbolic equations," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 12, pages 1-7, January.
    7. Пигнастый, Олег & Koжевников, Георгий, 2019. "Распределенная Динамическая Pde-Модель Программного Управления Загрузкой Технологического Оборудования Производственной Линии [Distributed dynamic PDE-model of a program control by utilization of t," MPRA Paper 93278, University Library of Munich, Germany, revised 02 Feb 2019.
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