Second-order difference scheme for hyperbolic equations with unbounded delay

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Abstract

The present paper is devoted to the study the initial value problem for the hyperbolic equation with unbounded time delay term \( \begin{equation*}
\begin{cases}
\dfrac{d^{2}v(t)}{dt^{2}}+A^{2}v(t)=a\left( \dfrac{dv(t-\omega )}{dt}
+Av(t-\omega )\right) +f(t), &  t>0, \\
v(t)=\varphi (t), & -\omega \leq t\leq 0
\end{cases}
\end{equation*} \) in a Hilbert space H with a self-adjoint positive definite operator A. The second order of accuracy difference scheme for the numerical solution of the differential problem is presented. The main theorem on stability estimates for the solutions of this difference scheme is established. In practice, the stability estimates for solutions of four problems for hyperbolic difference equations with time delay are proved.

About the authors

A. Ashyralyev

Bahcesehir University; RUDN University; Institute of Mathematics and Mathematical Modeling

Author for correspondence.
Email: allaberen.ashyralyev@bau.edu.tr
ORCID iD: 0000-0002-4153-6624
Scopus Author ID: 6602401828
ResearcherId: K-4377-2017
Istanbul, Turkiye; Moscow, Russia; Almaty, Kazakhstan

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