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.
Keywords:hyperbolic equation, unbounded time delay, numerical solution, difference scheme, second order of accuracy, stability of solutions.
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