A New Unconditionally Stable Second Order of Accuracy Difference Scheme for the Time Delay Telegraph Equation

In this paper, we study a new unconditionally stable second order of accuracy difference scheme for the approximate solution of the initial value problem for the time delay telegraph equation in a Hilbert space with self-adjoint positive definite operator. We prove the main theorem on stability of this difference scheme. As an application, we present absolutely stable difference schemes for the approximate solution of two initial-boundary value problems for one-dimensional delay telegraph equation with nonlocal conditions and multidimensional delay telegraph equation with Dirichlet condition. Finally, to support the theoretical result, a numerical example of the initial-boundary value problem for the two-dimensional delay telegraph equation with Dirichlet condition is presented. © 2025 John Wiley & Sons Ltd.