Analysis of higher order difference method for a pseudo-parabolic equation with delay

Abstract

In this paper, the author considers the one dimensional initial-boundary problem for a pseudo-parabolic equation with time delay in second spatial derivative. To solve this problem numerically, the author constructs higher order difference method and obtain the error estimate for its solution. Based on the method of energy estimates the fully discrete scheme is shown to be convergent of order four in space and of order two in time. Some numerical examples illustrate the convergence and effectiveness of the numerical method.

A second-order numerical method for pseudo-parabolic equations having both layer behavior and delay argument

“…I. Amirali et. al [1] have constructed two-level difference scheme for semilinear pseudo-parabolic initial-boundary value problems with delay parameter (Please, see also a series of the papers [2,3,8]). C. Zhang and Z. Tan [65] have used linearized compact finite difference methods for solving nonlinear delay Sobolev partial differential equations.…”

Linearized compact difference methods for solving nonlinear Sobolev equations with distributed delay

“…This fact has aroused the interest of researchers to construct numerical methods for solving DSEs. For example, Okcu and Amiraliyev [20] derived a fourth‐order differential‐difference scheme for DSEs and obtained the error estimate of the scheme, Amiraliyev et al [21] considered a finite difference method for solving linear DSEs, Chiyaneh and Duru [22, 23] dealt with two finite difference methods based on the adaptive and uniform meshes, respectively, for solving linear singularly perturbed DSEs, Amirali [24] gave the error estimate of a higher order difference method for a class of linear DSEs, and Zhang and Tan [25] studied linearized compact difference methods (LCDMs) combined with Richardson extrapolation for solving one‐dimensional (1D) and two‐dimensional (2D) nonlinear DSEs.…”

“…dealt with two finite difference methods based on the adaptive and uniform meshes, respectively, for solving linear singularly perturbed DSEs, Amirali [24] gave the error estimate of a higher order difference method for a class of linear DSEs, and Zhang and Tan [25] studied linearized compact difference methods (LCDMs) combined with Richardson extrapolation for solving one-dimensional (1D) and two-dimensional (2D) nonlinear DSEs. Whereas, the above researches devoted only to the Sobolev equations with constant delay.…”