A parameter-uniform numerical method for a Sobolev problem with initial layer

Abstract

The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a linear Sobolev or pseudo-parabolic equation with initial jump. In order to obtain an efficient method, to provide good approximations with independence of the perturbation parameter, we have developed a numerical method which combines a finite difference spatial discretization on uniform mesh and the implicit rule on Shishkin mesh(S-mesh) for the time variable. The fully discrete scheme is shown to be convergent of order two in space and of order one expect for a logarithmic factor in time, uniformly in the singular perturbation parameter. Some numerical results confirming the expected behavior of the method are shown.

Error Estimates for Differential Difference Schemes to Pseudo-Parabolic Initial-Boundary Value Problem with Delay

“…For a discussion of existence and uniqueness results of pseudo-parabolic equations see [6,8,13,18]. Various finite difference schemes have been constructed to treat such problems [1][2][3][4] For example in [10] two difference approximation schemes to a nonlinear pseudo-parabolic equation are developed. Each of these schemes possesses a unique solution which can be obtained by an iterative procedure.…”

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

“…L. Govindarao and J. Mohapatra [30] have suggested a numerical scheme comprised of implicit-trapezoidal scheme on temporal direction and hybrid type scheme on spatial direction for solving singularly perturbed delay parabolic initial-boundary value problems. In the paper [6], a fully discrete scheme has been generated on Shishkin mesh to solve singularly perturbed Sobolev initialboundary value problem with initial jump. S. Kumar and M. Kumar [40] have discretized singularly perturbed nonlinear delay parabolic type partial differential equations on a generalized Shishkin mesh by using quasilinearization techniques.…”

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

“…Such problems are interesting not only because they are generalizations of a standard parabolic problem, but also because they arise naturally in a large variety of applications. Various numerical schemes have been constructed to treat PPEs in [2,3,5,6,10,12,14,15,23] (see also c 2019 Miskolc University Press the references cited in them). Not only the existence, uniqueness and nonexistence results for pseudo-parabolic equations were obtained, but also the asymptotic behavior, regularity and others properties of solutions were investigated.…”

“…The equivalence of the three different formulations for the PPEs and different time discrete (implicit or semi-implicit) numerical schemes has been discussed in [14]. The one-dimensional initial-boundary value problem for a linear PPEs with initial jump is studied in [5]. They developed a numerical method which combines a finite difference spatial discretization on uniform mesh and the implicit rule on Shishkin mesh (S-mesh) for the time variable.…”