Uniformly convergent numerical method for a singularly perturbed differential difference equation with mixed type


Çimen E.

BULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY-SIMON STEVIN, vol.27, no.5, pp.755-774, 2020 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 27 Issue: 5
  • Publication Date: 2020
  • Doi Number: 10.36045/j.bbms.200128
  • Title of Journal : BULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY-SIMON STEVIN
  • Page Numbers: pp.755-774
  • Keywords: Singular perturbation, differential difference equation, fitted difference method, Shishkin mesh, uniform convergence, BOUNDARY-VALUE-PROBLEMS, SMALL SHIFTS, DELAY

Abstract

In this paper, we deal with the singularly perturbed problem for a linear second order differential difference equation with delay as well as advance. In order to solve the problem numerically, we construct a new difference scheme by the method of integral identities with the use interpolating quadrature rules with remainder terms in integral form. Using an appropriately non-uniform mesh of Shishkin type, we find that the method is almost first order convergent in the discrete maximum norm with respect to the perturbation parameter. Furthermore, we present the numerical experiments that their results support of the theory.