Parameter-Robust Numerical Scheme for Time-Dependent Singularly Perturbed Reaction–Diffusion Problem with Large Delay. (25th January 2018)

• Record Type: Journal Article
• Authors: Bansal, Komal; Sharma, Kapil K.
• Is Part Of: Numerical functional analysis and optimization. Volume 39:Number 2(2018)
• Page Start: 127
• Page End: 154
• Publication Date: 2018-01-25
• Subjects: Convection–diffusion parabolic problem -- differential-difference equations -- finite difference scheme -- fitted mesh -- interior layer -- singular perturbation; 65L11 -- 65M12 -- 35K20; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2016.1277742
• Languages: English
• ISSNs: 0163-0563

Abstract

ABSTRACT: This work presents the development of numerical scheme for second-order time-dependent singularly perturbed reaction-diffusion problem with large delay in the undifferentiated term. These types of problems arise frequently in many areas of science and engineering that take into consideration the effect of present situation as well as the past history of the physical system. As the characteristics of the reduced problem ( 𝜀 = 0) corresponding to the original singularly perturbed problem considered here are parallel to the boundary of the domain this implies, parabolic layers exhibit in the solution. In this paper, we initiate the study of parabolic layers together with interior layers in the solution of singularly perturbed parabolic partial differential-difference equations due to propagation of singularity. Proposed numerical scheme comprised of finite difference scheme and piecewise uniform Shishkin mesh. The method is shown to be accurate of order ( M − 1 + N − 2 ( l n N ) 2 ), where M and N are the number of mesh elements in time and spatial direction, respectively. Proposed numerical scheme is proved to be parameter uniform convergent in the maximum norm. Numerical experiments have been performed to show the existence of interior layer due to large state-dependent delay argument in the reaction term and to confirm the predicted theory.