Numerical analysis of singularly perturbed delay differential equations with layer behavior
type: Item
instance of: scholarly article
title: Numerical analysis of singularly perturbed delay differential equations with layer behavior
zbMATH Open document ID: 1069.65086
author: M. K. Kadalbajoo
author: Kapil K. Sharma
published in: Applied Mathematics and Computation
publication date: 2004-11-23T00:00:00Z
review text: The problem under consideration is the singularly perturbed boundary value problem (BVP) for the delay differential equation \[ \varepsilon y''(x)+a(x)y'(x-\delta)+b(x)y(x)=f(x), \quad 0 < x < 1, \] under the boundary conditions \[ {y(x)}=\phi(x),\quad -\delta\leq x\leq 0, \quad y(1)=\gamma, \] where \(\varepsilon\) and \(\delta\) are small positive parameters. The stated BVP for the delay differential equation is approximated by one for the ordinary differential equation (ODE), created by replacing the retarded term \(y'(x-\delta)\) by its first order Taylor approximation \(y'(x)-\delta y''(x)\). The approximate BVP for the ODE is approximated by a standard three points difference scheme. The stability and convergence of the method is discussed for two cases corresponding to the location the boundary layer, on the left side (when \(a(x)>0\)) and on the right (when \(a(x)< 0\)). Numerical examples are presented.
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 34K28
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34K26
Mathematics Subject Classification ID: 34K10
zbMATH DE Number: 2117974
zbMATH Keywords: delay differential equation
zbMATH Keywords: negative shift
zbMATH Keywords: boundary layer
zbMATH Keywords: singular perturbation
zbMATH Keywords: differential-difference equation
zbMATH Keywords: difference scheme
zbMATH Keywords: stability
zbMATH Keywords: convergence
zbMATH Keywords: numerical examples
reviewed by: Vladimir Gorbunov
MaRDI profile type: Publication
full work available at URL: j.amc.2003.06.012
OpenAlex ID: W2086475810
cites work: Q4124657
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DOI: 10.1016/J.AMC.2003.06.012
description: scientific article; zbMATH DE number 2117974