IJPAM: Volume 55, No. 2 (2009)

AN $\epsilon-$UNIFORM RITZ-GALERKIN FINITE
ELEMENT METHOD FOR NUMERICAL SOLUTION OF
SINGULARLY PERTURBED DELAY
DIFFERENTIAL EQUATIONS

Mohan K. Kadalbajoo$^1$, Arjun Singh Yadaw$^2$
$^{1,2}$Department of Mathematics and Statistics
Indian Institute of Technology
Kanpur, Kanpur, 208016, INDIA
$^1$e-mail: kadal@iitk.ac.in
$^2$e-mail: asyadaw@iitk.ac.in


Abstract.A boundary value problem for second order singularly perturbed delay differential equation is considered with the delay and advance arguments that are sufficiently small. Such problems have earlier been tackled asymptotically by the researchers Lange and Miura [#!Lange1!#], [#!Lange2!#]. The numerical treatment of the problem is given in Kadalbajoo and Sharma[#!Kadalbajoo1!#], [#!Kadalbajoo2!#], [#!Kadalbajoo3!#], they have used fitted mesh finite difference scheme and shown the order of convergence is one. In this paper, we have taken a piecewise-uniform fitted mesh (Shishkin mesh) to resolve the boundary layer and we have shown that Ritz-Galerkin method has almost second order parameters-uniform convergence. Several test examples are solved to demonstrate the efficiency of the method and how the size of the delay and advance arguments affect the layer behavior of the solution.

Received: August 14, 2009

AMS Subject Classification: 34D15, 65L10, 76N20, 65L60

Key Words and Phrases: singularly perturbed, two point boundary value problems, boundary layer, delay differential equation, Shishkin mesh, finite element method

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 55
Issue: 2