IJPAM: Volume 55, No. 2 (2009)

ELEMENT METHOD FOR NUMERICAL SOLUTION OF
SINGULARLY PERTURBED DELAY
DIFFERENTIAL EQUATIONS



Indian Institute of Technology
Kanpur, Kanpur, 208016, INDIA


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