A parameter uniform numerical method for a singularly perturbed initial value problem with Robin initial condition

Dinesh Selvaraj; Joseph Paramasivam Mathiyazhagan

doi:10.26637/MJM0802/0015

Abstract

In this paper an initial value problem for a singularly perturbed first order differential equation with Robin initial condition is considered on the interval (0;1]. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be first - order convergent in the maximum norm uniformly in the perturbation parameters. A numerical illustration is provided to support the theory.

Keywords:

Singular perturbation problems, Robin initial condition, Finite difference schemes, Shishkin mesh, Parameter uniform convergence.

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Dinesh Selvaraj Department of Mathematics, Bishop Heber College, Tiruchirappalli-620017, Tamil Nadu, India.
Joseph Paramasivam Mathiyazhagan Department of Mathematics, Bishop Heber College, Tiruchirappalli-620017, Tamil Nadu, India.

Pages: 414-420
Date Published: 01-04-2020
Vol. 8 No. 02 (2020): Malaya Journal of Matematik (MJM)

H.G.Roos, M.Stynes and L.Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer Verlag, 1996.

J.J.H. Miller, E.O'Riordan, and G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, World Scientific publishing Co.Pvt.Ltd., Singapore, 1996.

R. E. O'Malley, Introduction to Singular Perturbations, Academic Press, New York, 1974.

E. P. Doolan, J.J.H. Miller, W. H. A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, 1980 .

P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E.O'Riordan, and G.I.Shishkin, Robust Computational Techniques for Boundary Layers, Chapman and hall/CRC, Boca Raton, Florida,USA, 2000.

P. Maragatha Meenakshi, S. Valarmathi, and J.J.H. Miller, Solving a partially singularly perturbed initial value problem on shishkin meshes, Applied Mathematics and Computation, 215(2010), 3170-3180.

S. Valarmathi and J.J.H. Miller, A parameter-uniform finite difference method for a singularly perturbed initial value problem: a special case, Lecture Notes in Computational Science and Engineering, Springer-Verlag, 29(2009), 267-276.

N. Shivaranjani, N.Ramanujam, J.J.H. Miller, and S.Valarmathi, A parameter uniform method for an initial value problem for a system of singularly perturbed delay differential equations, Springer Proceedings in Mathematics and Statistics, 87(2014), 127-138.