Crank -Nicolson scheme for solving a system of singularly perturbed partial differential equations of parabolic type

S.Parthibana,* S.Valarmathib and V.Franklinc

Crank -Nicolson scheme for solving a system of singularly perturbed partial differential equations of parabolic type

Authors :

S.Parthiban^a,* S.Valarmathi^b and V.Franklin^c

Author Address :

^a,b,cDepartment of Mathematics, Bishop Heber College, Tiruchirappalli-620 017, Tamil Nadu, India.

Abstract :

A singularly perturbed boundary value problem (SPBVP) for a system of two linear parabolic second order differential equations of convection-diffusion type is considered. Since the second order space derivative of each equation is multiplied by distinct singular perturbation parameters, the components of the solution exhibit overlapping layers. In this work, a method which comprises the Crank- Nicolson scheme to discretise time variable on a uniform mesh and standard central difference scheme on a Shishkin piecewise uniform mesh to discretise space variable is suggested to obtain numerical approximations to the solution of the continuous problem. The numerical solution obtained using the suggested method is second order convergent in time and first order convergent in space.

Keywords :

Singular perturbation problems, parabolic problems, boundary layers, Crank - Nicolson scheme, Shishkin mesh.

DOI :

Article Info :

Received : July 10, 2015; Accepted : September 23, 2015.