A numerical method to solve singularly perturbed linear parabolic second order delay differential equation of reaction-diffusion type

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

A numerical method to solve singularly perturbed linear parabolic second order delay differential equation of reaction-diffusion type

Authors :

S.Parthiban^a,* S.Valarmathi^band V.Franklin^c

Author Address :

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

*Corresponding author.

Abstract :

A singularly perturbed boundary value problem (SPP) for a linear parabolic second order delay differential equation of reaction-diffusion type is considered. Due to a singular perturbation parameter which multiplies the second order space derivative and the delay term that occurs in the space variable, the solution gives raise to boundary and interior layers respectively. As classical numerical methods fail to solve SPP, this work suggests 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. It is verified numerically that the solution obtained using the suggested method is second order convergent in time and first order convergent in space. Numerical illustrations are presented. Related works are found in [2] and [4].

Keywords :

Singular Perturbation problems, Delay differential equations, parabolic problems, boundary layers, interior layers, Crank - Nicolson scheme, Shishkin mesh.

DOI :

Article Info :

Received : June 23, 2015; Accepted : August 23, 2015.