Numerical treatment of singularly perturbed delay reaction-diffusion equations with twin layers and oscillatory behaviour

Abstract

In this thesis, second and fourth order parametric uniform numerical methods are presented for solving singularly perturbed delay reaction-diffusion equations with twin layers and oscillatory behaviour for which a small shift ( )  is in the reaction term. First, the given singularly perturbed delay reaction-diffusion equation is converted into an asymptotically equivalent singularly perturbed boundary value problem by using the Taylor series expansion for the delay term as the delay parameter is sufficiently small. Using the finite difference approximations the given differential equation is transformed to a three-term recurrence relation, which can easily be solved by using Thomas Algorithm. The stability and ε-uniform convergence of the methods have been established. To validate the applicability of the proposed methods, four model examples without exact solution have been considered and solved for different values of parameters  and  and mesh sizes h . Both theoretical error bounds and numerical rate of convergence have been established for the methods. The numerical results have been presented in tables and further to examine the effect of delay on the twin boundary layer and oscillatory behavior of the solution, graphs have been given for different values of  . In a nutshell, the present methods gives better results than some existing numerical methods reported in the literature.