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.
