Parameter uniform finite difference formulation with oscillation free for solving singularly perturbed delay parabolic differential equation via exponential spline.

Abstract

Objective: In this work, singularly perturbed time dependent delay parabolic convection-diffusion problem with Dirichlet boundary conditions is considered. The solution of this problem exhibits boundary layer at the right of special domain. In this layer the solution experiences steep gradients or oscillation so that traditional numerical methods may fail to provide smooth solutions. We developed oscillation free parameter uniform exponentially spline numerical method to solve the considered problem.

Results: In the temporal direction, the implicit Euler method is applied, and in the spatial direction, an exponential spline method with uniform mesh is applied. To handle the effect of perturbation parameter, an exponential fitting factor is introduced. For the developed numerical scheme, stability and uniform error estimates are examined. It is shown that the scheme is uniformly convergent of linear order in the maximum norm. Numerical examples are provided to illustrate the theoretical findings.