A Parameter Uniform Numerical Scheme for Singularly Perturbed Parabolic Differential Equation with Discontinuous Coefficients and Negative Shift

In this study, a parameter uniform numerical scheme for a singularly perturbed delay parabolic partial differential equation with discontinuous coefficients involving large negative shift is presented. Due to the presence of singular perturbation parameter and discontinuity in the problem, it is a difficult task to solve this problem analytically. Consequently, a direct or analytical method for singularly perturbed delay parabolic partial differential equation with discontinuous coefficients involving large negative shift is still lacking, so one has to rely on suitable numerical methods to solve such problems. Therefore, it is noteworthy to develop ε-uniform convergent numerical methods for such types of problems. The main aim of this study is to develop and analyze a parameter uniform numerical scheme based on fitted techniques for solving singularly perturbed delay parabolic partial differential equation with discontinuous coefficients involving large negative shift. The proposed method consists of the implicit Euler method for temporal discretization and special designed finite difference method for spatial discretization on a uniform mesh step size. The stability and parameter uniform convergence of the proposed method is carried out. The proposed scheme gives an accurate, stable and -uniform numerical result. The error analysis of the proposed scheme was carried out and proved that the scheme is uniformly convergent of first order in temporal and second order in spatial variables. To demonstrate applicability of the method some numerical examples are considered. The maximum absolute errors and rate of convergence for various perturbation parameters and uniform mesh size values are tabulated for two model examples. Further, the numerical results depict that the present method is more convergent than some methods available in the literature.