Uniformly convergent numerical scheme for singularly perturbed parabolic delay differential equations

¹ Department of Applied Mathematics, Adama Science and Technology University, Adama, Ethiopia
² Department of Mathematics, Jimma University, Jimma, Ethiopia

^* e-mail: msfnmkr02@gmail.com
^** e-mail: gammeef@gmail.com

Abstract

This paper deals with numerical treatment of singularly perturbed parabolic differential difference equations having small shifts on the spatial variable. The considered problem contain small perturbation parameter (ε) multiplied on the diffusion term of the equation. For small values of ε the solution of the problem exhibits a boundary layer on the left or right side of the spatial domain depending on the sign of the convective term. The terms involving the shifts are approximated using Taylor’s series approximation. The resulting singularly perturbed parabolic partial differential equation is solved using implicit Euler method in the temporal discretization with exponentially fitted operator finite difference method in the spatial discretization. The uniform stability of the scheme investigated using comparison principle and discrete solution bound by constructing barrier function. Uniform convergence analysis has been carried out. The scheme gives second order convergence for the case ε > N⁻¹ and first order convergence for the case ε « N⁻¹, where N is number of mesh interval. Test examples and numerical results are considered for validating the theoretical analysis of the scheme.