A Robust Numerical Approach for Singularly Perturbed Time Delayed Parabolic Partial Differential Equations

A numerical study for a class of singularly perturbed partial functional differential equation has been initiated. The solution of the problem, being contaminated by a small perturbation parameter, exhibits layer behavior. There exist narrow regions, in the neighborhood of outflow boundary, where the solution has steep gradient. The presence of parasitic parameters, perturbation parameter and time delay, is often the source for the increased order and stiffness of these systems. The stiffness, attributed to the simultaneous occurrence of slow and fast phenomena and on their dependence on the past history of the physical systems. A numerical method based on standard finite difference operator is presented. The first step involves a discretization of time variable using backward Euler method. This results into a set of stationary singularly perturbed semi-discrete problems which are further discretized in space using standard finite difference operators. A priori explicit bounds on the solution of problem are established. Extensive amount of analysis is carried out in order to establish the convergence and stability of the method proposed. © 2016, Foundation for Scientific Research and Technological Innovation.