Numerical computation for the class of singularly perturbed delay parabolic reaction diffusion equations with integral boundary condition has been considered. A parameter-uniform numerical method is constructed via the nonstandard finite difference method for the spatial direction, and the backward Euler method for the resulting system of initial value problems in temporal direction is used. The integral boundary condition is treated using numerical integration techniques. Maximum absolute errors and the rate of convergence for different values of perturbation parameter ε and mesh sizes are tabulated for two model examples. The proposed method is shown to be parameter-uniformly convergent.

This paper deals with numerical solution of singularly perturbed delay parabolic reaction diffusion problem having large delay on the spatial variable with non-local boundary condition. The solution of the problem exhibits parabolic boundary layer on both sides of the spatial domain and interior layer is also created. Introducing a fitting parameter into asymptotic solution and applying average finite difference approximation, a fitted operator finite difference method is developed for solving the problem under consideration. To treat the non-local boundary condition, Simpson's rule is applied. The stability and $\varepsilon$ uniform convergence analysis has been carried out. To validate the applicability of the scheme, numerical examples are presented and solved for different values of the perturbation parameter $\varepsilon$ and mesh sizes. The numerical results are tabulated in terms of maximum absolute errors and rate of convergence and it is observed that the present method is more accurate and shown to be second order Uniformly convergent in both direction, and it also improves the results of the methods existing in the literature.

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The motive of this paper is, to develop accurate and parameter uniform numerical method for solving singularly perturbed delay parabolic differential equation with non-local boundary condition exhibiting parabolic boundary layers. Also, the delay term that occurs in the space variable gives rise to interior layer. Fitted operator finite difference method on uniform mesh that uses the procedures of method of line for spatial discretization and backward Euler method for the resulting system of initial value problems in temporal direction is considered. To treat the non-local boundary condition, Simpsons rule is applied. The stability and parameter uniform convergence for the proposed method are proved. To validate the applicability of the scheme, numerical examples are presented and solved for different values of the perturbation parameter. The method is shown to be accurate of O(h2 + △t) . Finally, conclusion of the work is included at the end.

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