A nonstandard fitted operator finite difference method for two-parameter singularly perturbed time-delay parabolic problems

Abstract

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In this article, a class of singularly perturbed time-delay two-parameter second-order parabolic problems are considered. The presence of the two small parameters attached to the derivatives causes the solution of the given problem to exhibit boundary layer(s). We have developed a uniformly convergent nonstandard fitted operator finite difference method (NSFOFDM) to solve the considered problems. The Crank-Nicolson scheme with a uniform mesh is used for the discretization of the time derivative, while for the spatial discretization, we have applied a fitted operator finite difference method following the nonstandard methodology of Mickens. Moreover, the solution bounds of the governing equation are shown by asymptotic analysis. The convergence of the proposed numerical scheme is investigated using truncation error and the barrier function approach. The study shows that our proposed scheme is uniformly convergent independent of the perturbation parameters, quadratically in time, and linearly in space. Numerical experiments are carried out, and the results are presented in tables and graphically.

Keywords

• singular perturbation
• delay differential equation
• non-standard finite difference method
• uniformly convergent scheme
• boundary layers