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In this paper, fitted operator finite difference methods are presented for two-parameter singularly perturbed one-dimensional parabolic partial differential equations with a delay in the time variable. Such boundary value problems are frequently encountered in the spatial diffusion of reactants and in control systems. To approximate the solution of the problem, we consider the Crank–Nicolson method for time discretization and the non-standard scheme for space discretization on uniform meshes. We prove that the proposed numerical method is uniformly convergent, having convergence of order two in time and one in space. Further, we present numerical results for two test problems in support of our theoretical findings and to demonstrate that the non-standard scheme is more effective than the hybrid scheme comprising central difference, upwind, and mid-point.

This article proposes a parameter uniform numerical method for solving a singularly perturbed delay parabolic initial boundary-value problem involving mixed space shifts. The model also involves a large delay in time. Taylor’s series expansion is applied to approximate the retarded terms in the spatial direction. For the time discretization, the implicit trapezoidal scheme is applied on uniform mesh, and for the spatial discretization, we use a proper combination of the mid-point upwind and the central difference scheme on Shishkin mesh. The proposed scheme provides a second-order convergence rate uniformly with respect to the perturbation parameter. Some comparison results are presented by using the proposed method to support our claim. [ABSTRACT FROM AUTHOR]

The purpose of this work is to provide robust numerical scheme for singularly perturbed time delay (large) convection-reaction-diffusion problem with two small parameters. The work comprises of the discretization of time derivative with the implicit Euler scheme on uniform mesh and the discretization of spatial derivative with upwind difference scheme on Shishkin type meshes namely, Shishkin mesh and Bakhvalov–Shishkin mesh. Again for boosting up the order of convergence, Richardson extrapolation technique is applied. The prescribed scheme is proved to be uniformly convergent and robust providing almost a second-order convergence. Also, Thomas algorithm is used which takes less computational time. The study is also extended by investigating the application and effectiveness of the proposed method on Duran-Shishkin mesh for the given model in the numerical section. The convergence and error analysis along with the numerical findings are provided which show that the scheme is independent of both the perturbation parameters. [ABSTRACT FROM AUTHOR]