This paper proposes a new fitted operator strategy for solving singularly perturbed parabolic partial differential equation with delay on the spatial variable. We decomposed the problem into three piecewise equations. The delay term in the equation is expanded by Taylor series, the time variable is discretized by implicit Euler method, and the space variable is discretized by central difference methods. After developing the fitting operator method, we accelerate the order of convergence of the time direction using Richardson extrapolation scheme and obtained O h 2 + k 2 uniform order of convergence. Finally, three examples are given to illustrate the effectiveness of the method. The result shows the proposed method is more accurate than some of the methods that exist in the literature.

In this paper, we design and investigate a higher order ε -uniformly convergent method to solve singularly perturbed parabolic reaction-diffusion problems with a large time delay. We use the Crank–Nicolson method for the time derivative, while the spatial derivative is discretized using a nonstandard finite difference approach on a uniform mesh. Furthermore, to improve the order of convergence, we used the Richardson extrapolation technique. The designed scheme converges independent of the perturbation parameter ( ε -uniformly convergent) and also achieves fourth-order convergent in both time and spatial variables. Two model examples are considered to demonstrate the applicability of the suggested method. The proposed method produces better accuracy and a higher rate of convergence than some methods that appear in the literature.

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In this paper, we produce ϵ , μ − uniform numerical method for a singularly perturbed parabolic differential equation with two parameters. To approximate the solution, we consider the implicit Euler method for time direction, the finite difference method for spatial direction on a uniform mesh, and the fitted operator method with multiple fitting factors. To accelerate the convergence of the method, the Richardson extrapolation method is applied. The results show that the proposed method is second-order convergent in both temporal and spatial directions. The convergence of the scheme is insensitive to the two perturbation parameters. Two model examples are considered to validate the applicability of the proposed method and produced more accurate results compared to some methods that appear in the literature. Matlab software is used to manipulate the results.

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