Abstract
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.
Highlights
The spatial delay parabolic singularly perturbed differential equation is a differential equation in which the perturbation parameter multiply the highest order derivative, and it has at least one retarded term on the spatial variable
For solving singular perturbation problems, some ε-uniform numerical schemes have been developed in the literature
We have presented the method for solving spatial delayed singularly perturbed parabolic partial differential equation
Summary
The spatial delay parabolic singularly perturbed differential equation is a differential equation in which the perturbation parameter multiply the highest order derivative, and it has at least one retarded term on the spatial variable. Mbroh et al [3] proposed parameter uniform method for solving a time delay nonautonomous singularly perturbed parabolic differential equation. A few scholars have examined numerical solution for spatial delay singularly perturbed parabolic partial differential equations. Gupta et al [9] examined spatial delay parabolic singularly perturbed partial differential equations and Abstract and Applied Analysis its solution using higher order fitted mesh method. Fitted operator method was developed to solve differential-difference singularly perturbed problem by Woldaregay and Duressa [10]. Das and Natesan [11] presented the solution of delay singularly perturbed partial differential equation using second-order convergent method. A singularly perturbed delayed partial differential equation with small spatial shift right boundary layer problem is decomposed into three piecewise equations which are treated using fitted operator difference methods. The numerical results of the examples considered shows that the present method has better accuracy compared to some results that appear in the literature
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