Uniformly convergent numerical solution for caputo fractional order singularly perturbed delay differential equation using extended cubic B-spline collocation scheme

Document Type : Research Article

Authors

• N.A. Endrie ¹
• G.F. Duressa ²

¹ Department of Mathematics,College of Natural and Computational Science, Arba Minch University, Arba Minch, Ethiopia.

² Department of Mathematics, College of Natural and Computational Science, Jimma University, Jimma, Ethiopia.

10.22067/ijnao.2024.86894.1393

Abstract

This article presents a parameter uniform convergence numerical scheme for solving time fractional order singularly perturbed parabolic convection-diffusion differential equations with a delay. We give a priori bounds on the exact solution and its derivatives obtained through the problem’s asymp-totic analysis. The Euler’s method on a uniform mesh in the time direction and the extended cubic B-spline method with a fitted operator on a uniform mesh in the spatial direction is used to discretize the problem. The fitting factor is introduced for the term containing the singular perturbation pa-rameter, and it is obtained from the zeroth-order asymptotic expansion of the exact solution. The ordinary B-splines are extended into the extended B-splines. Utilizing the optimization technique, the value of μ (free param-eter, when the free parameter μ tends to zero the extended cubic B-spline reduced to convectional cubic B-spline functions) is determined. It is also demonstrated that this method is better than some existing methods in the literature.

Keywords

• Singularly perturbed problem
• Fractional derivative
• Artificial viscosity
• Delay differential equation

Main Subjects

• Numerical methods for differential equations and its applications

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