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

Articles in Press

Document Type : Research Article

Authors

• Nuru Ahmed Endrie ¹
• Gemechis File 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 Sciences, 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 asymptotic 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 are used to discretize the problem. The fitting factor is introduced for the term containing the singular perturbation parameter, and it is obtained from the zeroth-order asymptotic expansion of the exact solution. The ordinary B-splines extended into the extended B-splines. Utilizing the optimization technique, the  value of $ \mu $ (free parameter, when the free parameter $\mu$ 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

---