An efficient spline technique for solving time-fractional integro-differential equations

Muhammad Abbas; Sadia Aslam; Farah Aini Abdullah; Muhammad Bilal Riaz; Khaled A Gepreel

doi:10.1016/j.heliyon.2023.e19307

An efficient spline technique for solving time-fractional integro-differential equations

Heliyon. 2023 Aug 23;9(9):e19307. doi: 10.1016/j.heliyon.2023.e19307. eCollection 2023 Sep.

Authors

Muhammad Abbas¹, Sadia Aslam¹, Farah Aini Abdullah², Muhammad Bilal Riaz^{3

4

5}, Khaled A Gepreel⁶

Affiliations

¹ Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan.
² School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia.
³ Department of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon.
⁴ Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Narutowicza, 11/1280-233 Gdańsk, Poland.
⁵ Department of Mathematics, University of Management and Technology, 54770 C-II Johar Town Lahore, Pakistan.
⁶ Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia.

Abstract

Spline curves are very prominent in the mathematics due to their simple construction, accuracy of assessment and ability to approximate complicated structures into interactive curved designs. A spline is a smooth piece-wise polynomial function. The primary goal of this study is to use extended cubic B-spline (ExCuBS) functions with a new second order derivative approximation to obtain the numerical solution of the weakly singular kernel (SK) non-linear fractional partial integro-differential equation (FPIDE). The spatial and temporal fractional derivatives are discretized by ExCuBS and the Caputo finite difference scheme, respectively. The present study found that it is stable and convergent. The validity of the current approach is examined on a few test problems, and the obtained outcomes are compared with those that have previously been reported in the literature.

Keywords: Approximation; Caputo's fractional derivative; ExCuBS; Finite difference scheme; Fractional order partial integro-differential equation; Stability and convergence.