Explicit scheme for solving variable-order time-fractional initial boundary value problems

Asia Kanwal; Salah Boulaaras; Ramsha Shafqat; Bilal Taufeeq; Mati Ur Rahman

doi:10.1038/s41598-024-55943-4

Explicit scheme for solving variable-order time-fractional initial boundary value problems

Sci Rep. 2024 Mar 5;14(1):5396. doi: 10.1038/s41598-024-55943-4.

Authors

Asia Kanwal¹, Salah Boulaaras², Ramsha Shafqat³, Bilal Taufeeq⁴, Mati Ur Rahman^{5

6}

Affiliations

¹ School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, People's Republic of China.
² Department of Mathematics, College of Science, Qassim University, 51452, Buraydah, Saudi Arabia. s.boularas@qu.edu.sa.
³ Department of Mathematics and Statistics, The University of Lahore, Sargodha, 40100, Pakistan.
⁴ Department of Mathematics, Government College University Lahore Pakistan, Lahore, Punjab, Pakistan.
⁵ School of Mathematical Sciences, Jiangsu University, Zhenjiang, 212013, Jiangsu, People's Republic of China.
⁶ Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon.

Abstract

The creation of an explicit finite difference scheme with the express purpose of resolving initial boundary value issues with linear and semi-linear variable-order temporal fractional properties is presented in this study. The rationale behind the utilization of the Caputo derivative in this scheme stems from its known importance in fractional calculus, an area of study that has attracted significant interest in the mathematical sciences and physics. Because of its special capacity to accurately represent physical memory and inheritance, the Caputo derivative is a relevant and appropriate option for representing the fractional features present in the issues this study attempts to address. Moreover, a detailed Fourier analysis of the explicit finite difference scheme's stability is shown, demonstrating its conditional stability. Finally, certain numerical example solutions are reviewed and MATLAB-based graphic presentations are made.

Keywords: Caputo derivative; Explicit scheme; Fractional derivatives; Fractional diffusion equations; Initial boundary value problem; Stability analysis.