An efficient technique based on cubic B-spline functions for solving time-fractional advection diffusion equation involving Atangana-Baleanu derivative

Madiha Shafiq; Muhammad Abbas; Khadijah M Abualnaja; M J Huntul; Abdul Majeed; Tahir Nazir

doi:10.1007/s00366-021-01490-9

An efficient technique based on cubic B-spline functions for solving time-fractional advection diffusion equation involving Atangana-Baleanu derivative

Eng Comput. 2022;38(1):901-917. doi: 10.1007/s00366-021-01490-9. Epub 2021 Aug 4.

Authors

Madiha Shafiq¹, Muhammad Abbas¹, Khadijah M Abualnaja², M J Huntul³, Abdul Majeed⁴, Tahir Nazir¹

Affiliations

¹ Department of Mathematics, University of Sargodha, Sargodha, 40100 Pakistan.
² Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif, 21944 Saudi Arabia.
³ Department of Mathematics, Faculty of Science, Jazan University, Jazan, Saudi Arabia.
⁴ Department of Mathematics, Division of Science and Technology, University of Education, Lahore, 54770 Pakistan.

Abstract

The present paper deals with cubic B-spline approximation together with $θ$ -weighted scheme to obtain numerical solution of the time fractional advection diffusion equation using Atangana-Baleanu derivative. To discretize the Atangana-Baleanu time derivative containing a non-singular kernel, finite difference scheme is utilized. The cubic basis functions are associated with spatial discretization. The current discretization scheme used in the present study is unconditionally stable and the convergence is of order $O (h^{2} + Δ t^{2})$ . The proposed scheme is validated through some numerical examples which reveal the current scheme is feasible and quite accurate.

Keywords: Advection diffusion equation; Atangana–Baleanu time fractional derivative; Cubic B-spline functions; Finite difference formulation; Spline interpolation; Stability and convergence.