Numerical Solution to the Time-Fractional Burgers–Huxley Equation Involving the Mittag-Leffler Function

Afzaal Mubashir Hayat, Muhammad Bilal Riaz, Muhammad Abbas (), Moataz Alosaimi, Adil Jhangeer and Tahir Nazir
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Afzaal Mubashir Hayat: Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan
Muhammad Bilal Riaz: IT4Innovations, VSB-Technical University of Ostrava, 70800 Ostrava, Czech Republic
Muhammad Abbas: Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan
Moataz Alosaimi: Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
Adil Jhangeer: IT4Innovations, VSB-Technical University of Ostrava, 70800 Ostrava, Czech Republic
Tahir Nazir: Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan

Mathematics, 2024, vol. 12, issue 13, 1-22

Abstract: Fractional differential equations play a significant role in various scientific and engineering disciplines, offering a more sophisticated framework for modeling complex behaviors and phenomena that involve multiple independent variables and non-integer-order derivatives. In the current research, an effective cubic B-spline collocation method is used to obtain the numerical solution of the nonlinear inhomogeneous time-fractional Burgers–Huxley equation. It is implemented with the help of a θ -weighted scheme to solve the proposed problem. The spatial derivative is interpolated using cubic B-spline functions, whereas the temporal derivative is discretized by the Atangana–Baleanu operator and finite difference scheme. The proposed approach is stable across each temporal direction as well as second-order convergent. The study investigates the convergence order, error norms, and graphical visualization of the solution for various values of the non-integer parameter. The efficacy of the technique is assessed by implementing it on three test examples and we find that it is more efficient than some existing methods in the literature. To our knowledge, no prior application of this approach has been made for the numerical solution of the given problem, making it a first in this regard.

Keywords: nonlinear time-fractional Burgers–Huxley equation; cubic B-spline interpolation; Atangana–Baleanu operator; convergence; stability; finite difference formulation; Mittag-Leffler function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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