Eur. Phys. J. Plus (2020) 135: 237
https://doi.org/10.1140/epjp/s13360-020-00158-5

Regular Article

An operational matrix method for nonlinear variable-order time fractional reaction–diffusion equation involving Mittag-Leffler kernel

¹ Department of Mathematics, Shiraz University of Technology, Shiraz, Iran
² Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein, South Africa
³ Department of Medical Research China Medical University Hospital, China Medical University, Taichung, Taiwan
⁴ Laboratory for Intelligent Computing and Financial Technology, Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, Suzhou, 215123, Jiangsu, China
⁵ Institute of Research and Development, Duy Tan University, Da Nang, 550000, Vietnam
⁶ Department of Statistics, Faculty of Science, Fasa University, Fasa, Iran

^* e-mail: mohammadrezamahmoudi@duytan.edu.vn

Received: 15 July 2019
Accepted: 16 November 2019
Published online: 10 February 2020

Abstract

This paper is concerned with an operational matrix (OM) scheme based on the shifted Chebyshev cardinal functions (SCCFs) of the second kind for numerical solution of the variable-order time fractional nonlinear reaction–diffusion equation. The fractional derivative operator is defined in the sense of Atangana–Baleanu–Caputo. Through the way, a new OM of variable-order fractional derivative is derived for the mentioned cardinal functions. More precisely, the unknown solution is expanded by the SCCFs with undetermined coefficients. Then the expansion substituted in the equation and the generated OM is utilized to extract some algebraic equations. The precision of the established approach is examined through various types of test examples. Numerical simulations confirm the suggested approach is highly accurate to provide satisfactory results.