Eur. Phys. J. Plus (2019) 134: 484
https://doi.org/10.1140/epjp/i2019-12871-y

Regular Article

Innovative operational matrices based computational scheme for fractional diffusion problems with the Riesz derivative

¹ School of Mathematical Sciences, Peking University, 100871, Beijing, China
² BIC-ESAT, College of Engineering, Peking University, 100871, Beijing, China
³ State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, Peking University, 100871, Beijing, China
⁴ Institute of Ocean Research, Peking University, 100871, Beijing, China
⁵ Department of Electrical Engineering, Bahria University, 44000, Islamabad, Pakistan

^* e-mail: muhammadhamid@pku.edu.cn
^** e-mail: musman@pku.edu.cn

Received: 2 May 2019
Accepted: 4 July 2019
Published online: 2 October 2019

Abstract

The computational methods based on operational matrices are promising tools to tackle the fractional order differential equations and they have gained increasing interest among the mathematical community. Herein, an efficient and precise computational algorithm based on a new kind of polynomials together with the collocation technique is presented for time-space fractional partial differential equations with the Riesz derivative. The method is proposed with the aid of a new operational matrix of the derivative using Chelyshkov polynomials (CPs) in the Caputo sense. The operational matrices of the derivative, exact and approximate, are derived via two different ways for integer and non-integer orders. The fractional problems under study have been converted into the corresponding nonlinear algebraic system of equations and solved by means of the collocation technique. The convergence and error bound are analyzed for the suggested computational method while a comparative study is included in our work to show the accuracy and efficiency of said method. The attained results confirm that the suggested technique is very accurate, efficient and reliable. As a suitable tool, it could be adopted to obtain the solutions for a class of the fractional order partial differential (linear and nonlinear) equations arising in engineering and applied sciences.