Eur. Phys. J. Plus (2018) 133: 327
https://doi.org/10.1140/epjp/i2018-12172-1

Regular Article

An efficient algorithm based on Gegenbauer wavelets for the solutions of variable-order fractional differential equations

¹ School of Mathematical Sciences, Peking University, 100871, Beijing, China
² Department of Electrical Engineering, Bahria University, Islamabad Campus, Islamabad, Pakistan

^* e-mail: ideal_riz@hotmail.com

Received: 12 May 2018
Accepted: 3 July 2018
Published online: 17 August 2018

Abstract

The article is devoted to a new computational algorithm based on the Gegenbauer wavelets (GWs) to solve the linear and nonlinear variable-order fractional differential equations. The novel operational matrices for derivatives of positive integer and variable order are derived. New piecewise functions are introduced to obtain the said operational matrices. In the proposed method, the given problem via Gegenbauer wavelets is transformed to a system of algebraic equations. The obtained solutions are endorsing the accuracy and efficiency of the suggested method and are in excellent agreement with the existing literature. The convergence and error bound analysis are presented in our study to show the credibility of the computational method and support the mathematical formulation of the algorithm. The discussed problems reconfirm the appropriateness of said algorithm and perceived that the proposed algorithm is an efficient tool to tackle the nonlinear fractional order problems of complex nature.