Fractional Gegenbauer wavelets operational matrix method for solving nonlinear fractional differential equations

Abstract

The main aim of the paper is to introduce the shifted fractional-order Gegenbauer wavelets (SFGWs) and the development of a method for solving fractional nonlinear initial and boundary value problems on a semi-infinite domain. The proposed method is the combination of SFGWs and parametric iteration method. We have derived and constructed the new operational matrices for the SFGWs, which are utilized for the solutions of nonlinear fractional differential equations. We have constructed the weight function and normalizing factor for SFGWs. The operational matrices for the SFGWs are derived and constructed, to make the calculations fast. Furthermore, we work out an error analysis for the method. The procedure of implementation for both fractional nonlinear initial and boundary value problems is presented. Numerical simulation is provided to illustrate the reliability and accuracy of the method. Many engineers can utilize the presented method for solving their nonlinear fractional differential models. To the best of the author’s knowledge, the shifted fractional-order Gegenbauer wavelets have never been introduced and implemented for nonlinear fractional differential equations.