Some Dynamical Models Involving Fractional-Order Derivatives with the Mittag-Leffler Type Kernels and Their Applications Based upon the Legendre Spectral Collocation Method

Hari M Srivastava,Abedel-Karrem N Alomari,Khaled M Saad,Waleed M Hamanah

doi:10.3390/fractalfract5030131

Abstract

Fractional derivative models involving generalized Mittag-Leffler kernels and opposing models are investigated. We first replace the classical derivative with the GMLK in order to obtain the new fractional-order models (GMLK) with the three parameters that are investigated. We utilize a spectral collocation method based on Legendre’s polynomials for evaluating the numerical solutions of the pr. We then construct a scheme for the fractional-order models by using the spectral method involving the Legendre polynomials. In the first model, we directly obtain a set of nonlinear algebraic equations, which can be approximated by the Newton-Raphson method. For the second model, we also need to use the finite differences method to obtain the set of nonlinear algebraic equations, which are also approximated as in the first model. The accuracy of the results is verified in the first model by comparing it with our analytical solution. In the second and third models, the residual error functions are calculated. In all cases, the results are found to be in agreement. The method is a powerful hybrid technique of numerical and analytical approach that is applicable for partial differential equations with multi-order of fractional derivatives involving GMLK with three parameters.

Highlights

It is presumably the first time that the numerical results, which are presented in this paper, are based upon the use of the fractional derivatives and upon the properties of the Legendre polynomials as well as GMLK
Three fractional-order models have been presented: The first model belongs to fractional ordinary differential equations, and the other two models involve fractional partial differential equations
The treatment was achieved by directly converting the fractionalorder model with the help of the Legendre polynomials to a system of algebraic equations and finding approximate solutions by using the Newton-Raphson method, in addition to finding the solution analytically

Summary

Collocation Method

Hari M. Srivastava 1,2,3,4, *,† , Abedel-Karrem N. Alomari 5,† , Khaled M. Saad 6,7,† Citation: Srivastava, H.M.; Alomari, A.-K.N.; Saad, K.M.; Hamanah, W.M. Method. Fractal Fract. 2021, 5, 131. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku AZ1007, Azerbaijan Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz P.O. Box 6803, Yemen These authors contributed equally to this work.

Introduction

Preliminaries

Construction of the Schemes of the Proposed Models

Conclusions

Results