A New Operational Matrices-Based Spectral Method for Multi-Order Fractional Problems

M Hamid,Ilyas Khan,Oi Mean Foong,Wei Wang,Muhammad Usman

doi:10.3390/sym12091471

M Hamid, Ilyas Khan + Show 3 more

Open Access

https://doi.org/10.3390/sym12091471

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Abstract

The operational matrices-based computational algorithms are the promising tools to tackle the problems of non-integer derivatives and gained a substantial devotion among the scientific community. Here, an accurate and efficient computational scheme based on another new type of polynomial with the help of collocation method (CM) is presented for different nonlinear multi-order fractional differentials (NMOFDEs) and Bagley–Torvik (BT) equations. The methods are proposed utilizing some new operational matrices of derivatives using Chelyshkov polynomials (CPs) through Caputo’s sense. Two different ways are adopted to construct the approximated (AOM) and exact (EOM) operational matrices of derivatives for integer and non-integer orders and used to propose an algorithm. The understudy problems have been transformed to an equivalent nonlinear algebraic equations system and solved by means of collocation method. The proposed computational method is authenticated through convergence and error-bound analysis. The exactness and effectiveness of said method are shown on some fractional order physical problems. The attained outcomes are endorsing that the recommended method is really accurate, reliable and efficient and could be used as suitable tool to attain the solutions for a variety of the non-integer order differential equations arising in applied sciences.

Highlights

The subject of fractional calculus (FC) and fractional or non-integer differential equations (FDEs) is an inspirational mathematical topic of research among scientific community
The current work is dedicated to the construction of a new computational algorithm based on operational matrices of derivatives
A novel mathematical method is proposed depending upon developed matrices and successfully applied for some class of FDEs

Summary

Introduction

The subject of fractional calculus (FC) and fractional or non-integer differential equations (FDEs) is an inspirational mathematical topic of research among scientific community. The FC area is an old topic as traditional calculus proposed individually by Leibniz and Newton, but FC gained a vital importance among the researcher community. The devotion of scientific communities is pretty realistic due to its specific appearance in mathematical as well as other areas of science and engineering [1]. It converts the symbolization of derivatives for those sections where the order of the derivatives is non-integer or fractional. The subject of FC has been developed in different domains of sciences, from pure mathematical theories to modeling

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