A fractional-order Legendre collocation method for solving the Bagley-Torvik equations

Fakhrodin Mohammadi,Syed Tauseef Mohyud-Din

doi:10.1186/s13662-016-0989-x

Fakhrodin Mohammadi, Syed Tauseef Mohyud-Din

Open Access

https://doi.org/10.1186/s13662-016-0989-x

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Abstract

In this article, a numerical method based on the fractional-order shifted Legendre polynomials (FSLPs) and their operational matrix of fractional integration is introduced for solving the fractional Bagley-Torvik equations. The main advantage of the presented method is that it can reduce a solution of the initial and boundary value problems for the fractional Bagley-Torvik differential equations to a system of algebraic equations. In order to confirm the efficiency and superiority of the presented method, some numerical examples are provided and a comparison is presented between the obtained results and those results achieved from other existing methods in the literature.

Highlights

Fractional calculus, the theory of differentiation and integration to non-integer order, is very useful for the description of various physical phenomena, such as damping laws, diffusion process, etc
The fractional Bagley-Torvik equation was originally formulated in a description of a real material by the use of fractional calculus
2 Preliminaries we review some basic definitions and preliminaries of the fractional calculus which are used

Summary

Introduction

Fractional calculus, the theory of differentiation and integration to non-integer order, is very useful for the description of various physical phenomena, such as damping laws, diffusion process, etc. The fractional Bagley-Torvik equation was originally formulated in a description of a real material by the use of fractional calculus. A fractional-order Legendre collocation method is proposed for solving the Bagley-Torvik equations. Spectral methods are efficient techniques for solving a different kind of fractional differential and integral equations accurately [ , , , ]. By using the operational matrices for basis functions, spectral methods reduce the solution of fractional differential and integral equations into a solution of systems of algebraic equations which produce highly accurate solutions for these equations [ , , , ]. For more details of fractional calculus and their applications please refer to [ – ]

The FSLPs and their properties

Discussion and conclusion