Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials

Ampol Duangpan,Matinee Juytai,Ratinan Boonklurb

doi:10.3390/fractalfract5030103

Ampol Duangpan, Matinee Juytai + Show 1 more

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https://doi.org/10.3390/fractalfract5030103

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Journal: Fractal and Fractional	Publication Date: Aug 25, 2021
Citations: 9	License type: CC BY 4.0

Affiliation: Chulalongkorn University

Abstract

In this paper, the finite integration method and the operational matrix of fractional integration are implemented based on the shifted Chebyshev polynomial. They are utilized to devise two numerical procedures for solving the systems of fractional and classical integro-differential equations. The fractional derivatives are described in the Caputo sense. The devised procedure can be successfully applied to solve the stiff system of ODEs. To demonstrate the efficiency, accuracy and numerical convergence order of these procedures, several experimental examples are given. As a consequence, the numerical computations illustrate that our presented procedures achieve significant improvement in terms of accuracy with less computational cost.

Highlights

In order to illustrate the effectiveness of the proposed numerical procedure in the preceding section, we present some experimental examples for solving the system of fractional integro-differential equation (FIDE) (1)
We extend the concept of solving system of FIDEs in Section 3 by studying in the common case, i.e., the order of the fractional derivative is focused on the positive integer which is called classical integro-differential equation (CIDE)
For the system of FIDEs (1), the fractional derivative is considered in the Caputo sense which is manipulated by the novel operational matrix of fractional integration (SCFM) as shown in Theorem 3

Summary

Introduction

Due to the fact that several real-world phenomena can be described successfully by developing mathematical models using fractional derivatives and integrations. One interesting issue regarding the fractional calculus is a fractional integro-differential equation (FIDE). It consists of both integral and differential operators involving derivatives of positive fractional order. The fractional order derivative of FIDEs can be reduced to a positive integer order It is called the classical integro-differential equation (CIDE) which is frequently used to describe many applications which can be seen in [11,12,13] for details of applications.

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Conclusion