The Fractional Residual Power Series Method for Solving a System of Linear Fractional Fredholm Integro-differential Equations

Prapart Pue-On

doi:10.13189/ms.2021.090519

Abstract

In this manuscript, the fractional residual power series (FRPS) method is employed in solving a system of linear fractional Fredholm integro-differential equations. The significant role of this system in various fields has attracted the attention of researchers for a decade. The definition of fractional derivative here is described in the Caputo sense. The proposed method relies on the generalized Taylor series expansion as well as the fact that the fractional derivative of stationary is zero. The process starts by constructing a residual function by supposing the finite order of an approximate power series solution that prescribes the initial conditions. Then, utilizing some conditions, the residual functions are converted to a linear system for the power series coefficients. Solving the linear system reveals the coefficients of the fractional power series solution. Finally, by substituting these coefficients into the supposed form of a solution, the approximate fractional power series solutions are derived. This technique has the advantage of being able to be applied directly to the problem and spending less time on computation. It is not only an easy method for implementation of the problem, but also provides productive results after a few iterations. Some problems with known solutions emphasize the procedure's simplicity and reliability. Moreover, the obtained exact solution demonstrated the efficiency and accuracy of the presented method.

Highlights

Fractional calculus came into existence from a question posed by L’Hopital in a message to Leibniz in 1695 [20]
In this manuscript, the fractional residual power series (FRPS) method is employed in solving a system of linear fractional Fredholm integro-differential equations
Solving the linear system reveals the coefficients of the fractional power series solution

Summary

Introduction

Fractional calculus came into existence from a question posed by L’Hopital in a message to Leibniz in 1695 [20]. This study aims to use the FRPS algorithm to solve a linear system of fractional Fredholm integro-differential equations (FIDEs), Dγi ui(x) = gi(x) + Ri(u1(x), u2(x), . The Caputo fractional derivative operator of order γ is defined by. Definition 2.1 [13] The Riemann-Liouville fractional integral operator of order γ ≥ 0 is defined by 1 x φ(τ ). To determine the coefficient ci, in (3), one substitutes the 1-st residual power series approximate solution, xγi ui,1(x) = ci + ci, Γ(1 + γi) , i = 1, 2, . Assume that the FPS solution of the system (1) with initial conditions of ui(0) = ci at x = 0 has the following form: ui(x). To follow the fractional power series method, let’s suppose the approximate solution of the system (1)is in the form a kthtruncated series: ui,k (x). −λi κi(x, τ )Fi(τ, u1(τ ), u2(τ ), . . . , un(τ ))dτ, a unknown coefficient, ci,k will be determined [1], [9]

Illustrative Examples

Conclusions