A New Attractive Method in Solving Families of Fractional Differential Equations by a New Transform

Ahmad Qazza,Rania Saadeh,Aliaa Burqan

doi:10.3390/math9233039

Abstract

In this paper, we use the ARA transform to solve families of fractional differential equations. New formulas about the ARA transform are presented and implemented in solving some applications. New results related to the ARA integral transform of the Riemann-Liouville fractional integral and the Caputo fractional derivative are obtained and the last one is implemented to create series solutions for the target equations. The procedure proposed in this article is mainly based on some theorems of particular solutions and the expansion coefficients of binomial series. In order to achieve the accuracy and simplicity of the new method, some numerical examples are considered and solved. We obtain the solutions of some families of fractional differential equations in a series form and we show how these solutions lead to some important results that include generalizations of some classical methods.

Highlights

In order to produce this paper, we deal with the ARA transform [21], which is a general form of the Laplace transform
We present some basic definitions and properties of fractional calculus theory and the ARA transform that are needed to construct the new formulas about the ARA solution of the fractional differential equations
In this work a new technique has been developed for solving families of fractional differential equations

Summary

Introduction

Fractional calculus is a field of mathematics that studies the theory and applications of integral and derivatives of a non-integer order. Researchers have been interested in establishing and refining new methods to solve fractional differential equations such as the Adomian decomposition method [7], the iteration method [8,9], residual power series [10], the finitedifference method [11,12], the Laplace transform method [12,13,14,15] and the Homotopy analysis method [16].

Definitions and Properties

The ARA Transform of Fractional Operators

Solutions of Families of Fractional Differential Equations

Conclusions