ISSN 2348-1218 (print) International Journal of Interdisciplinary Research and Innovations ISSN 2348-1226 (online) Vol. 8, Issue 4, pp: (100-105), Month: October - December 2020, Available at: www.researchpublish.com
A Study of Exact Fractional Differential Equations Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong Province, China
Abstract: In this article, a new multiplication of fractional functions and chain rule for fractional derivatives are used to obtain the general solution of exact fractional differential equation (FDE), regarding the Jumarie type of modified Riemann-Liouville (R-L) fractional derivatives. Furthermore, an example is given for demonstrating the advantage of our result. Keywords: New multiplication, Fractional functions, Chain rule, Exact FDE, Jumarie type of modified R-L fractional derivatives.
I. INTRODUCTION Fractional differential equations have excited in recent years a considerable interest both in mathematics and in applications. They were used in modeling of many physical and chemical processes and in engineering [1-4]. In its turn, mathematical aspects of fractional differential equations and methods of their solution were discussed by many authors: the iteration method in [5], the series method in [1], the Fourier transform technique in [6-7], special methods for fractional differential equations of rational order or for equations of special type in [8-13], the operational calculus method in [14-15]. Unlike standard calculus, there is no unique definition of derivative in fractional calculus. The definition of fractional derivative is given by many authors. The commonly used definitions are the Riemann-Liouvellie (R-L) fractional derivative [16], Caputo definition of fractional derivative [17], the Grunwald-Letinikov (G-L) fractional derivative [16], and Jumarie’s modified R-L fractional derivative is used to avoid nonzero fractional derivative of a constant functions [18]. After the first congress at the University of New Haven, in 1974, fractional calculus has developed and several applications emerged in many areas of scientific knowledge. As a consequence, distinct approaches to solve problems involving the derivative were proposed and distinct definitions of the fractional derivative are available in the literature. In this paper, the general solution of exact fractional differential equation (FDE), regarding the Jumarie type of modified R-L fractional derivatives can be obtained by using a new multiplication of fractional functions and chain rule for fractional derivatives. In fact, the result we obtained is the generalization of general solution of exact ordinary differential equations. On the other hand, an example is proposed to demonstrate the advantage of our result.
II. PRELIMINARIES In the following, fractional calculus used in this paper is introduced. Definition 2.1: If is a real number and fractional derivatives of Jumarie type ([16]) (
[ ( )]
(
{
)
)
(
is a positive integer Then we define the modified Riemann-Liouville
∫ (
)
∫ (
)
( ) [ ( )
( )]
(1)
)[ ( )]
Page | 100 Research Publish Journals