International Journal of Electrical and Electronics Research ISSN 2348-6988 (online) Vol. 10, Issue 1, pp: (8-13), Month: January - March 2022, Available at: www.researchpublish.com
Using Fractional Laplace Transform to Solve Fractional Differential Equations Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong, China
Abstract: The purpose of this paper is to solve fractional differential equations by using fractional Laplace transforms based on Jumarie’s modified Riemann-Liouville (R-L) fractional derivative. A new multiplication of fractional analytic functions plays a vital role in this research. We give some examples to illustrate the application of fractional Laplace transform in solving fractional differential equations. In fact, the method we use is a natural generalization of the Laplace transforms of analytic functions. Keywords: fractional differential equations, fractional Laplace transforms, Jumarie’s modified R-L fractional derivative, new multiplication, fractional analytic functions.
I. INTRODUCTION Fractional derivatives of non-integer orders are widely used in physics, mechanics, dynamics, and mathematical economics [1-8]. Until now, the rules of fractional derivative are not unique. Many authors have given the definition of fractional derivative. The commonly used definition is the Riemann-Liouvellie (R-L) definition [9-12]. Other useful definitions include Caputo definition of fractional derivative, Grunwald Letinikov (G-L) fractional derivative, conformable fractional derivative, and Jumarie’s modified R-L fractional derivative [9-12]. Jumarie’s modification of R-L fractional derivative helps to avoid non-zero fractional derivative of constant function [13]. This paper introduces some fractional analytic functions such as the fractional exponential function, fractional cosine and sine functions. Based on Jumarie type of modified R-L fractional derivative, some fractional differential equations are solved by using the fractional Laplace transform of some fractional analytical functions. A new multiplication of fractional analytic functions plays an important role in this study. On the other hand, the method used in this paper is the extension of classical Laplace transform. In addition, the introduction and application of fractional Laplace transform can be referred to [14-16].
II. DEFINITIONS AND PROPERTIES First, the fractional calculus used in this article is introduced below. Definition 2.1: Suppose that is a real number, and Liouville fractional derivative [17] is defined by (
(
), ( )-
(
{ where (
( )
)
)
∫ (
)
∫ ( (
)
is a positive integer. The Jumarie type of modified Riemann-
( ) , ( )
( )-
(1)
), ( )-
is the gamma function. Moreover, we define the
), ( )-, where
. If (
-fractional integral of
( ) by (
), ( )-
), ( )- exists, then ( ) is called an -fractional integrable function. For any
positive integer , we define ( ) , ( )We have the following properties.
(
)(
)
(
), ( )-, the -th order fractional derivative of ( ).
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