International Journal of Engineering Research and Reviews
ISSN 2348-697X (Online) Vol. 10, Issue 1, pp: (8-13), Month: January - March 2022, Available at: www.researchpublish.com
Study of Fractional Laplace Transform Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong, China
Abstract: In this article, we study the fractional Laplace transforms of fractional analytic functions based on Jumarie type of modified Riemann-Liouville (R-L) fractional derivative. A new multiplication of fractional analytic functions plays an important role in this paper. And the results we obtained are generalizations of Laplace transforms of classical analytic functions. Keywords: fractional Laplace transforms, fractional analytic functions, Jumarie type of modified R-L fractional derivative, new multiplication.
I. INTRODUCTION In 1695, the concept of fractional derivative first appeared in a famous letter between L’Hospital and Leibniz. Many great mathematicians have further developed this field. We can mention Euler, Lagrange, Laplace, Fourier, Abel, Liouville, Riemann, Hardy, Littlewood, and Weyl. In the past decades, fractional calculus has been considered as one of the best tools to describe the process of long memory. Such models are interesting for physicists, engineers, and mathematicians. Fractional calculus has important applications in various fields such as physics, mechanics, electricity, biology, economics, control theory, and so on. The introduction and application of fractional calculus can refer to [1-9]. Fractional calculus includes the derivative and integral of any real or complex order. There is no unique definition of fractional derivative and integral. Common definitions include Riemann Liouville (R-L) fractional derivative, Caputo fractional derivative, Grunwald Letinikov (G-L) fractional derivative, and Jumarie’s modified R-L fractional derivative [1-4]. In this article, based on Jumarie type of modified R-L fractional derivative, we evaluate the fractional Laplace transforms of some fractional analytic functions such as fractional exponential function, fractional sine and cosine functions, and fractional hyperbolic sine and cosine functions. A new multiplication of fractional analytic functions plays an important role in this paper, and the results we obtained are natural generalizations of the results in classical Laplace transform. For the introduction and application of fractional Laplace transform can refer to [10-11]
II. DEFINITIONS AND PROPERTIES Firstly, we introduce the fractional calculus used in this article. Definition 2.1: If is a real number, and derivative [12] is defined by
is a positive integer. The Jumarie’s modified Riemann-Liouville fractional
(
(
), ( )-
(
)
∫ (
∫ (
)
(
{
) )
( ) , ( )
( )-
), ( )-
where ( ) is the gamma function. On the other hand, we define the -fractional integral of ( ) by ( ( ), ( )-, where following properties. Proposition 2.2: If
. If (
(1)
), ( )-
), ( )- exists, then ( ) is called an -fractional integrable function. We have the
are real numbers and
then [ ]
( (
) )
,
(2)
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