ISSN 2348-1196 (print) International Journal of Computer Science and Information Technology Research ISSN 2348-120X (online) Vol. 10, Issue 1, pp: (38-42), Month: January - March 2022, Available at: www.researchpublish.com
Some Applications of Integration by Parts for Fractional Calculus Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong, China
Abstract: In this paper, we make use of the integration by parts for fractional calculus to solve some fractional integrals 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 study. In fact, the method we used is a natural generalization of the integration by parts for classical calculus. Keywords: integration by parts for fractional calculus, fractional integrals, Jumarie type of modified R-L fractional derivative, new multiplication, fractional analytic functions.
I. INTRODUCTION Fractional calculus is a mathematical analysis tool used to study arbitrary order derivatives and integrals. It unifies and extends the concepts of integer order derivatives and integrals [1-5]. Generally, many scientists do not know these fractional integrals and derivatives, and they have not been used in pure mathematical context until recent years. However, in the past few decades, the fractional integrals and derivatives have frequently appeared in many scientific fields such as fluid mechanics, viscoelasticity, physics, image processing, economics and engineering [6-13]. Until now, the definition of fractional derivative is not unique. The commonly used definitions are Riemann-Liouvellie (R-L) fractional derivative, Caputo definition of fractional derivative, Grunwald-Letnikov (G-L) fractional derivative, conformable fractional derivative, and Jumarie’s modified R-L fractional derivative [1-5]. Jumarie’s modification of R-L fractional derivative helps to avoid non-zero fractional derivative of constant function [14-15]. In this article, we use integration by parts for fractional calculus to evaluate several fractional integrals based on Jumarie’s modification of R-L fractional derivative. Some fractional analytic functions are introduced such as the fractional exponential function, cosine function, sine function, and logarithmic function. A new multiplication of fractional analytic functions plays an important role in this research. In fact, the method we used is the natural generalization of integration by parts for classical calculus.
II. DEFINITIONS AND PROPERTIES At First, the fractional calculus used in this paper is introduced below. Definition 2.1: Suppose that is a real number, and Liouville fractional derivative [15] is defined as (
(
)[ ( )]
(
{
)
)
∫ (
∫ ( (
is a positive integer. The Jumarie type of modified Riemann) )
( ) [ ( )
( )]
(1)
)[ ( )]
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