ISSN 2348-1218 (print) International Journal of Interdisciplinary Research and Innovations ISSN 2348-1226 (online) Vol. 10, Issue 1, pp: (37-41), Month: January - March 2022, Available at: www.researchpublish.com
A Study on Two Types of Improper Fractional Integrals CHII-HUEI YU School of Mathematics and Statistics, Zhaoqing University, Guangdong, China
Abstract: In this paper, the concept of fractional analytic function and a new multiplication of fractional analytic functions play important roles. We use Jumarie type of modified Riemann-Liouville (R-L) fractional derivative to study two types of improper fractional integrals. Keywords: fractional analytic function, new multiplication, Jumarie type of modified R-L fractional derivative, improper fractional integrals.
I. INTRODUCTION Fractional calculus includes the derivative and integral of any real order or complex order. In recent years, fractional calculus has achieved considerable popularity and attention, due to its application in various fields such as elasticity, mechanics, dynamics, electronics, modelling, physics, mathematical economics, and control theory [1-11]. Fractional calculus is different from traditional calculus. There is no unique definition of fractional derivative and integral. Commonly used 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 [4, 12, 23]. On the other hand, the application of fractional calculus in fractional differential equations can be referred to [13-22]. In this article, we solve the following two types of improper fractional integrals: (
)[
],
(1)
(
)[
].
(2)
Where , and are real numbers with . The fractional analytic functions such as fractional exponential function, fractional sine and cosine functions are discussed. The new multiplication of fractional analytic functions is a natural generalization of ordinary multiplication in calculus. And Jumarie′s modified Riemann-Liouville fractional derivative is used to study the above two improper fractional integrals.
II. PRELIMINARIES At first, we introduce the fractional calculus used in this paper. Definition 2.1: Suppose that is a real number, and Liouville fractional derivative [12] is defined as
is a positive integer. The Jumarie type of modified Riemann-
∫ (
)[
]
[
∫ {
(
)[
]
(3)
]
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