ISSN 2348-1196 (print) International Journal of Computer Science and Information Technology Research ISSN 2348-120X (online) Vol. 10, Issue 4, pp: (53-57), Month: October - December 2022, Available at: www.researchpublish.com
Some Type of Improper Fractional Integral Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong, China DOI: https://doi.org/10.5281/zenodo.7476989
Published Date: 23-December-2022
Abstract: In this paper, based on Jumarie type of Riemann-Liouville (R-L) fractional calculus, we evaluate some type of improper fractional integral. A new multiplication of fractional analytic functions plays an important role in this paper. The main methods we used are integration by parts for fractional calculus and fractional L’Hospital’s rule. At the same time, some examples are given to illustrate our result. In fact, our result is a generalization of the classical calculus result. Keywords: Jumarie type of R-L fractional calculus, improper fractional integral, new multiplication, fractional analytic functions, integration by parts, fractional L’Hospital’s rule.
I. INTRODUCTION During the 18th and 19th centuries, there were many famous scientists such as Euler, Laplace, Fourier, Abel, Liouville, Grunwald, Letnikov, Riemann, Laurent, Heaviside, and some others who reported interesting results within fractional calculus. In recent years, fractional calculus has become an increasingly popular research area due to its effective applications in different scientific fields such as economics, mechanics, biology, control theory, electrical engineering, viscoelasticity, and so on [1-8]. However, the definition of fractional derivative is not unique. Commonly used definitions include Riemann-Liouville (RL) fractional derivative, Caputo fractional derivative, Grunwald-Letnikov (G-L) fractional derivative, Jumarie’s modified R-L fractional derivative [9-14]. Since Jumarie type of R-L fractional derivative helps to avoid non-zero fractional derivative of constant function, it is easier to use this definition to connect fractional calculus with classical calculus. In this paper, based on Jumarie type of R-L fractional calculus, we solve the following improper fractional integral: 𝛼 ( 0𝐼+∞ )[
1
Γ(𝛼+1)
𝑥 𝛼 ⨂[𝐸𝛼 (𝑡𝑥 𝛼 ) − 1]⨂ −1 ].
(1)
Where 0 < 𝛼 ≤ 1 and 𝑡 > 0 . Integration by parts for fractional calculus, fractional L’Hospital’s rule, and a new multiplication of fractional analytic functions play important roles in this paper. In fact, our result is a generalization of the result of ordinary calculus.
II. DEFINITIONS AND PROPERTIES Firstly, the fractional calculus used in this paper is introduced below. Definition 2.1 ([15]): Suppose that 0 < 𝛼 ≤ 1, and 𝑥0 is a real number. The Jumarie’s modified Riemann-Liouville (R-L) 𝛼-fractional derivative is defined by ( 𝑥0𝐷𝑥𝛼 )[𝑓(𝑥)] =
1 𝑑 𝑥 𝑓(𝑡)−𝑓(𝑥0 ) 𝑑𝑡 ∫ Γ(1−𝛼) 𝑑𝑥 𝑥0 (𝑥−𝑡)𝛼
,
(2)
And the Jumarie type of Riemann-Liouville 𝛼-fractional integral is defined by ( 𝑥0𝐼𝑥𝛼 )[𝑓(𝑥)] =
𝑥 1 𝑓(𝑡) 𝑑𝑡 ∫ Γ(𝛼) 𝑥0 (𝑥−𝑡)1−𝛼
,
(3)
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