Research on Some Fractional Integral Problems

International Journal of Engineering Research and Reviews

ISSN 2348-697X (Online) Vol. 10, Issue 3, pp: (24-28), Month: July - September 2022, Available at: www.researchpublish.com

Research on Some Fractional Integral Problems Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong, China DOI: https://doi.org/10.5281/zenodo.7032499

Published Date: 29-August-2022

Abstract: In this paper, we solve two type of fractional integrals based on Jumarie’s modified Riemann-Liouville (RL) fractional calculus. The main two methods used in this article are change of variables for fractional integral and integration by parts for fractional calculus. On the other hand, a new multiplication of fractional analytic functions plays an important role in this paper. And these two types of fractional integrals are generalizations of the integrals in traditional calculus. Keyword: fractional integrals, Jumarie’s modified R-L fractional calculus, change of variables, integration by parts, new multiplication, fractional analytic functions.

I. INTRODUCTION Fractional calculus is a branch of mathematical analysis, which studies several different possibilities of defining real order or complex order. In the second half of the 20th century, a large number of studies on fractional calculus were published in engineering literature. Fractional calculus is widely welcomed and concerned because of its applications in many fields such as mechanics, dynamics, modelling, physics, economics, viscoelasticity, biology, electronics, signal processing, and so on [1-9]. However, fractional calculus is different from ordinary calculus. The definition of fractional derivative and integral is not unique. Commonly used definitions include Riemann-Liouville (R-L) fractional derivative, Caputo fractional derivative, Grunwald-Letnikov (G-L) fractional derivative, and Jumarie’s modified R-L fractional derivative [10-13]. In this paper, we study the following two type of fractional integrals: ( 0𝐼𝑥𝛼 ) [( ( 0𝐼𝑥𝛼 ) [(

1 Γ(𝛼+1)

1 Γ(𝛼+1)

𝑥𝛼)

𝑥𝛼)

⨂𝑝

⨂𝑚

⨂(𝐸𝛼 (𝑥 𝛼 ))⨂𝑛 ],

⨂ (𝐿𝑛𝛼 (1 +

1 Γ(𝛼+1)

(1)

𝑥 𝛼 ))

⨂𝑞

].

(2)

Where 0 < α ≤ 1, and 𝑚, 𝑛, 𝑝, 𝑞 are positive integers. The change of variables for fractional integral and the integration by parts for fractional calculus are the main methods used in this article. In addition, a new multiplication of fractional analytic functions plays an important role in this paper. In fact, the above two types of fractional integrals are generalizations of the integrals in classical calculus.

II. PRELIMINARIES First, the fractional calculus used in this paper and its properties are introduced below. Definition 2.1 ([14]): Let 0 < 𝛼 ≤ 1, and 𝑥0 be a real number. The Jumarie type of Riemann-Liouville (R-L) 𝛼-fractional derivative is defined by ( 𝑥0 𝐷𝑥𝛼 )[𝑓(𝑥)] =

1 𝑑 𝑥 𝑓(𝑡)−𝑓(𝑥0 ) 𝑑𝑡 ∫ Γ(1−𝛼) 𝑑𝑥 𝑥0 (𝑥−𝑡)𝛼

.

(3)

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