A Study of Fractional Line Integral

ISSN 2348-1218 (print) International Journal of Interdisciplinary Research and Innovations ISSN 2348-1226 (online) Vol. 10, Issue 3, pp: (46-50), Month: July 2022 - September 2022, Available at: www.researchpublish.com

A Study of Fractional Line Integral Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong, China DOI: https://doi.org/10.5281/zenodo.7027259

Published Date: 27-August-2022

Abstract: In this paper, we study the fractional line integral based on Jumarie type of Riemann-Liouville (R-L) fractional calculus. The major method we used is a new multiplication of fractional analytic functions. On the other hand, two examples are provided to illustrate the fractional line integral. In fact, the fractional line integral is the generalization of line integral in classical calculus. Keyword: Fractional line integral, Jumarie type of R-L fractional calculus, new multiplication, fractional analytic functions.

I. INTRODUCTION The history of fractional calculus is almost as long as the development of ordinary calculus theory. As early as 1695, L’Hospital wrote to Leibniz to discuss the fractional derivative of a function. However, it was not until 1819 that Lacroix first proposed the result of a simple function with fractional derivative. Then, after hundreds of years of development, mathematicians such as Euler, Laplace, Fourier, Abel, Riemann, Liouville, Grunwald, Letnikov, Weyl, and Ritz conducted in-depth research, which promoted the development of this discipline. In the past few decades, fractional calculus has been applied to many fields, such as physics, dynamics, signal processing, robotics, electrical engineering, viscoelasticity, economics, bioengineering, control theory, and electronics [1-10]. However, the definition of fractional derivative is not unique, there are many useful definitions, including Riemann-Liouville (R-L) fractional derivative, Caputo fractional derivative, Grunwald-Letnikov (G-L) fractional derivative, Jumarie’s modified RL fractional derivative [11-15]. Jumarie modified the definition of R-L fractional derivative with a new formula, and we obtained that the modified fractional derivative of a constant function is zero. Therefore, it is easier to connect fractional calculus with classical calculus by using this definition. This paper studies the fractional line integral based on Jumarie’s modification of R-L fractional calculus. A new multiplication of fractional analytic functions plays an important role in this paper. Moreover, some examples are given to illustrate the fractional line integral. In fact, the fractional line integral is the natural generalization of line integral in ordinary calculus.

II. PRELIMINARIES First, we introduce the fractional calculus used in this paper and some important properties. Definition 2.1 ([16]): Suppose that 0 < 𝛼 ≤ 1, and 𝑡0 is a real number. The Jumarie’s modified R-L 𝛼-fractional derivative is defined by ( 𝑡0𝐷𝑡𝛼 )[𝑓(𝑡)] =

1 𝑑 𝑡 𝑓(𝑥)−𝑓(𝑡0 ) 𝑑𝑥 , ∫ Γ(1−𝛼) 𝑑𝑡 𝑡0 (𝑡−𝑥)𝛼

(1)

And the Jumarie′s modified R-L 𝛼-fractional integral is defined by ( 𝑡0𝐼𝑡𝛼 )[𝑓(𝑡)] =

𝑡 1 𝑓(𝑥) 𝑑𝑥 ∫ Γ(𝛼) 𝑡0 (𝑡−𝑥)1−𝛼

,

(2)

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