International Journal of Electrical and Electronics Research ISSN 2348-6988 (online) Vol. 10, Issue 4, pp: (17-22), Month: October - December 2022, Available at: www.researchpublish.com
Fractional Integral Curves of Some Fractional Differential Equations Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong, China DOI: https://doi.org/10.5281/zenodo.7298955
Published Date: 07-November-2022
Abstract: In this paper, based on Jumarie’s modification of Riemann-Liouville (R-L) fractional calculus, we find the fractional integral curves of some fractional differential equations. A new multiplication of fractional analytic functions and product rule for fractional derivatives play important roles in this article. In fact, our results are generalizations of the results in ordinary differential equations. Keywords: Jumarie’s modification of R-L fractional calculus, fractional integral curves, fractional differential equations, new multiplication, fractional analytic functions, product rule.
I. INTRODUCTION Fractional calculus is a field of mathematical analysis. It studies and applies integrals and derivatives of any order. Fractional calculus originated in 1695, almost at the same time as classical calculus. In the second half of the 20th century, a large number of studies on fractional calculus were published in engineering literature. In fact, the latest development of fractional calculus is widely used in differential and integral equations, physics, mechanics, control theory, economics, viscoelasticity, biology, electrical engineering, and other fields [1-10]. However, the definition of fractional derivative is not unique. Common 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 [11-14]. Based on Jumarie’s modified R-L fractional calculus, we study the fractional integral curves of two kinds of fractional differential equations. A new multiplication of fractional analytic functions and product rule for fractional derivatives play important roles in this paper. In fact, our results are generalizations of these results in ordinary differential equations.
II. DEFINITIONS AND PROPERTIES First, we introduce the fractional calculus used in this paper. Definition 2.1 ([15]): Assume 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
,
(1)
And the Jumarie type of R-L 𝛼-fractional integral is defined by ( 𝑥0 𝐼𝑥𝛼 )[𝑓(𝑥)] =
𝑥 1 𝑓(𝑡) 𝑑𝑡 ∫ Γ(𝛼) 𝑥0 (𝑥−𝑡)1−𝛼
,
(2)
𝑛
where Γ( ) is the gamma function. Moreover, we define ( 𝑥0𝐷𝑥𝛼 ) [𝑓(𝑥)] = ( 𝑥0𝐷𝑥𝛼 )( 𝑥0𝐷𝑥𝛼 ) ∙∙∙ ( 𝑥0𝐷𝑥𝛼 )[𝑓(𝑥)], and it is called the 𝑛-th order 𝛼-fractional derivative of 𝑓(𝑥), where 𝑛 is any positive integer. In the following, some properties of Jumarie’s fractional derivative are proposed.
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