International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online) Vol. 10, Issue 2, pp: (33-39), Month: October 2022 - March 2023, Available at: www.researchpublish.com
Method of Fractional Variation of Parameter and Its Applications Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong, China DOI: https://doi.org/10.5281/zenodo.7255065
Published Date: 26-October-2022
Abstract: Based on Jumarie’s modified Riemann-Liouville (R-L) fractional calculus, this paper studies the method of fractional variation of parameter. The product rule for fractional derivatives and a new multiplication of fractional analytic functions play important roles in this article. In addition, two examples are provided to illustrate how to use the method of fractional variation of parameter to find the particular solution of fractional differential equations. In fact, our results are generalizations of these results in ordinary differential equations. Keywords: Jumarie’s modified R-L fractional calculus, Method of fractional variation of parameter, Product rule, New multiplication, Fractional analytic functions, Particular solution, Fractional differential equations.
I. INTRODUCTION Fractional calculus comes from the generalization of differential and integral operators, which are applied to non- integer orders. In the past decades, fractional calculus has been widely used in quantum mechanics, electronic engineering, viscoelasticity, control theory, dynamics, economics and other fields [1-6]. 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 modification of R-L fractional derivative [7-11]. 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 traditional calculus. In this paper, based on the Jumarie type of R-L fractional calculus and a new multiplication of fractional analytic functions, the method of fractional variation of parameter is studied. The product rule for fractional derivatives plays an important role in this article. Moreover, we give some examples to illustrate how to use the method of fractional variation of parameter to find the particular solution of fractional differential equations. In fact, these results we obtained are generalizations of those results in ordinary differential equations.
II. PRELIMINARIES First, we introduce the fractional calculus used in this paper. Definition 2.1 ([12]): If 0 < 𝛼 ≤ 1, and 𝑥0 is a real number. The Jumarie type of 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 Riemann-Liouville 𝛼-fractional integral is defined by ( 𝑥0𝐼𝑥𝛼 )[𝑓(𝑥)] =
𝑥 1 𝑓(𝑡) 𝑑𝑡 ∫ Γ(𝛼) 𝑥0 (𝑥−𝑡)1−𝛼
,
(2)
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