Exact Solution of Linear System of Fractional D

International Journal of Mechanical and Industrial Technology

ISSN 2348-7593 (Online) Vol. 10, Issue 2, pp: (1-7), Month: October 2022 - March 2023, Available at: www.researchpublish.com

Exact Solution of Linear System of Fractional Differential Equations with Constant Coefficients Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong, China DOI: https://doi.org/10.5281/zenodo.7391108

Published Date: 02-December-2022

Abstract: In this paper, based on Jumarie type of Riemann-Liouville (R-L) fractional derivative, the exact solution of linear system of fractional differential equations with constant coefficients is obtained. A new multiplication of fractional analytic functions plays an important role in this paper. In addition, we also provide some examples to illustrate the application of our results. In fact, our results are generalizations of these results in ordinary differential equations. Keywords: Jumarie type of R-L fractional derivative, exact solution, linear system of fractional differential equations with constant coefficients, new multiplication, fractional analytic functions.

I. INTRODUCTION The history of fractional calculus is almost as long as the development of traditional calculus. In 1695, the concept of fractional derivative first appeared in a famous letter between L’Hospital and Leibniz. Many great mathematicians have further developed this field, such as Euler, Lagrange, Laplace, Fourier, Abel, Liouville, Riemann, and Weyl. In the past few decades, fractional calculus has played a very important role in physics, electrical engineering, economics, biology, control theory, and other fields [1-7]. 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, Jumarie type of R-L fractional derivative [8-12]. Since the Jumarie type of R-L fractional derivative makes the derivative of constant function equal to zero, it is easier to use this definition to connect fractional calculus with classical calculus. In this paper, based on Jumarie’s modified R-L fractional derivative, we obtain the exact solution of linear system of fractional differential equations with constant coefficients. A new multiplication of fractional analytic functions plays an important role in this article. Moreover, two examples are provided to illustrate the application of our results. And our results are generalizations of these results in ordinary differential equations.

II. PRELIMINARIES Firstly, the fractional calculus used in this paper and some important properties are introduced below. Definition 2.1 ([13]): Assume 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)

where Γ( ) is the gamma function. Proposition 2.2 ([14]): If 𝛼, 𝛽, 𝑡0 , 𝑐 are real numbers and 𝛽 ≥ 𝛼 > 0, then ( 𝑡0𝐷𝑡𝛼 )[(𝑡 − 𝑡0 )𝛽 ] =

Γ(𝛽+1) Γ(𝛽−𝛼+1)

(𝑡 − 𝑡0 )𝛽−𝛼 ,

(2)

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