Application of Fractional Laplace Transform Method in Solving Linear Systems of Fractional Differe

ISSN 2348-1218 (print) International Journal of Interdisciplinary Research and Innovations ISSN 2348-1226 (online) Vol. 10, Issue 4, pp: (28-34), Month: October 2022 - December 2022, Available at: www.researchpublish.com

Application of Fractional Laplace Transform Method in Solving Linear Systems of Fractional Differential Equations Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong, China DOI: https://doi.org/10.5281/zenodo.7274410

Published Date: 02-November-2022

Abstract: Based on Jumarie type of Riemann-Liouville (R-L) fractional calculus, this paper provides several examples to illustrate how to use fractional Laplace transform to find the solution of linear system of fractional differential equations. A new multiplication of fractional analytic functions plays an important role in this article. In fact, our results are generalizations of those results in ordinary differential equations. Keywords: Jumarie type of R-L fractional calculus, fractional Laplace transform, linear system of fractional differential equations, new multiplication, fractional analytic functions.

I. INTRODUCTION 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, Hardy, Littlewood, and Weyl. In the last decades, fractional calculus has played a very important role in physics, dynamics, electrical engineering, viscoelasticity, economics, biology, control theory, 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 [1115]. 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 Jumarie’s modification of R-L fractional calculus, some examples are provided to illustrate how to use fractional Laplace transform method to solve linear system of fractional differential equations. A new multiplication of fractional analytic functions plays an important role in this paper. Moreover, our results are natural generalizations of those results in ordinary differential equations.

II. DEFINITIONS AND PROPERTIES Firstly, 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 type of 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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