ISSN 2348-1196 (print) International Journal of Computer Science and Information Technology Research ISSN 2348-120X (online) Vol. 10, Issue 4, pp: (17-23), Month: October - December 2022, Available at: www.researchpublish.com
Picard Iterative Method for Solving Fractional Differential Equations Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong, China DOI: https://doi.org/10.5281/zenodo.7257957
Published Date: 27-October-2022
Abstract: In this paper, based on Jumarie’s modified Riemann-Liouville (R-L) fractional calculus, some examples are provided to illustrate how to use Picard iterative method to find the approximation solution of fractional differential equation. A new multiplication of fractional analytic functions plays an important role in this paper. In fact, our results are generalization of these results of ordinary differential equations. Keywords: Jumarie’s modified R-L fractional calculus, Picard iterative method, approximation solution, fractional differential equation, new multiplication, fractional analytic functions.
I. INTRODUCTION Fractional calculus is a branch of mathematical analysis, which studies several different possibilities of defining real or complex order. In the past decades, fractional calculus has developed rapidly in mathematics and applied science. Fractional calculus is very popular in many fields, such as mechanics, dynamics, control theory, physics, economics, viscoelasticity, biology, electrical engineering, etc [1-8]. However, the definition of fractional derivative is not unique. Commonly used definitions include Riemann-Liouville (R-L) fractional derivative, Caputo fractional derivative, Grunwald-Letnikov (G-L) fractional derivative, Jumarie’s modified R-L fractional derivative [9-14]. 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 classical calculus. In this paper, based on Jumarie type of R-L fractional calculus, we provide some examples to illustrate how to use Picard iterative method to find the approximation solution of fractional differential equation. A new multiplication of fractional analytic functions plays an important role in this article. In fact, our results are generalization of these results of ordinary differential equations.
II. PRELIMINARIES Firstly, we introduce the fractional calculus used in this paper. Definition 2.1 ([15]): Let 0 < 𝛼 ≤ 1, and 𝑥0 be 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)
where Γ( ) is the gamma function. In the following, some properties of Jumarie type of fractional derivative are introduced.
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