ISSN 2348-1218 (print) International Journal of Interdisciplinary Research and Innovations ISSN 2348-1226 (online) Vol. 10, Issue 4, pp: (48-53), Month: October 2022 - December 2022, Available at: www.researchpublish.com
Fractional Differential Problem of Some Fractional Trigonometric Functions Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong, China DOI: https://doi.org/10.5281/zenodo.7458073
Published Date: 19-December-2022
Abstract: In this paper, based on Jumarie’s modified Riemann-Liouville (R-L) fractional calculus, we study the fractional differential problem of two types of fractional trigonometric functions. Using chain rule for fractional derivatives and a new multiplication of fractional analytic functions, we can obtain the fractional derivatives of any order of these two types of fractional trigonometric functions. On the other hand, we provide some examples to illustrate our methods. In fact, our results are generalizations of those results in classical calculus. Keywords: Jumarie’s modified R-L fractional calculus, fractional trigonometric functions, chain rule for fractional derivatives, new multiplication, fractional analytic functions.
I. INTRODUCTION 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 progress of fractional calculus is mainly in physics, mechanics, electrical engineering, economics, viscoelasticity, biology, control theory, and other fields [1-10]. There is no doubt that fractional calculus has become an exciting new mathematical tool to solve various problems in mathematics, science and engineering. 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 [11-15]. 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 type of R-L fractional calculus, the fractional derivatives of any order of two types of fractional trigonometric functions are obtained. A new multiplication of fractional analytic functions and chain rule for fractional derivatives play important roles in this article. Moreover, we give two examples to illustrate the application of our results. In fact, our results are natural generalizations of those results in ordinary calculus.
II. DEFINITIONS AND PROPERTIES Firstly, the fractional calculus used in this paper and its properties are introduced below. Definition 2.1 ([16]): Let 0 < 𝛼 ≤ 1, and 𝜃0 be a real number. The Jumarie type of Riemann-Liouville (R-L) 𝛼-fractional derivative is defined by ( 𝜃0𝐷𝜃𝛼 )[𝑓(𝜃)] =
1 𝑑 𝜃 𝑓(𝑡)−𝑓(𝜃0 ) 𝑑𝑡 ∫ Γ(1−𝛼) 𝑑𝜃 𝜃0 (𝜃−𝑡)𝛼
.
(1)
And the Jumarie’s modified R-L 𝛼-fractional integral is defined by ( 𝜃0𝐼𝜃𝛼 )[𝑓(𝜃)] =
𝜃 1 𝑓(𝑡) 𝑑𝑡 ∫ Γ(𝛼) 𝜃0 (𝜃−𝑡)1−𝛼
,
(2)
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