International Journal of Electrical and Electronics Research ISSN 2348-6988 (online) Vol. 9, Issue 2, pp: (19-24), Month: April - June 2021, Available at: www.researchpublish.com
Fractional Mean Value Theorem and Its Applications Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong Province, China
Abstract: Based on the Jumarie type of modified Riemann-Liouville (R-L) fractional derivatives, the method used in this paper is to first transform the definition of modified R-L fractional derivatives into the form of limit, and then use fractional Fermat’s theorem and fractional Rolle’s theorem to prove our main result: fractional mean value theorem. In fact, this result is the generalization of mean value theorem for classical calculus. On the other hand, we provide some examples to illustrate the applications of fractional mean value theorem. Keywords: Jumarie type of modified R-L fractional derivatives, form of limit, fractional Fermat’s theorem, fractional Rolle’s theorem, fractional mean value theorem.
I. INTRODUCTION Fractional calculus belongs to the field of mathematical analysis, involving the research and applications of arbitrary order integrals and derivatives. Fractional calculus originated from a problem put forward by L’Hospital and Leibniz in 1695. Therefore, the history of fractional calculus was formed more than 300 years ago, and fractional calculus and classical calculus have almost the same long history. Since then, fractional calculus has attracted the attention of many contemporary great mathematicians, such as N. H. Abel, M. Caputo, L. Euler, J. Fourier, A. K. Grunwald, J. Hadamard, G. H. Hardy, O. Heaviside, H. J. Holmgren, P. S. Laplace, G. W. Leibniz, A. V. Letnikov, J. Liouville, B. Riemann, M. Riesz, and H. Weyl. With the efforts of researchers, the theory of fractional calculus and its applications have developed rapidly. On the other hand, fractional calculus has wide applications in continuum mechanics, quantum mechanics, electrical engineering, fluid science, viscoelasticity, control theory, dynamics, finance, and so on [4-20, 29]. Moreover, the applications of fractional calculus to fractional differential equations can refer to [21-28]. However, different from the traditional calculus, the rule of fractional derivative is not unique, many scholars have given the definitions of fractional derivatives. The common definition is Riemann-Liouville (R-L) fractional derivatives [1-2]. Other useful definitions include Caputo fractional derivatives, Grunwald-Letnikov (G-L) fractional derivatives [1], and Jumarie type of R-L fractional derivatives to avoid non-zero fractional derivative of constant function [3]. In this paper, we make use of fractional Fermat’s theorem and fractional Rolle’s theorem to prove our major result: fractional mean value theorem. In fact, the fractional mean value theorem is the generalization of mean value theorem in traditional calculus. In addition, two examples are proposed to illustrate its applications.
II. METHODS AND RESULTS The following is the fractional calculus used in this article. Definition 2.1: Assume that is a real number and derivatives of Jumarie type ([12]) is defined by (
[ ( )]
(
{
)
)
(
∫ (
)
∫ (
)
is a positive integer The modified Riemann-Liouville fractional
( ) [ ( )
( )]
(1)
)[ ( )]
Page | 19 Research Publish Journals