International Journal of Engineering Research and Reviews
ISSN 2348-697X (Online) Vol. 8, Issue 4, pp: (33-37), Month: October - December 2020, Available at: www.researchpublish.com
Research on First Order Linear Fractional Differential Equations Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong Province, China
Abstract: In this paper, we use product rule of fractional functions, integrating factor, and constant variation method to obtain the general solution of first order linear fractional differential equation (LFDE), regarding Jumarie’s modified Riemann-Liouville (R-L) fractional derivative. Moreover, an example is proposed for demonstrating the advantage of our result. Keywords: product rule, integrating factor, constant variation method, first order LFDE, Jumarie’s modified R-L fractional derivative.
I. INTRODUCTION The classical calculus provides a power tool to model and explain many important dynamically processes in most parts of applied areas of the sciences. But There are many complex systems in nature with anomalous dynamics, including biology, chemistry, physics, geology, astrophysics and social sciences, and more in particular in transport of chemical contaminant through water around rocks, dynamics of viscoelastic materials as polymers, signals theory, control theory, electromagnetic theory, and many more their dynamics cannot be characterized by classical derivative models. (for detail [1-6]). Fractional calculus is the calculus of differentiation and integration of non-integer orders. During last three decades or so, fractional calculus has gained much attention due to its demonstrated applications in various fields of science and engineering [6-9]. There are many good textbooks of fractional calculus and fractional differential equations, such as [1012]. For various applications of fractional calculus in physics, see [3, 6, 7, 8, 9]. Unlike standard calculus, there is no unique definition of derivation and integration in fractional calculus. The commonly used definition is the RiemannLiouville (R-L) fractional derivative [5]. Other useful definitions include Caputo definition of fractional derivative (1967) [13], the Grunwald-Letinikov (G-L) fractional derivative [5], and Jumarie’s modified R-L fractional derivative is used to avoid nonzero fractional derivative of constant functions [14]. In this paper, the first order linear fractional differential equation (LFDE), regarding the Jumarie type of modified R-L fractional derivatives is the generalization of first order linear ordinary differential equation. We define a new multiplication of fractional functions and use the product rule and the integrating factor method to obtain the general solution of the first order LFDE. On the other hand, an example is given to demonstrate the advantage of our result.
II. MATERIALS AND METHODS At first, the fractional calculus adopted in this paper is introduced below. Definition 2.1: Suppose that is a real number and derivatives of Jumarie type ([15]) is defined by (
[ ( )]
(
{
)
)
(
∫ (
)
∫ (
)
is a positive integer The modified Riemann-Liouville fractional ( )
[ ( )
( )]
(1)
)[ ( )]
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