International Journal of Engineering Research and Reviews
ISSN 2348-697X (Online) Vol. 9, Issue 2, pp: (13-17), Month: April - June 2021, Available at: www.researchpublish.com
A New Insight into Fractional Logistic Equation Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong Province, China
Abstract: In this article, based on Jumarie type of modified Riemann-Liouville (R-L) fractional derivatives, we make use of a new multiplication and some techniques include separation of variables, partial fraction integration and chain rule for fractional derivatives to obtain the closed form solution of the fractional logistic equation we defined. In fact, the fractional logistic equation is a generalization of the classical logistic equation. Moreover, the Mittag-Leffler function plays an important role in this paper. Keywords: Jumarie type of modified R-L fractional derivatives, new multiplication, closed form solution, fractional logistic equation, Mittag-Leffler function.
I. INTRODUCTION For centuries, mathematics has played a vital role in the development of human civilization, because in other fields, it allows the description and prediction of events in the real world, through mathematical representations. In this regard, it is reasonable to emphasize the importance of differential calculus and integral calculus for the study of many of the laws of nature. Fractional calculus is the study of derivatives and integrals of arbitrary orders. For a long time, the theory of fractional calculus developed only as a theoretical field of mathematics. However, in the last decades, it was shown that some fractional operators can better describe some complex physical phenomena, so fractional calculus has been paid more and more attention by mathematicians. On the other hand, physicists and engineers are also very interested in the applications of this nice theory. Many real life phenomena have been described using fractional differential equations, such as viscoelasticy, continuum mechanics, optimal control, hydrologic modelling, variational problems, fluid mechanics, finance, and others [1-17]. Furthermore, the applications of fractional calculus to fractional differential equations can refer to [18-25]. Logistic equation is a famous population growth model introduced by mathematical biologist Pierre Francois Verhulst [26]. It is the extension of Malthus population model. The logistic population model is considered as an important type of nonlinear differential equations because it can be widely used in biology, medicine, economics and management. The classical logistic (or Verhulst’s) equation is the nonlinear initial value problem: ( )
{
( )(
( ))
,
(1)
( ) where ( ) denotes the population at time , is the rate of maximum population growth, is the population at time , and is the carrying capacity, i.e., the maximum attainable value of population. By dividing both sides of Eq. (1) by
and defining
( )
( ) as the normalization of population to its maximum attainable value,
we obtain the differential equation with initial value: { For each realization of
( )
( )( ( )
( ))
.
(2)
, Eq. (2) has an exact closed form solution:
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