Modified Fractional Logistic Equation

TL;DR

This paper introduces a modified fractional logistic equation with a new solution, the West function, which involves Mittag-Leffler functions and can be applied to nonlinear fractional differential equations in physics.

Contribution

The paper proposes a modified fractional logistic equation where the West function is an exact solution, extending the applicability of fractional integro-differential equations in physics.

Findings

The West function accurately solves the modified fractional logistic equation.

The method can analyze solutions to nonlinear fractional differential equations.

The equation includes a nonlinear additive term related to the classical logistic equation.

Abstract

In the article [B.J.West, Exact solution to fractional logistic equation, Physica A: Statistical Mechanics and its Applications 429 (2015) 103-108], the author has obtained a function as the solution to fractional logistic equation (FLE). As demonstrated later in [I. Area, J. Losada, J. J. Nieto, A note on the fractional logistic equation, Physica A: Statistical Mechanics and its Applications 444 (2016) 182-187], this function (West function) is not the solution to FLE, but nevertheless as shown by West, it is in good agreement with the numerical solution to FLE. The West function indicates a compelling feature, in which the exponentials are substituted by Mittag-Leffler functions. In this paper, a modified fractional logistic equation (MFLE) is introduced, to which the West function is a solution. The proposed fractional integro-differential equation possesses a nonlinear additive term related to the solution of the logistic equation (LE). The method utilized in this article, may be applied to the analysis of solutions to nonlinear fractional differential equations of mathematical physics.