Solutions of fractional logistic equations by Euler's numbers

TL;DR

This paper introduces a novel method for solving fractional logistic equations within the convergence set by leveraging Euler's numbers, providing explicit solutions and extending classical logistic models to fractional calculus.

Contribution

The paper presents a new approach connecting Euler's numbers with fractional logistic equations, offering explicit solutions within the convergence set, which was previously an open problem.

Findings

Explicit solutions within the convergence set using Euler's numbers

Connection established between coefficients and Euler's numbers

Extension of logistic equations to fractional calculus context

Abstract

In this paper, we solve in the convergence set, the fractional logistic equation making use of Euler's numbers. To our knowledge, the answer is still an open question. The key point is that the coefficients can be connected with Euler's numbers, and then they can be explicitly given. The constrained of our approach is that the formula is not valid outside the convergence set, The idea of the proof consists to explore some analogies with logistic function and Euler's numbers, and then to generalize them in the fractional case.