Fractional Euler numbers and generalized proportional fractional logistic differential equation

We solve a logistic differential equation for generalized proportional Caputo fractional derivative. The solution is found as a fractional power series. The coefficients of that power series are related to the Euler polynomials and Euler numbers as well as to the sequence of Euler’s fractional numbers recently introduced. Some numerical approximations are presented to show the good approximations obtained by truncating the fractional power series. This generalizes previous cases including the Caputo fractional logistic differential equation and Euler’s numbers.

The solution, for a given initial condition u(0) = u 0 , is B Juan J. Nieto juanjose.nieto.roig@usc.es For u 0 = 1/2, we have the classical logistic function Different versions and generalizations of the logistic equation have been considered and, in particular, the fractional versions of the logistic differential equation [2-5, 12, 15].
For example, a fractional version has been studied: with α ∈ (0, 1) and D α the Caputo fractional derivative [7]. Although an analytical expression for the solutions is not known, it has been solved using different techniques such Euler's numbers [4,6], implicit solutions [10] or fractional power series [3].
We study a new generalization of the fractional differential equation (1.1) that includes as a particular case the Caputo fractional logistic differential equation.
Moreover, we use fractional generalized proportional derivative having a singular kernel.
We introduce a novel class of Euler's numbers, the generalized proportional fractional Euler numbers. We recall that Euler's polynomials and Euler's numbers are related to the Riemann's zeta function and to the logistic function [3]. The relevance is apparent due to the importance of solving the famous Riemann Hypotheses. This paper is organized as follows. In the next section we introduce the generalized proportional fractional calculus with its basic concepts and properties. Then, it is solved a simple linear fractional differential equations to motivate our technique in order to solve a generalized fractional logistic differential equation. Finally in the last section, we present the generalized proportional fractional Euler's numbers. Euler's numbers appear in connection to the most important function in mathematics: the zeta function.

Generalized proportional calculus
Let T > 0, α > 0 be the order of the fractional integral and ρ ∈ (0, 1] be the proportion. For a function u ∈ L 1 (0, T ) we define the generalized proportional integral of the function u as The corresponding Caputo generalized proportional fractional derivative for a function u ∈ L 1 (0, T ) such that u ∈ AC[0, T ] is defined as [1] We note that for ρ = 1 we obtain the classical Caputo fractional derivative [7]: We recall that [1] ( Also, for β > 0, consider the function We now study some differential equations under this generalized fractional calculus. Indeed, consider the nonlinear differential equation of the type with the initial condition Here f : [0, T ] × R → R is a nonlinear function satisfying appropriate conditions.

Linear generalized proportional differential equations
Let σ ∈ L 1 (0, T ) so that the corresponding generalized proportional integral of σ exists.
We begin with the simple case By applying the generalized proportional fractional integral, we have Therefore, Now, for λ ∈ R, let us study the following linear differential equation The solution is known [1] where E α is the classical Mittag-Leffler function defined for any z ∈ C as .
We now re-obtain this solution as a fractional power series. Moreover, this will serve as a clear introduction to our methodology. Indeed, take r = ρ−1 ρ and let us assume that the solution is given formally as the following fractional power series Thus, formally, Equivalently, e rt ∞ n=0 a n (t α ) n = u 0 e rt + λ ∞ n=1 a n−1 Γ (α(n − 1) + 1) ρ α Γ (αn + 1) e rt t αn .

Logistic generalized proportional differential equations
We now consider a logistic-type equation corresponding to the nonlinear equation where λ, μ ∈ R and r = ρ−1 ρ , that is, the logistic fractional generalized proportional differential equation (4.1) For μ = 0 we have the previous linear equation (3.1).
In the case ρ = 1, that is, r = 0, we obtain the following Caputo fractional logistic differential equation that has been solved recently [3].
The solution of new the logistic equation (4.1) is given by (4.2) and can be approximated by (See Fig. 3) p m (t) = e rt m n=0 a n (t α ) n , m ≥ 1.
This logistic function is the solution of the logistic problem It is well-known that the coefficients a n are related to the Euler numbers E n by a n = E n 2 · n! .

Conclusions
We have introduced a new generalization of the fractional logistic differential equation.
To find an explicit solution as a fractional power series, one is lead to the corresponding general fractional Euler's numbers. Some figures are plotted to illustrate the results in order to compare the solutions of the classical logistic equation, of the Caputo fractional logistic differential equations and the new generalized proportional fractional logistic differential equation.
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