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.

Introduction

The logistic ordinary differential equation appears in many contexts and have different applications in physics [3, 8, 9, 11], medicine [17], economy [16], and even to study the evolution of the COVID-19 epidemic [13, 14].

The solution, for a given initial condition isFor we have the classical logistic functionDifferent 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 and 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 be the order of the fractional integral and be the proportion. For a function we define the generalized proportional integral of the function u asThe corresponding Caputo generalized proportional fractional derivative for a function such that is defined as [1]whereWe note that for we obtain the classical Caputo fractional derivative [7]:We recall that [1]Also, for , consider the functionthenWe now study some differential equations under this generalized fractional calculus. Indeed, consider the nonlinear differential equation of the typewith the initial conditionHere is a nonlinear function satisfying appropriate conditions.

Linear generalized proportional differential equations

Let so that the corresponding generalized proportional integral of exists.

We begin with the simple caseBy applying the generalized proportional fractional integral, we haveTherefore,Now, for , let us study the following linear differential equationThe solution is known [1]where is the classical Mittag-Leffler function defined for any asWe now re-obtain this solution as a fractional power series. Moreover, this will serve as a clear introduction to our methodology. Indeed, take and let us assume that the solution is given formally as the following fractional power seriesThus, formally,andEquivalently,Identifying the coefficients, we get and the recurrence formulaThis implies thatso thatSee Fig. 1. For , that is we have the classical fractional Caputo equationwhose solution is indeed given by

Fig. 1

Solutions of the linear generalized proportional fractional differential equations (3.1) with for the initial condition and . For in blue. For the Caputo fractional differential equation (3.2) in the middle (orange) and the classical logistic function below (green)

Logistic generalized proportional differential equations

We now consider a logistic-type equation corresponding to the nonlinear equation (2.1) withwhere and that is, the logistic fractional generalized proportional differential equationFor we have the previous linear equation (3.1).

In the case , that is, , we obtain the following Caputo fractional logistic differential equationthat has been solved recently [3].

As for the linear situation, we assume thatThen, using the Cauchy product we getTherefore,Identifying the coefficients corresponding to the powers of , we getand for , the recurrence relationorTaking the initial condition so that Thus, for example,andFor example, for , and we have the following values of for (See Fig. 2)

Fig. 2

Coefficients for the generalized proportional fractional logistic equation for and

The solution of the classical ordinary differential equationwith the initial conditionis the logistic functionIn this case, and the recurrence relation isThe solution of new the logistic equation (4.1) is given by (4.2) and can be approximated by (See Fig. 3)

Fig. 3

Approximate solution of the logistic generalized proportional fractional differential equations (4.1) for the initial condition and . For in blue. For the corresponding approximate solution of the Caputo fractional logistic differential equation (3.2) (orange) and the classical logistic function below (green)

Generalized proportional Euler numbers

We recall the Euler polynomials defined asTaking we derivewhereThis logistic function is the solution of the logistic problemIt is well-known that the coefficients are related to the Euler numbers byFor a given and in view of (4.3), we define the generalized proportional fractional Euler numbers by the recurrence relationFor we denote . If in addition then

Obviously,Also,whereare the Euler fractional numbers introduced in [12] and studied in [3].

We therefore have generalized the Euler numbers and the Euler fractional numbers to the generalized proportional Euler’s fractional numbers

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.