Fractional Bernoulli equation and logistic function

Question

I'm investigating the solution of the special case of the Bernoulli differential equation $$ \dfrac{dy}{dt} = \dfrac{y(1-y)}{\tau}, \tag{1} $$ with $\tau$ a time constant, and which models innovation processes fairly well, and whose solution is the logistic curve $$ y = \dfrac{1}{1+e^{-t/\tau}}.\tag{2}$$ Recently I've become acquainted with fractional calculus, and I'm interested in finding out if the solution to the corresponding fractional Bernoulli differential equation $$D^{\alpha} y = \dfrac{y(1-y)}{\tau}, \ \alpha\in(0,1) \tag{3}$$ is given by substituting the exponential with the one-parameter Mittag-Leffler function $$E_{\alpha}\left(-\dfrac{t^\alpha}{\tau}\right) \tag{4}$$ that is $$ y = \frac{1}{1+E_{\alpha}\left(-\dfrac{t^\alpha}{\tau}\right)}. \tag{5}$$ I've consulted a couple of papers that for ordinary differential equations show an almost direct correspondence between exponential and Mittag-Leffler, but do not have a solution to my question. I would be grateful for pointing me to the correct solution and a paper showing it.

Mainardi, Francesco, On some properties of the Mittag-Leffler function (E_\alpha(-t^\alpha)), completely monotone for (t>0) with (0<\alpha<1), Discrete Contin. Dyn. Syst., Ser. B 19, No. 7, 2267-2278 (2014). ZBL1303.26007.

Luchko, Yuri, [Operational method for fractional ordinary differential equations; in Handbook of fractional calculus with applications], vol.2 - Fractional Differential Equations](2022).

Answer 1

The conjecture of the OP is wrong. The attached graph proves it; it compares the numerical resolution of equation (3) with the hypothesis (5), showing that they are different functions.

The calculations are preformed using the following code:Garrappa, Roberto, Numerical solution of fractional differential equations: a survey and a software tutorial, ZBL06916890., and code by the same author to calculate the Mittag-Leffler function.

I then added two scripts to calculate and plot and ran them with Octave 7.3.0:

%FCadoptionFDE -- Copyright Giovanni Ciriani
%FDE resolution of fractional adoption funtion
%Evaluates fractional differential equation y^(alpha) = y(1-y)
%
% Functions utilized
% https://www.dm.uniba.it/it/members/garrappa/software
% References
%  [1] Garrappa R.: Numerical Solution of Fractional Differential
%  Equations: a Survey and a Software Tutorial, Mathematics 2018, 6(2), 16
%  doi: https://doi.org/10.3390/math6020016
%% Initialization %
clear ; close all; clc
addpath(genpath("C:/Program Files/GNU Octave/Fractional Calculus"))
labels = {}; %initialize legend
alpha = [.5, 0.7, 0.9]; lambda = -1; param = lambda ;
f_fun = @(t,y,lam) -lam .* y.*(1-y); %RHS of Bernoulli function
J_fun = @(t,y,lam) -lam + lam * 2 .* y; %Jacobian of RHS of function
t0 = 0 ; T = 1 ; y0 = 0.5 ; h = 2^(-8) ;

%plot function solving FDE for various alphas
alpha_color = ['r', 'b', 'g'] %colors for the different alphas
for i = 1:size(alpha,2)
  [t, y] = flmm2(alpha(i), f_fun, J_fun, t0, T, y0, h, param);
  plot(t, y, alpha_color(i))
  hold on
end

%plot presumed Mittag-Leffler solution
for i = 1:size(alpha,2)
  mittag = 1 ./ (1 + FCealphaML (t, alpha(i), lambda));
  plot(t, mittag, 'color', alpha_color(i),  'linestyle', '--')
  hold on
end

hold off
title('Fractional Logistic eq.: FDE vs Mittag-Leffler hypothesis')
xlabel('time')
ylabel('Adoption fraction')
legend (num2str (alpha'), 'title', strcat('\alpha', "\n", '- - ML'), 'location', 'southeast')
legend(labels);

%FCealphaML -- Copyright Giovanni Ciriani
%calculates Queen ML funtion e_alpha = ML_alpha(-t^alpha) as defined in (2) of
%Mainardi, F. (2020). Why the Mittag-Leffler Function Can Be Considered
%the Queen Function of the Fractional Calculus? Entropy, 22(12), 1359.
%https://doi.org/10.3390/e22121359
   
function E = FCealphaML (t, alpha, lambda) ;
  E = ml (t.^alpha.*lambda, alpha, alpha, 1);
endfunction

Answer 2

I'm adding separately another answer. The paper Area, I.; Nieto, J. J., Power series solution of the fractional logistic equation, ZBL07459876. proves that the solution to the fractional differential equation (FDE) (3), is given by a series of fractional powers of the type: $$ \sum_{n=0}^{\infty}b_n(\alpha)(t^\alpha)^n,\;\alpha\in(0,1,)\tag6$$ $$b_n(\alpha)=\frac{E^\alpha_n}{\Gamma(\alpha n+1)}\tag7$$ where $E^\alpha_n$ is the sequence of Euler's fractional numbers.

There are several papers addressing the FDE; the key is searching for fractional logistic differential equation. The bibliography of the paper cited above lists many, and the most important ones, even those that later on have been proven incorrect.

It's important to note though, that power series (6), which generates the correct solution, diverges quickly as $t>1$. As a consequence more $b_n(\alpha)$ terms need to be added to the series for accuracy; but since the $\Gamma$ function is involved, the numbers very quickly exceed floating point capabilities. The result is that invariably for $t>2$ my results swing to ${}^+_-\infty$, instead of approaching 1.

Therefore if one needs practical results, the numerical integration of the FDE, shown in the other answer I posted, is the way to go.