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.

Automated summary

We solve the logistic differential equation for generalized proportional Caputo fractional derivative using a fractional power series solution. The coefficients of the power series are connected to Euler polynomials, Euler numbers, and a recently introduced sequence of Euler's fractional numbers. Numerical approximations are provided to demonstrate the accuracy of truncating the fractional power series. This extends previous studies on the Caputo fractional logistic differential equation and Euler numbers.