Caputo-Hadamard fractional differential equations on time scales: Numerical scheme, asymptotic stability, and chaos.

Abstract

This study investigates Caputo-Hadamard fractional differential equations on time scales. The Hadamard fractional sum and difference are defined for the first time. A general logarithm function on time scales is used as a kernel function. New fractional difference equations and their equivalent fractional sum equations are presented by the use of fundamental theorems. Gronwall inequality, asymptotical stability conditions, and two discrete-time Mittag-Leffler functions of Hadamard type are obtained. Numerical schemes are provided and chaos in fractional discrete-time logistic equation and neural network equations are reported.