Caputo–Hadamard fractional differential equations on time scales: Numerical scheme, asymptotic stability, and chaos
Guo–Cheng Wu, Ting-Ting Song, Shuqiang Wang
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.
Topics & Concepts
MathematicsHadamard transformMittag-Leffler functionFractional calculusApplied mathematicsMathematical analysisStability (learning theory)LogarithmHadamard three-lines theoremDifferential equationGronwall's inequalityHadamard matrixInequalityComputer scienceMachine learningFractional Differential Equations SolutionsNonlinear Differential Equations AnalysisDifferential Equations and Numerical Methods