Hadamard type fractional time-delay semilinear differential equations: Delayed Mittag-Leffler function approach

TL;DR

This paper introduces a new delayed Mittag-Leffler matrix function with logarithm to solve and analyze the stability of Hadamard type fractional time-delay differential equations, extending classical functions and methods.

Contribution

It develops a novel delayed Mittag-Leffler type matrix function with logarithm and applies it to explicit solutions, existence, uniqueness, and stability analysis of fractional time-delay equations.

Findings

Explicit solutions for nonhomogeneous equations derived

Established existence and uniqueness results

Proved stability in Ulam-Hyers sense

Abstract

We propose a delayed Mittag-Leffler type matrix function with logarithm, which is an extension of the classical Mittag-Leffler type matrix function with logarithm and delayed Mittag-Leffler type matrix function. With the help of the delayed Mittag-Leffler type matrix function with logarithm, we give an explicit form a of solutions to nonhomogeneous Hadamard type fractional time-delay linear differential equations. Moreover, we study existence uniqueness and stability in Ulam-Hyers sense of the Hadamard type fractional time-delay nonlinear equations.