Exploring the Dynamics of Impulsive Fractional Langevin Equations Via Mittag-Leffler Functions

Abstract

In this paper, we study impulsive fractional Langevin equations, deriving solutions by incorporating the Mittag-Leffler functions through an analysis of linear Langevin equations with distinct fractional derivatives. We investigate both a general class of impulsive fractional Langevin equations and nonlinear implicit impulsive switched coupled systems with four fractional derivatives. The existence of solutions is established using mathematical tools such as boundedness, continuity, monotonicity, and nonnegativity properties of the Mittag-Leffler functions, along with fixed point methods. Stability aspects, including the Ulam-Hyers, the generalized Ulam-Hyers, the Ulam-Hyers-Rassias, and the generalized Ulam-Hyers-Rassias stability, are explored under appropriate conditions using fixed point theorems. The theoretical findings are illustrated through practical examples, with detailed graphical analysis and three-dimensional mesh plots that highlight the behavior of the solutions over time and their dependence on fractional order, demonstrating the applicability of the proposed models.