RESEARCH ARTICLE
Exploring the Dynamics of Impulsive Fractional Langevin Equations Via Mittag-Leffler Functions
Rizwan Rizwan,
Fengxia Liu,
Choonkil Park,
Corresponding Author
Rizwan Rizwan
School of Interdisciplinary Studies, Renmin University of China, Beijing, China
Institute of Mathematics, Henan Academy of Sciences, Zhengzhou, China
Department of Mathematics, University of Buner, Buner, Pakistan
Correspondence:
Rizwan Rizwan ([email protected])
Contribution: Investigation, Methodology, Software, Formal analysis
Search for more papers by this authorFengxia Liu
Institute of Artificial Intelligence, Beihang University, Beijing, China
Contribution: Funding acquisition, Writing - review & editing, Supervision
Search for more papers by this authorChoonkil Park
Research Institute for Natural Sciences, Hanyang University, Seoul, Korea
Contribution: Writing - review & editing, Supervision
Search for more papers by this authorRizwan Rizwan,
Fengxia Liu,
Choonkil Park,
Corresponding Author
Rizwan Rizwan
School of Interdisciplinary Studies, Renmin University of China, Beijing, China
Institute of Mathematics, Henan Academy of Sciences, Zhengzhou, China
Department of Mathematics, University of Buner, Buner, Pakistan
Correspondence:
Rizwan Rizwan ([email protected])
Contribution: Investigation, Methodology, Software, Formal analysis
Search for more papers by this authorFengxia Liu
Institute of Artificial Intelligence, Beihang University, Beijing, China
Contribution: Funding acquisition, Writing - review & editing, Supervision
Search for more papers by this authorChoonkil Park
Research Institute for Natural Sciences, Hanyang University, Seoul, Korea
Contribution: Writing - review & editing, Supervision
Search for more papers by this authorFirst published: 16 July 2025
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.
Conflicts of Interest
The authors declare no conflicts of interest.
Open Research
Data Availability Statement
The authors have nothing to report.
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