Existence Results for Langevin Equation Involving Atangana-Baleanu Fractional Operators

Abstract

A new form of nonlinear Langevin equation (NLE), featuring two derivatives of non-integer orders, is studied in this research. An existence conclusion due to the nonlinear alternative of Leray-Schauder type (LSN) for the solution is offered first and, following that, the uniqueness of solution using Banach contraction principle (BCP) is demonstrated. Eventually, the derivatives of non-integer orders are elaborated in Atangana-Baleanu sense.

Qualitative Analysis of Langevin Integro-Fractional Differential Equation under Mittag–Leffler Functions Power Law

“…In fact, our outcomes generalize those in [32]. Due to the wide recent investigations and applications of Mittag-Leffler power law, we believe that the acquired results here are important for future investigations on the theory of fractional calculus and fractional inequalities.…”

“…It has been demonstrated that the solution for a fractional generalised Langevin equation presents the correct short and long time behaviour for the mean square displacement of single-file diffusion for an external force that changes with power-law if an appropriate choice is made of parameters of fractional generalised Langevin equation. Recently, Baleanu et al in [32] studied the existence and uniqueness of solution for the following nonlinear fractional Langevin equation:…”

“…The target of this paper is to extend the previous work studied by [32] under a new fractional operator with respect to another function. We will combine the definition of the Atangana-Baleanu derivative and an increasing function.…”

“…The proposed problem (1) for different values of a function φ includes the study of problems involving the results in [32] and many other results that are not studied yet.…”

Nonlocal boundary value problems for integro-differential Langevin equation via the generalized Caputo proportional fractional derivative

“…It is worth mentioning here that all of the above cited work has been conducted in the frame of the classical Riemann-Liouville, Caputo, and Hadamard fractional operators. Further, the problem of Langevin has been considered using some generalized fractional derivatives in which, for instance, Atangana-Baleanu and Hilfer fractional derivatives were employed [30,31].…”