Machine learning

Mathematical analysis

Langevin equation

Stability (learning theory)

Physics

Nonlinear system

Uniqueness

Kernel (algebra)

Pure mathematics

Mathematics

Fractional calculus

Quantum mechanics

Computer science

Statistical physics

Invertible matrix

Applied mathematics

Fractal and Fractional

The fractional Langevin equation is a very effective mathematical model for depicting the random motion of particles in complex viscous elastic liquids. This manuscript is mainly concerned with a class of nonlinear fractional Langevin equations involving nonsingular Mittag–Leffler (ML) kernel. We first investigate the existence and uniqueness of the solution by employing some fixed-point theorems. Then, we apply direct analysis to obtain the Ulam–Hyers (UH) type stability. Finally, the theoretical analysis and numerical simulation of some interesting examples show that there is a great difference between the fractional Langevin equation and integer Langevin equation in describing the random motion of free particles.

Existence, Stability and Simulation of a Class of Nonlinear Fractional Langevin Equations Involving Nonsingular Mittag–Leffler Kernel