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

Abstract

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.

Coincidence Theory of a Nonlinear Periodic Sturm–Liouville System and Its Applications

“…Based on this paper, we will further continue to study the dynamics of Equation (1) under pulse, delay and random effects. In addition, inspired by the papers [13][14][15][16][17][18][19][20][22][23][24][42][43][44][45][46][47][48][49][50][51][52], we will also study the Sturm-Liouville equation involving fractional differential as well as reaction-diffusion terms in the future.…”

On Coupled System of Langevin Fractional Problems with Different Orders of μ-Caputo Fractional Derivatives

“…which implies that K 2 is a contraction mapping by (13). The continuity of h i , i = 1, 2 implies that the operator K 1 is continuous.…”

“…Otherwise, the extended Langevin equation represents the particle motion (see, e.g., [11,12]). In fractal media, Langevin's equation has become widely used to represent dynamical operations (see [13][14][15][16][17][18] for more recent interesting results). In [19], the authors utilized the fractional Langevin equation to recreate Brownian motion.…”

Solvability, Approximation and Stability of Periodic Boundary Value Problem for a Nonlinear Hadamard Fractional Differential Equation with p-Laplacian

“…For (28), we can apply the appropriate ODE toolbox in MATLAB to perform numerical solutions and simulations.…”

“…Subsequently, extensive and in-depth research was conducted on the Ulam-Hyers stability of various systems. Especially, many excellent research results have emerged regarding the Ulam-Hyers stability of fractional order differential systems (see some of them [3,21,[24][25][26][27][28][29][30][31][32]). Moreover, it is rare to combine the Hadamard fractional derivative with Laplacian operator.…”