Stability of nonlocal fractional Langevin differential equations involving fractional integrals

2022

Fractal Fract

The Langevin system is an important mathematical model to describe Brownian motion. The research shows that fractional differential equations have more advantages in viscoelasticity. The exploration of fractional Langevin system dynamics is novel and valuable. Compared with the fractional system of Caputo or Riemann–Liouville (RL) derivatives, the system with Mittag–Leffler (ML)-type fractional derivatives can eliminate singularity such that the solution of the system has better analytical properties. Therefore, we concentrate on a nonlinear Langevin system of ML-type fractional derivatives affected by time-varying delays and differential feedback control in the manuscript. We first utilize two fixed-point theorems proposed by Krasnoselskii and Schauder to investigate the existence of a solution. Next, we employ the contraction mapping principle and nonlinear analysis to establish the stability of types such as Ulam–Hyers (UH) and Ulam–Hyers–Rassias (UHR) as well as generalized UH and UHR. Lastly, the theoretical analysis and numerical simulation of some interesting examples are carried out by using our main results and the DDESD toolbox of MATLAB.

Section: Introductionmentioning

confidence: 88%

“…There have many papers dealing with the UH-stability [20][21][22][23][24][25][26] and generalized UH-stability [27][28][29] of fractional system. However, only a few papers [30][31][32][33][34] have discussed that the fractional Langevin system is UH-stable.…”

Section: Introductionmentioning

confidence: 99%

Stability of a Nonlinear Langevin System of ML-Type Fractional Derivative Affected by Time-Varying Delays and Differential Feedback Control

2022

Fractal Fract

“…Now, we apply Arzelá-Ascoli theorem to prove that Q: X ⟶ X is completely continuous. Indeed, for all t ∈ [− η, T], u ∈ X, when t ∈ [0, T], it follows from (14) and…”

Section: Existence Of Solutionsmentioning

confidence: 99%

Stability of a Nonlinear Fractional Langevin System with Nonsingular Exponential Kernel and Delay Control

Discrete Dynamics in Nature and Society

2022

Fractional Langevin system has great advantages in describing the random motion of Brownian particles in complex viscous fluid. This manuscript deals with a delayed nonlinear fractional Langevin system with nonsingular exponential kernel. Based on the fixed point theory, some sufficient criteria for the existence and uniqueness of solution are established. We also prove that this system is UH- and UHR-stable attributed to the nonlinear analysis and inequality techniques. As applications, we provide some examples and simulations to illustrate the availability of main findings.

“…Ulam and Hyers [7,8] proposed a concept of system stability called UH-stability in the 1940s. Over the past decade, there have been many works published (some of which can be found in [9][10][11][12][13][14][15]) on the UH-stability of a fractional system.…”

Section: Introductionmentioning

confidence: 99%

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

2022

Fractal Fract

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.