Analysis of a Coupled System of Nonlinear Fractional Langevin Equations with Certain Nonlocal and Nonseparated Boundary Conditions

Zaid Laadjal,Qasem M Al-Mdallal,Fahd Jarad,Xiangfeng Yang

doi:10.1155/2021/3058414

Abstract

In this article, we use some fixed point theorems to discuss the existence and uniqueness of solutions to a coupled system of a nonlinear Langevin differential equation which involves Caputo fractional derivatives of different orders and is governed by new type of nonlocal and nonseparated boundary conditions consisting of fractional integrals and derivatives. The considered boundary conditions are totally dissimilar than the ones already handled in the literature. Additionally, we modify the Adams-type predictor-corrector method by implicitly implementing the Gauss–Seidel method in order to solve some specific particular cases of the system.

Highlights

Ey acquainted a new form of Langevin equations involving two different fractional order for the sake of describing the viscoelastic anomalous diffusion in the complex liquids
Lozinski et al [9] discussed the applications of Langevin and Fokker–Planck equations in polymer rheology and stochastic simulation techniques for solving this equation
Using the tools in mathematical analysis and the theory of fixed points, discussing the qualitative specification encapsuling the behaviors of solutions of differential equations in fractional derivatives settings has attracted the attention of many scientists

Summary

Introduction

Ey acquainted a new form of Langevin equations involving two different fractional order for the sake of describing the viscoelastic anomalous diffusion in the complex liquids. Laadjal et al [10] presented the Journal of Mathematics existence and uniqueness of solutions for the multiterm fractional Langevin equation with boundary conditions. Using the tools in mathematical analysis and the theory of fixed points, discussing the qualitative specification encapsuling the behaviors of solutions of differential equations in fractional derivatives settings has attracted the attention of many scientists.

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Conclusion