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

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.


Introduction
e fractional calculus is the ramification of mathematics concerning the integrals and derivatives of functions with arbitrary orders. It has a long history that goes back to more than three hundred years. Nonetheless, researchers discovered the importance and effectiveness of this calculus just a mere in the last few decades. It turned out that the fractional integrals and derivatives are very good tools in modeling some phenomena.
is was concluded simply because of the amazing results obtained when some of the researchers used the implements in the fractional calculus for the sake of understanding real world problems happening in the environment surrounding. Recently, differential equations of fractional order have been applied in various fields like physical, biology, chemistry, control theory, electrical circuits, blood flow phenomena, and signal and image processing; for more details, see [1][2][3] and references cited therein.
In 1908, Langevin [4] formulated his famous equation containing derivative of integer order.
is equation describes the evolution of certain physical phenomena in fluctuating environments [5]. e Langevin equation was used in large part to describe some phenomena such as anomalous transport [6]. e Langevin equation has been recently extended to the fractional order by Lim et al. [7]. 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. We refer the reader to Subsection 2.1 in [3] and the references cited therein for further details. Uranagase and Munakata [8] discussed the generalized Langevin equation with emphasis on a mechanical random force whose time evolution is not natural due to the presence of a projection operator in a propagator. Lozinski et al. [9] discussed the applications of Langevin and Fokker-Planck equations in polymer rheology and stochastic simulation techniques for solving this equation. Laadjal et al. [10] presented the 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. To get an update about the works in the literature, we ask the readers to investigate [11] and the references cited there. On the top of this, classes of systems of fractional differential equations with separated (or nonseparated) boundary conditions have been studied intensively in literatures [12][13][14].
Motivated by what are mentioned above and the recent development on Langevin equations, in this paper, we discuss the existence and the uniqueness of solutions to a coupled system of fractional Langevin equations in the form as follows: subject to a new type of nonlocal nonseparated boundary conditions as follows: p are the Caputo fractional derivatives of order α 1 , β 1 , α 2 , β 2 , and p, respectively, f, g: J × R× R ⟶ R are given functions, and I q is the Riemann-Liouville fractional integral of order q. By using the Banach contraction principle and Leray-Schauder alternative fixed point theorem, we investigate the existence of solutions for problems (1) and (2). We remark that the boundary value problem discussed here is distinctive of the ones discussed in literatures [12][13][14]. is article is organized as follows. In Section 2, we present some definitions, theorems, and related lemmas used in next sections. Section 3 discusses the existence and uniqueness of the system under consideration. In Section 4, we furnish some numerical examples. Section 5 is devoted to our concluding remarks.

Preliminaries
where Γ is the Euler Gamma function.
Definition 2 (see [1,2]). e Caputo fractional derivative of order α ∈ R + for a function f ∈ C n [a, b] is defined by where n − 1 < α ≤ n, n ∈ N, and D n � d n /dt n .
For the sake of simplicity, henceforwards we will write I α and c D α instead of I α 0 and c D α 0 , respectively.

Main Results
In this section, we will discuss the existence and uniqueness of the solution to systems (1)- (2).
en, the solution of the following coupled system of fractional Langevin equations is as follows: equipped with the boundary condition (2) which is equivalent to the coupled system of the following integral equations: where Proof. Applying the operator I α 1 +β 1 and I α 2 +β 2 on (10) and (11), respectively, we get Journal of Mathematics where c i , c i ∈ R (i � 0, 1, 2). From the boundary conditions , we obtain that Solving the above system, we find that where Δ is the determinant of the matrix associated with systems (17)- (18) in the two variables c 2 and c 2 , and it is given by (9).
Substituting the values of c 0 , c 0 , c 1 , c 1 , c 2 , and c 2 in (15) and (16), we obtain the system of the integral equations (12) and (13). e proof is completed.
For computation convenience, we set the following constants: Journal of Mathematics In the following step, we present the following result about the uniqueness of solutions for problems (1)-(2) by applying the Banach contraction principle.
Proof. Choose a positive real constant R where For all (ψ 1 , ψ 2 ) ∈ B R , we have On the other hand, we have 6 Journal of Mathematics Consequently,

Journal of Mathematics
In the other respect, we have Consequently, (36) erefore, the operator N has a unique fixed point. us, we conclude that problems (1)-(2) have a unique solution on [0, 1]. e proof is complete. Now, we apply the Leray-Schauder alternative theoerm to obtain the following result about the existence of solutions for problems (1)-(2).
where Proof. First, we show that the operator N is completely continuous.

Conclusion
e most important features of differential equations subject to either initial or boundary conditions are the existence and uniqueness of their solutions. In this paper, we discussed the existence and uniqueness of solutions of specific type of the couple system of the Langevin differential equation in the framework of Caputo fractional derivatives and under the suzerainty of nonlocal and nonseparated boundary conditions. e boundary value problem we studied contained 6 different parameters. Because of the complexity, we were forced to use computer programs in order to find examples that would support our results. We discussed these examples from the theoretical point and solved numerically using the Adams-type predictorcorrector method by implicitly implementing the Gauss-Seidel method.
It is recommended to consider the same problem in the frame of other fractional derivatives especially the ones with no singular kernels and compare their results to the ones discussed in this paper.

Data Availability
e data used to support the findings of this study are included within the article in the references.

Conflicts of Interest
e authors declare that they have no conflicts of interest.