Existence and Numerical Simulation of Solutions for Fractional Equations Involving Two Fractional Orders with Nonlocal Boundary Conditions

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in various fields, such as physics,mechanics, chemical technology, population dynamics, biotechnology, and economics (see, e.g., [1–7]). As one of the important topics in the research on differential equations, the boundary value problem has attained a great deal of attention from many researchers (see [8–18]) and the references therein. As pointed out in [19], the nonlocal boundary condition can be more useful than the standard condition to describe some physical phenomena. There are several noteworthy papers (see [20–22]) dealing with nonlocal boundary value problems of fractional differential equations. In [19], Benchohra et al. investigated the existence and uniqueness of the solutions for the differential equations with nonlocal conditions:


Introduction
Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in various fields, such as physics, mechanics, chemical technology, population dynamics, biotechnology, and economics (see, e.g., [1][2][3][4][5][6][7]). As one of the important topics in the research on differential equations, the boundary value problem has attained a great deal of attention from many researchers (see [8][9][10][11][12][13][14][15][16][17][18]) and the references therein. As pointed out in [19], the nonlocal boundary condition can be more useful than the standard condition to describe some physical phenomena. There are several noteworthy papers (see [20][21][22]) dealing with nonlocal boundary value problems of fractional differential equations.
In this paper we will study the fractional Langevin equation where the fractional derivative is in Caputo sense. In 1908 the French physicist Langevin introduced the concept of the equation of motion with a random variable, which reads as where is the mass of the particle, is the coefficient of viscosity, ( ) is the external force, and ( ) is the random force. The Langevin equation is always regarded as the first stochastic differential equation.
Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environm-2 Journal of Applied Mathematics ents [23][24][25]. For instance, Brownian motion is well described by the Langevin equation when the random fluctuation force is assumed to be white noise. In case the random fluctuation force is not white noise, the motion of the particle is described by the generalized Langevin equation [26]. For systems in complex media, ordinary Langevin equation does not provide the correct description of the dynamics. Various generalizations of Langevin equations have been proposed to describe dynamical processes in a fractal medium. One such generalization is the generalized Langevin equation [27][28][29][30][31][32] which incorporates the fractal and memory properties with a dissipative memory kernel into the Langevin equation.
Fractional order models are more accurate than integerorder models as fractional order models allow more degrees of freedom. The presence of memory term in such models not only takes into account the history of the process involved but also carries its impact to present and future development of the process. Fractional differential equations are also regarded as an alternative model to nonlinear differential equations [33]. In consequence, the subject of fractional differential equations is gaining much importance and attention. For some recent work on fractional differential equations, see [1,[34][35][36][37][38][39][40][41][42][43][44][45][46].
In [47], Ahmad et al. studied nonlinear Langevin equation involving two fractional orders in different intervals: where and denote Caputo's fractional derivative of order and with the lower limit zero.
The fractional calculus has been studied for more than three hundred years. In recent few decades, the fractional calculus has been widely used in many fields such as chaotic dynamics, viscoelasticity, acoustics, and physical chemistry. In [49], Guo studied the numerical solution of fractional partial differential equations. In [50], Guo studied the numerical simulation of the fractional Langevin equation.
As far as we know, there are no papers discussing the existence and numerical simulation of solutions for fractional equations involving two fractional orders with nonlocal boundary conditions. Motivated by the works mentioned above, in this paper, we establish the existence and uniqueness of solutions by the fixed point theorem and use 2 algorithm to describe the dynamic behaviors for the following problem: ( + ) ( ) = ( , ( ) , ( )) , 0 < < 1, 1 < ≤ 2, 0 < ≤ 1, where and denote Caputo's fractional derivative of order and with the lower limit zero, : [0, 1] × 2 → is a given continuous function and is a real number, and are two continuous functions, . Evidently, problem (6) not only includes boundary value problems mentioned above but also extends them to a much wider case.
The organization of this paper is as follows. In Section 2, we will give some lemmas which are essential to prove our main results. In Section 3, main results are given. In Section 4, we will give the numerical simulation for the fractional Langevin equation.

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts. Throughout this paper, set C = ([0, 1], ) denotes the Banach space of all continuous functions from For the convenience of the readers, let us recall the following useful definitions and fundamental facts of fractional calculus theory.
Lemma 5 (see [1] We also easily prove the following lemmas.
is a solution of the following integral equation: Proof. According to Lemma 5 and applying the operator to both sides of (12), for some constants 0 , 1 , and 2 , we get then the above equation can be written as and applying the operator to both sides of the above equation, we obtain then the above equation can be written as that can be written as (13). The proof is completed. (13) is a generalized solution of the nonlocal boundary value problem (6).

Lemma 8 (Krasnoselskii). Let B be a closed convex and nonempty subset of . Suppose that L and N are general nonlinear operators which map B into such that
(2) L is a contraction mapping; (3) N is compact and continuous.
Then there exists ∈ B such that = L + N .

Main Results
In order to apply Lemma 8 to prove our main results, we first give , , as follows.

Journal of Applied Mathematics
Define an operator : 1 → 1 by Proof. Firstly, we show that ∈ 1 . Assuming ∈ 1 is a generalized solution of the problem (6), there exist three constants 0 , 1 , and 2 . Equation (13) can be written as and differentiating both sides of the above equation, we get It is clear that every term of the above equation belongs to ; then ∈ 1 . Secondly, we show that is the generalized solution of the problem (6).
Let be a generalized solution of the problem (6) and Applying the operator to both sides of the above equation, we obtain and then applying the operator to both sides of the above equation, we obtain By simple calculations, it is clear that satisfies conditions (6); then it is a generalized solution for the problem (6). The proof is completed.
Journal of Applied Mathematics 5 For convenience, let us set Clearly, for any ∈ [0, 1], Now, we make the following hypotheses.
Proof. The proof will be given in two steps.
Clearly, we also can get

− ( ) ]
Journal of Applied Mathematics 7 For convenience, we let where we have used the Hölder inequality and the following equalities: Therefore, ‖( )( )‖ ≤ .
Step 2. is a contraction operator. For convenience, we get Journal of Applied Mathematics Clearly, we can also get For , V ∈ 1 ([0, 1], ) and for each ∈ [0, 1], we obtain Journal of Applied Mathematics Since Λ < 1, we have ‖ ( ) − (V)‖ ≤ Λ‖ − V‖; that is, is a nonlinear contraction. Hence, by using Lemma 8, the conclusion of the theorem holds by Banach fixed point theorem.
The proof is completed. Proof.
Step 1. There exists a positive constant > 0 such that + ∈ Ω . For ∈ Ω , by the same arguments of the first step of the proof in Theorem 10, we have ‖ + ‖ ≤ . In virtue of the definition of and a simple calculation, we obtain where is a constant. By the assumptions, < 1. Therefore, there exists a positive constant large enough such that Hence, there exists a positive constant such that + ∈ Ω .
Step 2. is a contraction operator.
Step 3. is continuous and compact.
Thus, all the assumptions of Lemma 8 are satisfied and the conclusion of Lemma 8 implies that the boundary value problem (6) has at least one solution on [0, 1].
The proof is completed.

Algorithm for the Fractional Langevin Equation and Examples
In this paper, we will give the numerical simulation for the fractional Langevin equation.
The definition of fractional order has many kinds; the different definitions will bring different algorithm forms and will cause different proof of the algorithm stability and different method of accuracy analysis. In the practical application, there are three kinds of fractional derivative definitions, such as Grünwald-Letnikov, Riemann-Liouvlle, and Caputo Fractional derivatives.
Thus, by Theorem 10, we can get that the problem (60) has at most one solution.
Thus, by Theorem 10, we can get that the problem (63) has at most one solution.
With the above algorithm we get Figures 3 and 4.