Journal of Applied Mathematics

Volume 2013 (2013), Article ID 268347, 15 pages

http://dx.doi.org/10.1155/2013/268347

Research Article

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

Jing Zhao, Peifen Lu, and Yiliang Liu

Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis and College of Sciences, Guangxi University for Nationalities, Nanning, Guangxi 530006, China

Received 26 March 2013; Accepted 6 June 2013

Academic Editor: Naseer Shahzad

Copyright © 2013 Jing Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study a boundary value problem for fractional equations involving two fractional orders. By means of a fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for the fractional equations. In addition, we describe the dynamic behaviors of the fractional Langevin equation by using the algorithm.

1. 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., [1–7]). 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:

where denotes Caputo’s fractional derivative of order with the lower limit zero.

In [22], Zhong and Lin studied the existence and uniqueness of solutions in the nonlocal and multiple-point boundary value problem for fractional differential equation:

where denotes Caputo’s fractional derivative of order with the lower limit zero.

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 environments [23–25]. 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 [27–32] which incorporates the fractal and memory properties with a dissipative memory kernel into the Langevin equation.

Fractional order models are more accurate than integer-order 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, 34–46].

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.

In [48], A. Chen and Y. Chen studied existence of solutions to nonlinear Langevin equation involving two fractional orders with boundary value conditions:

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 algorithm to describe the dynamic behaviors for the following problem:

where and denote Caputo’s fractional derivative of order and with the lower limit zero, 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.

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts. Throughout this paper, set denotes the Banach space of all continuous functions from with the norm . We also introduce the Banach space endowed with the norm defined by .

For the convenience of the readers, let us recall the following useful definitions and fundamental facts of fractional calculus theory.

Definition 1 (see [1, 6]). The Riemann-Liouville derivative of order with the lower limit zero for a function can be written as

Definition 2 (see [1, 6]). The fractional (arbitrary) order integral of the function of order is defined by

Definition 3 (see [1]). Let , . If , the Caputo fractional derivative of order of is defined by

Definition 4 (see [6]). Let , and the the Grünwald-Letnikov fractional derivative of order of defined by

where .

Lemma 5 (see [1]). Let , and the Caputo derivative of order for a function . If for , or ,

We also easily prove the following lemmas.

Lemma 6. Let , . , satisfying the following differential equation:

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 , , and , 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.

Definition 7. The function satisfying (13) is a generalized solution of the nonlocal boundary value problem (6).

Lemma 8 (Krasnoselskii). Let be a closed convex and nonempty subset of . Suppose that and are general nonlinear operators which map into such that

(1) whenever ;(2) is a contraction mapping;(3) is compact and continuous.

Then there exists such that .

3. Main Results

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

Define an operator by

Lemma 9. The function is a generalized solution of the nonlocal boundary value problem (6) if , for all .

Proof. Firstly, we show that .

Assuming is a generalized solution of the problem (6), there exist three constants , , and . 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 .

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.

For convenience, let us set

Clearly, for any ,

Now, we make the following hypotheses.

(H1) There exist two real-valued functions for some , such that

  for almost all , .(H2) There exist two positive constants , such that . Moreover, , and

Theorem 10. Let be a jointly continuous function and the assumptions (H1) and (H2) hold. In addition, assume that

where , , , , .

Then the problem (6) has at most one solution.

Proof. The proof will be given in two steps.

Step 1. is bounded.

Now we show that .

Let . For any , we have

Clearly, we also can get

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

Clearly, we can also get

For and for each , we obtain

Since , we have ; 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.

Theorem 11. Let be a jointly continuous function and the assumptions (H1) and (H2) hold. In addition,

(H3) assume that there exist a constant and a real-valued function such that

Then the problem (6) has at least one solution on if

Proof. Step 1. There exists a positive constant such that .

For , 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, . Therefore, there exists a positive constant large enough such that

Hence, there exists a positive constant such that .

Step 2. is a contraction operator.

For and for each , we obtain

Since , we have ; that is, is a nonlinear contraction.

Step 3. is continuous and compact.

Firstly, we show that the operator is continuous. For , such that in ; then

Similarly, we get

we get sequences and , which converge on with and .

Since

Combining (41) and (42), we can get . Thus is continuous in .

Secondly, we show that the operator is equicontinuous. Let . For any , for all , , we obtain

Clearly, we also easily get

Obviously the right hand side of the above inequality tends to zero independently of , as ; we get that is relatively compact on . Hence, by the Arzelá-Ascoli theorem, is compact on .

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 .

The proof is completed.

4. 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.

Remark 12 (see [51]). For , , ,

Remark 13 (see [49]). For , ,

In [52], shifted Grünwald-Letnikov formula is defined by

We get the following approximation:

We put a call shifted Grünwald discrete format, simply “ algorithm” for short.

In addition, Oldham and Spanier [53] found the following approximation format in 1974:

The approximation format has the rapid convergence properties. So they put forward an improved Grünwald-Letnikov fractional derivative definition (take to (48)):

For , the above equation can be written as

Therefore, they put forward “fractional center difference quotient” approximation format called in general “ algorithm.”

In this paper, we use the three-point interpolation formula:

Then “ algorithm” can be expressed as:

Remark 14 (see [49]). algorithm is based on Grünwald-Letnikov definition, not only used for numerical calculation of fractional derivative (), but also used for numerical calculation of fractional integral ().

As we all know, the fractional Langevin equation form is

where , , , is a constant, is an external force, and is a random force.

The above equation can be written as

According to algorithm, the Caputo fractional derivatives above can be written as

The previous equations are approximated by the three-point interpolation formula and can be written as

where

With the above algorithm we will give some examples.

Example 1. Consider the following fractional differential equations:

Obviously, we get

Letting , , , , , and , we have

Thus, by Theorem 10, we can get that the problem (60) has at most one solution.

With the above algorithm we get Figures 1 and 2.

Figure 1: The displacement without a Wiener process, the numerical simulation of (60), where , .

Figure 2: The displacement with a Wiener process, the numerical simulation of (60), where , .

Example 2. Consider the following fractional differential equations:

Obviously, we get

Letting , , , , , and , we have

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.

Figure 3: The displacement without a Wiener process, the numerical simulation of (63), where , .

Figure 4: The displacement with a Wiener process, the numerical simulation of (63), where , .

Acknowledgments

This project is supported by NNSF of China (Grants nos. 11271087 and 61263006) and Guangxi Scientific Experimental (China-ASEAN Research) Centre no. 20120116.

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