Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 329575, 11 pages

doi:10.1155/2012/329575

Research Article

A Numerical Algorithm for Solving a Four-Point Nonlinear Fractional Integro-Differential Equations

Er Gao,1,2 Songhe Song,1,2 and Xinjian Zhang¹

¹Department of Mathematics and Systems Science, College of Science, National University of Defense Technology, Changsha 410073, China

²State key Laboratory of High Performance Computing, National University of Defense Technology, Changsha 410073, China

Received 28 April 2012; Accepted 11 July 2012

Academic Editor: Hung Nguyen-Xuan

Copyright © 2012 Er Gao 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 provide a new algorithm for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order based on reproducing kernel space method. According to our work, the analytical solution of the equations is represented in the reproducing kernel space which we construct and so the n-term approximation. At the same time, the n-term approximation is proved to converge to the analytical solution. An illustrative example is also presented, which shows that the new algorithm is efficient and accurate.

1. Introduction

In recent years, differential equations of fractional order have been addressed by several researchers with the sphere of study ranging from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions. Several authors have used fixed point theory to show the existence of solution to differential equations of fractional order, see the monographs of Bai and Liu [1], Wu and Liu [2], Hamani et al. [3] and Ahmad and Sivasundaram [4]. At the same time, there may be several methods for solving differential equations of fractional order, such as the least squares finite-element method [5], collection method [6], fractional differential transform method [7], decomposition method [8], and variational iteration method [9]. Besides these cited works, few more contributions [10, 11] have been made to the analytical and numerical study of the solutions of fractional boundary value problems.

Ahmad and Sivasundaram [4] proved the existence and uniqueness of solutions for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order by applying some standard fixed point theorems:

where is the Caputo's fractional derivative and is continuous.

In this paper, we consider the following nonlinear fractional integro-differential equation with four-point nonlocal boundary conditions:

where is the Caputo's fractional derivative and is continuous, for ,

and . Here, is a Banach space and denotes the Banach space of all continuous functions from endowed with a topology of uniform convergence with the norm denoted by .

Actually, we remark that the boundary conditions in (1.2) arise in the study of heat flow problems involving a bar of unit length with two controllers at and adding or removing heat according to the temperatures detected by two sensors at and .

The rest of the paper is organized as follows. We begin by introducing some necessary definitions and mathematical preliminaries of the fractional calculus theory which are required for establishing our results. Then we construct some special reproducing kernel spaces, and the new reproducing kernel method is introduced in Section 3. In Section 4 we present one examples to demonstrate the efficiency of the method.

2. Preliminaries

Let us recall some basic definition and lemmas on fractional calculus.

Definition 2.1. For a function , the Caputo derivative of fractional order is defined as

where denotes the gamma function.

Definition 2.2. The Riemann-Liouville fractional integral of order is defined as

provided the integral exists.

Lemma 2.3 (see [4]). For a given , the unique solution of the boundary value problem

is given by

To introduce the next lemma, we need the following assumptions.

(A₁) There exist positive functions ,  ,   such that

Further,

(A₂) There exist a number such that , , where

Lemma 2.4 (see [4]). Assume that is a jointly continuous function and satisfies assumption (A₁). Then the boundary value problem (1.1) has an unique solution provided , where is given in assumption (A₂).

For more information on the mathematical properties of fractional derivatives and integrals one can consult the mentioned references.

3. Reproducing Kernel Method

3.1. Some Reproducing Kernel Spaces

Firstly, inner space is defined as is absolutely continuous real-valued functions, . The inner product in is given by

and the norm is denoted by . From [12], is a reproducing kernel Hilbert space and the reproducing kernel is

In order to solve (1.2) using RKM, we construct a reproducing kernel space in which every function satisfies the boundary conditions of (1.2). Inner space is defined as are absolutely continuous real valued functions, , and , , and the inner product is defined as follows:

Theorem 3.1 3.1. is a Hilbert reproducing kernel space.

Proof. Suppose is a Cauchy sequence in , that means

Therefore, we have and , which shows that is a Cauchy sequence in and is a Cauchy sequence in space . So, we have

where is a real constant and .

Let

where are determined by , and .

From , is absolutely continuous in and is almost true everywhere in . Consequently, . Moreover,

That means that, is complete.

Similar to [13], we can prove that the point-evaluation functional , of is bounded. So is a Hilbert reproducing kernel space.

From [12, 14], we have the following.

Theorem 3.2. The reproducing kernel of is

where

Actually, it is easy to prove that for every and , and holds, that is, is the reproducing kernel of .

3.2. The Reproducing Kernel Method

In recent years, there has been a growing interest in using a reproducing kernel to solve the operator equation. In this section, the representation of analytical solution of (1.2) is given in the reproducing kernel space .

Note and . We can convert (1.2) into an equivalent equation . It is clear that is a bounded linear operator.

Put , , where is the adjoint operator of . Then

Similar to [15], we can prove the following.

Lemma 3.3. Under the previous assumptions, if is dense on , then is the complete basis of .

The orthogonal system of can be derived from Gram-Schmidt orthogonalization process of , and

We also can prove the following theorem.

Theorem 3.4. If is dense on and the solution of (1.2) is unique, the solution can be expressed in the form

The approximate solution of the (1.2) is

If (1.2) is linear, that is , then the approximate solution of (1.2) can be obtained directly from (3.13). Else, the approximate process could be modified into the following form:

where .

4. Convergent Theorem of the Numerical Method

In this section, we will give the following convergent theorem of our algorithm.

Lemma 4.1. There exists a constant , satisfied , for all .

Proof. For all the and , there are

, and note that

That is, .

By Lemma 4.1, it is easy to obtain the following lemma.

Lemma 4.2. If (), is bounded,   () and is continuous, then .

Theorem 4.3. Suppose that is bounded in (3.13) and (1.2) has a unique solution. If is dense on , then the n-term approximate solution derived from the above method converges to the analytical solution of (1.2).

Proof. First, we will prove the convergence of .

From (3.14), we infer that

The orthonormality of yields that

That means . Due to the condition that is bounded, is convergent and there exists a constant such that

If , then

In view of , it follows that

The completeness of shows that as in the sense of .

Secondly, we will prove that is the solution of (1.2).

Taking limits in (3.12), we get

So

Therefore,

If , then

If , then

It is clear that

Moreover, it is easy to see by induction that

Since is dense on , for all , there exists a subsequence such that

It is easy to see that . Let ; by the continuity of and Lemma 4.2, we have

At the same time, ; clearly, satisfies the boundary conditions of (1.2).

That is, is the solution of (1.2).

The proof is complete.

In fact, is just the orthogonal projection of exact solution onto the space .

5. Numerical Example

To give a clear overview of the methodology as a numerical tool, we consider one example in this section. We apply the reproducing kernel method and results obtained by the method are compared with the analytical solution of each example and are found to be in good agreement with each other. Also, the numerical results obtained are compared with the corresponding experimental results obtained by the methods presented in [8, 9].

Example 5.1. Consider the following boundary value problem:

where . According to Lemma 2.4, the boundary value problem (5.1) has a unique solution on . is the solution of (5.1), so it is the one and the only one solution. Using our method, taking , , the numerical results are given in Table 1.

Table 1: Absolute errors for Example 5.1.

True solution
DM [8]
VIM [9]
RKM ()
RKM ()

0
0.05566

0.1
0.19798

0.2
0.39473

0.3
0.60560

0.4
0.77891

0.5
0.83516

0.6
0.70036

0.7
0.50144

0.8
0.29175

0.9
0.11797

1.0
0.01485

6. Conclusion

In this paper, RKM is presented to solve four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order . The results of numerical examples demonstrate that the present method is more accurate than the existing methods.

Acknowledgment

The work is supported by NSF of China under Grant no. 10971226.

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