Abstract
We present a new numerical algorithm for two-point boundary value problems. We first present the exact solution in the form of series and then prove that the n-term numerical solution converges uniformly to the exact solution. Furthermore, we establish the numerical stability and error analysis. The numerical results show the effectiveness of the proposed algorithm.
1. Introduction
It is well known that many problems can be presented by the following two-point boundary value problems: where , , and are finite constants.
Problem (1) arises from many fields of applied mathematics and physics, such as nuclear physics, economical system, chemical engineering, and underground water flow. Therefore, this problem has attracted considerable attention. For example, Aziz and Kumar [1, 2] presented a finite difference method based on nonuniform mesh to solve this problem. Kumar [3, 4] presented a second order spline finite difference method to solve (1) by using a spline function. Rashidinia et al. [5] presented a parametric spline method for (1). For these references, please see [6–10].
In this paper, we propose a new numerical algorithm to solve (1) by using the reproducing kernel theory. By homogenizing the boundary value conditions, (1) is converted into a nonlinear operator equation. We show that the solution of (1) is equivalent to the solution of the operator equation, and its exact solution can be represented in the form of series. Furthermore, we prove that the -term numerical solution converges uniformly to the exact solution. Then, numerical stability and error analysis of the method are presented. Numerical results show that this method has high accuracy.
The paper is organized as follows. In Section 2, fundamental definitions and theorems of the reproducing kernel theory are given. In Section 3, the nonlinear operator equation is constructed. The new numerical algorithm is presented in Section 4. In Section 5, we apply our method to linear and nonlinear numerical examples and illustrate the applicability of the presented method. Section 6 ends this paper with a brief conclusion.
2. Fundamental Definitions and Theorems
In this section, we show some fundamental theories of the reproducing kernel space [11, 12].
Definition 1. Let denote a space, which is composed of functions defined on an abstract set and admits reproducing kernel . That is, for each fixed , belongs to as a function in and for any , We call (2) the reproducing property of .
Theorem 2. Let be a space; let be a complete function system; that is, then is the reproducing kernel of .
Proof. In fact, ; , . In view of (2) and (3), we have
Definition 3. The reproducing kernel space is defined as follows.
is an absolutely continuous function, , .
The inner product and norm are defined as, respectively, ,
Theorem 4. is a complete space with respect to .
Proof. If the reproducing kernel of the space exists, in view of (2) and Cauchy-Schwartz’s inequality, we have which shows that is a bounded linear function on . Hence, there exists a Cauchy sequence . By (2) and Cauchy-Schwartz’s inequality, we obtain Therefore, there exists such that . Furthermore, we have and . So the proof of Theorem 4 is complete.
3. Structure of the Nonlinear Operator
Now, we show the method to solve (1). By transformation, we have That is, where and . Let with . By homogenizing the boundary value conditions, (1) can be converted into the equivalent nonlinear operator equation: For any and each fixed point , where is a different dense point set on , is the conjugate operator of , and is the reproducing kernel of . In terms of the property of (2) and the inner product, we obtain Therefore, we can see that the solution of (1) is equivalent to the solution of (11).
4. Solving the Problem
Through the normal orthogonal process, of can be derived from . That is, where are the orthogonalization coefficients.
Theorem 5. If is the different dense point set on and is the exact solution of (11) in , then
Proof. Since and is a normal complete orthogonal system, then can be expanded by Fourier series with the normal orthogonal basis; namely, Because is complete, is uniformly convergent in the sense of . Note that ; is absolutely continuous, in terms of (2) and (14); we obtain The proof is complete.
By truncating the right hand of (15), we obtain the approximate solution of (11); namely, where is the -term intercept of in (15). In view of the completeness of the reproducing kernel space, as .
Next, in order to discuss the uniform convergence of the approximate solution, for any fixed , with ; we construct an iterative sequence . That is,
Theorem 6. Assume the following.(a) is a different dense point set on .(b) is a normal orthogonal system.(c) is bounded.(d), for any , .Then the iterative formula converges uniformly to the exact solution of (11).
Proof. By , we have . In view of the orthonormality of , we have In view of the boundedness of , we have . Let and , owing to ; we obtain By the completeness of , there exists , such that in the sense of as . The proof is complete.
Theorem 7. Error Analysis. Assume that the conditions of Theorem 6 are satisfied; then the error of the numerical solution is monotonically decreasing with the increasing of nodes; that is, , as .
Proof. It is easy to see that Thus we complete the proof.
Now, we show the stability of the proposed method.
Theorem 8. For (11), if has a small perturbation , then the proposed method is stable.
Proof. For the problem (11), if has a small perturbation , then . Let be the numerical solution of ; in view of Theorems 5 and 6, we have Hence, That is, , , such that Therefore, the method is stable.
5. Numerical Examples
5.1. Example 1
Consider the following linear two-point boundary value problem [5]: The exact solution is with and . The maximum absolute errors (M.A. error) are tabulated in Table 1; the Root-mean-square errors (R.M.S) of the exact solution and the approximate solution () are shown in Figure 1. From the numerical results, we can see that the present method produces better approximate solution than [5] and the error of the numerical solution is monotonically decreasing with the increase of nodes.
5.2. Example 2
Consider the nonlinear singular two-point boundary value problem [14, 15]:
The exact solution is . When , the maximum absolute errors (M.A. error) are tabulated in Table 2; the Root-mean-square error (R.M.S) of the exact solution and the approximate solution () are shown in Figure 2. From the numerical results, we can see that the error of the numerical solution is monotonically decreasing with the increase of nodes.
6. Conclusions
In this paper, we use the reproducing kernel iterative method to solve a class of two-point boundary value problems. By homogenizing the boundary conditions, the two-point boundary value problem is converted into the equivalent nonlinear operator equation. We prove that their solutions are equivalent, the exact solution can be represented in the form of series, and the -term numerical solution converges uniformly to the exact solution . Furthermore, we show the analysis of error and stability for the method. At last, numerical results show the high accuracy and the validity of this method.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Science Foundation of China (11271100, 11301113, and 71303067), Harbin Science and Technology Innovative Talents Project of Special Fund (2013RFXYJ044), China Postdoctoral Science Foundation Fund Project (Grant no. 2013M541400), the Heilongjiang Postdoctoral Fund (Grant no. LBH-Z12102), and the Fundamental Research Funds for the Central Universities (Grant no. HIT.HSS.201201).