`Abstract and Applied AnalysisVolume 2012 (2012), Article ID 195310, 11 pageshttp://dx.doi.org/10.1155/2012/195310`
Research Article

## The Solution of a Class of Third-Order Boundary Value Problems by the Reproducing Kernel Method

1Department of Mathematics, Inner Mongolia University of Technology, Hohhot 010051, China
2Jining Teachers College, Jining 012000, China

Received 1 July 2012; Accepted 20 September 2012

Copyright © 2012 Zhiyuan Li 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

This paper expands the application of reproducing kernel method to a class of third-order boundary value problems with mixed nonlinear boundary conditions. The analytical solution is represented in the form of series in the reproducing kernel space. The -term approximation is obtained and is proved to converge to the analytical solution. The numerical examples are given to demonstrate the computation efficiency of the presented method. Results obtained by the method indicate that the method is simple and effective.

#### 1. Introduction

Third-order differential equations arise in a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross section, a three layer beam, electromagnetic waves, or gravity driven flows. Third-order boundary value problems were discussed in many papers in recent years, for instance, see [16] and references therein. In [13], the authors used the spline functions to solve boundary value problems. In [4], the authors developed a second-order method for solving third-order three-point boundary value problems based on Padé approximant in a recurrence relation. In [5], the authors introduced Adomian decomposition method for multipoint boundary value problems (BVPs). In this paper, we use reproducing kernel to solve singular third-order boundary value problems with mixed boundary conditions. Recently, the reproducing kernel methods [710] emerge one after another. Using the reproducing kernel methods, the authors discussed two-point boundary value problems and periodic boundary value problems. For third-order boundary value problems with mixed nonlinear boundary conditions, however, this method has not yet been applied. In previous work, the reproducing kernel method cannot be used directly to solve third-order boundary value problems with nonlinear boundary conditions. Our work is to present a numerical algorithm for solving a class of singular third-order boundary value problems. By using this method, the analytical solution and approximate solution are given and uniformly converge to the exact solution and its corresponding derivatives. The algorithms are efficiently applied to solve some model problems.

Let us consider the following singular problems of third-order ordinary differential equations: where are known functions. are linear independence conditions of determining the solution. We assume that has a unique solution which belongs to , where which is a reproducing kernel space is defined in the second section.

In order to solve , let . It is easy to prove that is bounded linear operator.On the other hand, we assume that the conditions of determining the solution can be homogenized; after homogenization of these conditions, we put the conditions into the reproducing kernel space constructed in the following section. Equation can be transformed into the following form in : To solve problem (1.2), we give a space . The inner product in is given by . Like in [8], we can get the following reproducing kernel space.

Theorem 1.1. The space is a reproducing kernel space and its reproducing kernel is , and where ,  the symbol indicates that the operator applies to the function of .

#### 2. The Reproducing Kernel Method

Let , Practise Gram-Schmidt orthonomalization for , we get where are coefficients of Gram-Schmidt orthonormalization.

Theorem 2.1. If is distinct points dense in and is existent, then is an analytical solution of the problem (1.2).

Proof. Since is an orthonormal systems, is expressed as

We denote the approximate solution of by

Theorem 2.2. Let where ,   are given by (2.2) and (2.4), then the sequence of number is monotone decreasing and .

Proof. Because clearly and consequently is monotone decreasing in the sense of . By Theorem 2.1, we know is convergent in the norm of , then we have Hence, .

Theorem 2.3 (convergence analysis). If , are given by (2.2) and (2.4), then and uniformly convergent to and , where .

Proof. For any then there exists such that, .

Theorem 2.4. If is distinct points dense in and are given by (2.2) and (2.4), then .

Proof. We may set projective operator . Hence,

Theorem 2.5 (error estimate). If is distinct points dense in and , are given by (2.2) and (2.4), then , where .

Proof. For every given , there is always satisfying and . By Theorem 2.5 and implying , so we obtain For application reproducing kernel property, we have We also have Moreover, It is noted that we take norm of for variable . The function is derived on in the interval of , so we have . Hence,

#### 3. Numerical Experiment

For showing the effectiveness of our method, we consider the following problems.

Example 3.1 (see [2, 3]). Considering the following third-order boundary values problem where the exact solution is . By the homogeneous process of the boundary condition, let , problem can be transformed into the equivalent form The numerical results are presented in Tables 1, 2, and 3.

Table 1: The absolute error of Example 3.1.
Table 2: The numerical results of Example 3.1.
Table 3: The numerical results of Example 3.1.

Example 3.2 (see [11, 14]). Considering the following third-order obstacle problems: where , the exact solution is where The numerical results are presented in Table 4.

Table 4: The observed maximum errors of Example 3.2.

Example 3.3. Considering the following boundary value problems with nonclassical condition: We determine to get the true solution, given by . The numerical results are presented in Table 5.

Table 5: The numerical results of Example 3.3.

Example 3.4. Considering the following singular third-order three points boundary value problems with nonlinear condition We determine to get the true solution, given by . The numerical results are presented in Table 6.

Table 6: The numerical results of Example 3.4, .

Example 3.5. Considering the following boundary value problems with nonlinear condition: We determine to get the true solution, given by . The numerical results are presented in Table 7.

Table 7: The numerical results of Example 3.5.

#### 4. Conclusions and Remarks

In this work, we present an algorithm for solving third-order mixed boundary value problems (BVPs) based on the reproducing kernel method. The method can be generalized to get reproducing kernel of problem with linear conditions. All computations are performed by the Mathematica 7.0 software package.

#### Acknowledgments

This paper is supported by the Natural Science Foundation of Inner Mongolia (no. 2009MS0103) and the project of Inner Mongolia University of Technology (no. ZS201036).

#### References

1. J. Rashidinia and M. Ghasemi, “B-spline collocation for solution of two-point boundary value problems,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 2325–2342, 2011.
2. H. N. Caglar, S. H. Caglar, and E. H. Twizell, “The numerical solution of third-order boundary-value problems with fourth-degree B-spline functions,” International Journal of Computer Mathematics, vol. 71, no. 3, pp. 373–381, 1999.
3. A. Khan and T. Aziz, “The numerical solution of third-order boundary-value problems using quintic splines,” Applied Mathematics and Computation, vol. 137, no. 2-3, pp. 253–260, 2003.
4. I. A. Tirmizi, E. H. Twizell, and Siraj-Ul-Islam, “A numerical method for third-order non-linear boundary-value problems in engineering,” International Journal of Computer Mathematics, vol. 82, no. 1, pp. 103–109, 2005.
5. M. Tatari and M. Dehghan, “The use of the Adomian decomposition method for solving multipoint boundary value problems,” Physica Scripta, vol. 73, no. 6, pp. 672–676, 2006.
6. M. Haque, M. H. Baluch, and M. F. N. Mohsen, “Solution of multiple point nonlinear boundary value problems by method of weighted residuals,” International Journal of Computer Mathematics, vol. 18, no. 3, pp. 341–354, 1986.
7. Y.-l. Wang and L. Chao, “Using reproducing kernel for solving a class of partial differential equation with variable-coefficients,” Applied Mathematics and Mechanics, vol. 29, no. 1, pp. 129–137, 2008.
8. Y. L. Wang, Z. Y. Li, Y. Cao, and X. H. Wan, “A new method for solving a class of mixed boundary value problems with singular coefficient,” Applied Mathematics and Computation, vol. 217, no. 6, pp. 2768–2772, 2010.
9. Z. Chen and Z.-J. Chen, “The exact solution of system of linear operator equations in reproducing kernel spaces,” Applied Mathematics and Computation, vol. 203, no. 1, pp. 56–61, 2008.
10. Y. L. Wang, C. L. Temuer, and J. Pang, “New algorithm for second-order boundary value problems of integro-differential equation,” Journal of Computational and Applied Mathematics, vol. 229, no. 1, pp. 1–6, 2009.
11. S. Islam, S. I. A. Tirmizi, and M. A. Khan, “Non-polynomial splines approachto the solution of asystem of third-order boundary-value problems,” Applied Mathematics Computation, vol. 168, no. 1, pp. 125–163, 2005.
12. M. A. Noor and E. E. Al-Said, “Quartic splines solutions of third-order obstacle problems,” Applied Mathematics and Computation, vol. 153, no. 2, pp. 307–316, 2004.
13. E. A. Al-Said, “Numerical solutions for system of third-order boundary value problems,” International Journal of Computer Mathematics, vol. 78, no. 1, pp. 111–121, 2001.
14. E. A. Al-Said and M. A. Noor, “Cubic splines method for a system of third-order boundary value problems,” Applied Mathematics and Computation, vol. 142, no. 2-3, pp. 195–204, 2003.
15. M. A. Noor and A. K. Khalifa, “A numerical approach for odd-order obstacle problems,” International Journal of Computer Mathematics, vol. 54, no. 1, pp. 109–116, 1994.
16. E. A. Al-Said, M. A. Noor, and A. K. Khalifa, “Finite difference scheme for variational inequalities,” Journal of Optimization Theory and Applications, vol. 89, no. 2, pp. 453–459, 1996.
17. F. Gao and C.-M. Chi, “Solving third-order obstacle problems with quartic B-splines,” Applied Mathematics and Computation, vol. 180, no. 1, pp. 270–274, 2006.