Research Article  Open Access
Yulan Wang, Shuai Lu, Fugui Tan, Mingjing Du, Hao Yu, "Solving a Class of Singular FifthOrder Boundary Value Problems Using Reproducing Kernel Hilbert Space Method", Abstract and Applied Analysis, vol. 2013, Article ID 925192, 6 pages, 2013. https://doi.org/10.1155/2013/925192
Solving a Class of Singular FifthOrder Boundary Value Problems Using Reproducing Kernel Hilbert Space Method
Abstract
We use the reproducing kernel Hilbert space method to solve the fifthorder boundary value problems. The exact solution to the fifthorder boundary value problems is obtained in reproducing kernel space. The approximate solution is given by using an iterative method and the finite section method. The present method reveals to be more effective and convenient compared with the other methods.
1. Introduction
The reproducing kernel Hilbert space method has been shown [1–7] to solve effectively, easily, and accurately a large class of linear and nonlinear, ordinary, partial differential equations. However, in [1–7], it cannot be used directly boundary value problems with mixed boundary conditions, since it is very difficult to obtain a reproducing kernel function satisfying mixed nonlinear boundary conditions. The aim of this work is to fill this gap. In [8], we give a new reproducing kernel Hilbert space for solving singular linear fourthorder boundary value problems with mixed boundary conditions. In this paper, we use the new reproducing kernel Hilbert function space method to solve the nonlinear fifthorder boundary value problems.
Singular fifthorder boundary value problems arise in the fields of gas dynamics, Newtonian fluid mechanics, fluid mechanics, fluid dynamics, elasticity, reactiondiffusion processes, chemical kinetics, and other branches of applied mathematics.
Let us consider the following class of singular fifthorder mixed boundary value problems: where () are known functions. () are linear conditions. We assume that (1) has a unique solution which belongs to , where is a reproducing kernel space.
Remark 1. If (), then (1) is an initial value problem. If (), then (1) is a multipoint problem and so on. That is, problem (1) has a rather general form.
Let and , .
Consider
It is easy to prove that is a bounded linear operator. On the other hand, we suppose that the linear conditions can also always be homogenized; after homogenization of these conditions, we put these conditions into the reproducing kernel space constructed in the following section. Equation (1) can be transformed into the following form in :
2. Reproducing Kernel Hilbert Space
Definition 2. Let be a real Hilbert space of functions . A function is called reproducing kernel for if(i) for all , (ii) for all and all .
Definition 3. A real Hilbert space of functions on a set is called a reproducing kernel Hilbert space if there exists a reproducing kernel of .
One defines that the inner product space are absolutely continuous function, .
The inner product in is given by
Theorem 4 (see [8]). The space is a reproducing kernel space, and its reproducing kernel is
For studying the solution of (1) in the homogenized form, we give space (6) as follows: is equipped with the same inner product . In the following, we construct a reproducing kernel for the space , and we give Lemmas 5 and 6.
Lemma 5. Let be a bounded linear operator; function is the reproducing kernel of space . Let ; then , where denotes any reproducing kernel space of functions over , the symbol indicates that the operator applies to functions of the variable , and the symbol indicates that the operator applies to function of the variable and .
Proof. Consider
Lemma 6. If , , and are defined as in Lemma 5, let ; consider the space , then, is the reproducing kernel of space .
Proof. For any , next, we will prove .
Consider
Let , , , , and , where the symbol () indicates that the operator () applies to functions of the variable . Using Lemma 6, we get Theorem 7.
Theorem 7. The space is a reproducing kernel space, and its reproducing kernel is
3. Analytical Solution
Let , . Via GramSchmidt orthonormalization for , we get where the are the coefficients resulting from GramSchmid orthonormalization.
Lemma 8. If are distinct points dense in and is existent, then is a complete function system in .
Proof. For each fixed , if , then Taking into account the density of , it results in . It follows that from the existence of .
Theorem 9. If are distinct points dense in and is existent, then is an analytical solution of (3).
Proof. can be expanded to Fourier series in terms of normal orthogonal basis in as follows:
4. Numerical Solution
We define an approximate solution by
Theorem 10. Let , where and are given by (12) and (14); then the sequence of real numbers is monotonously decreasing and .
Proof. We have Clearly, and consequently is monotonously decreasing in the sense of . By Theorem 9, we know that is convergent in the norm of ; then we have Hence, .
Theorem 11 (convergence analysis). and are uniformly convergent to and , , where and are given by (12) and (14).
Proof. For any , , Then there exists such that
The numerical solution to (3) can be obtained using the following method: where the coefficients , , are determined by the equation Using (19) and (20), we have , . So, is the approximation solution of (3).
5. Numerical Experiment
In this section, two numerical examples are studied to demonstrate the accuracy of the present method.
Example 12. Consider the following fifthorder boundary value problem with nonclassical side condition (the righthand side of this problem has a singularity at , ): where . The exact solution is . The numerical results are presented in Table 1.

Example 13 (see [9]). Consider the following fifthorder boundary value problem (the righthand side of this problem has a singularity at ): where the exact solution is . By the homogeneous process of the boundary condition, letting , the problem can be transformed into the equivalent form The numerical results are presented in Tables 2, 3, and 4.



6. Conclusions and Remarks
In this paper, a new reproducing kernel space satisfying mixed boundary value conditions is constructed skillfully. This makes it easy to solve such kind of problems. Furthermore, the exact solution of the problem can be expressed in series form. The numerical results demonstrate that the new method is quite accurate and efficient for singular problems of fifthorder ordinary differential equations. All computations have been performed using the Mathematica 7.0 software package.
Acknowledgments
The authors would like to express their thanks to the unknown referees for their careful reading and helpful comments. This paper is supported by the Natural Science Foundation of Inner Mongolia (2013MS0109) and Project Application Technology Research and Development Foundation of Inner Mongolia.
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Copyright
Copyright © 2013 Yulan Wang 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.