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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 745287, 9 pages
Numerical Solution of Seventh-Order Boundary Value Problems by a Novel Method
1Department of Mathematics, Science Faculty, Fırat University, 23119 Elazığ, Turkey
2Department of Mathematics, Education Faculty, Dicle University, 21280 Diyarbakır, Turkey
3Department of Mathematics, Texas A&M University Kingsville, Kingsville, TX 78363, USA
Received 25 November 2013; Revised 3 February 2014; Accepted 5 February 2014; Published 23 March 2014
Academic Editor: Hossein Jafari
Copyright © 2014 Mustafa Inc and Ali Akgül. 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.
We demonstrate the efficiency of reproducing kernel Hilbert space method on the seventh-order boundary value problems satisfying boundary conditions. These results have been compared with the results that are obtained by variational iteration method (VIM), homotopy perturbation method (HPM), Adomian decomposition method (ADM), variation of parameters method (VPM), and homotopy analysis method (HAM). Obtained results show that our method is very effective.
Consider the seventh-order boundary value problem [1–5]: with boundary conditions The analytical solution of seventh-order differential equations are rarely exists in literature. However, there are various numerical methods for the solution of (1)-(2). The aim of this work is to apply reproducing kernel Hilbert space method (RKHSM) [6–28] to solve the seventh-order boundary value problems. Numerical results of the seventh-order boundary value problems have been obtained by this method in our work. This study shows that the proposed method can be considered as an alternative technique for solving linear and nonlinear problems in science and engineering [29–31].
The paper is organized as follows. Section 2 introduces several reproducing kernel spaces. We provide the main results and the exact and approximate solutions of (1)-(2) in Section 3. We have proved that the approximate solution converges to the exact solution uniformly. Some numerical experiments are illustrated in Section 4. There are some conclusions in the last section.
2. Reproducing Kernel Spaces
In this section, we define some useful reproducing kernel spaces.
Definition 1. We define the space by The inner product and the norm in are defined, respectively, by The space is a reproducing kernel space and its reproducing kernel function is given by
Definition 2. We define the space by the following: The inner product and the norm in are defined, respectively, by The space is a reproducing kernel space; that is, for each fixed and any , there exists a function such that
Theorem 3. The space is a reproducing kernel Hilbert space whose reproducing kernel function is given by where and can be obtained by Maple 16 and proof of Theorem 3 is given in Appendix.
The solution of (1)-(2) is given in the reproducing kernel space . The linear operator is bounded. After homogenizing the boundary conditions, we obtain We choose a countable dense subset in and let where is conjugate operator of and is given by (5). Furthermore, for simplicity let ; namely, Now one can deduce the following lemmas.
Lemma 4. is complete system of .
Proof. For , let ; that is, Note that is the dense set in ; therefore, . It follows that from the existence of .
Lemma 5. The following formula holds: where the subscript of operator indicates that the operator applies to function of .
Proof. Consider the following: This completes the proof.
Remark 6. The orthonormal system of can be derived from Gram-Schmidt orthogonalization process of , where are orthogonal coefficients.
In the following, we will give the representation of the exact solution of (11) in the reproducing kernel space .
Theorem 7. If is the exact solution of (11), then where is a dense set in .
Now the approximate solution can be obtained by truncating the -term of the exact solution as
Lemma 8. Assume is the solution of (11) and is the error between the approximate solution and the exact solution . Then the error sequence is monotone decreasing in the sense of and .
Remark 9. The seventh-order boundary value problems have come out in construction engineering, beam column theory, and chemical reactions. Therefore solutions of the seventh-order boundary value problems are very important in the literature. The reproducing kernel function for seventh-order boundary value problem has not been calculated till now. All computations are performed by Maple 16. The RKHSM does not require discretization of the variables, that is, time and space, and it is not affected by computational round-off errors and one is not faced with necessity of large computer memory and time. The accuracy of the RKHSM for the seventh-order boundary value problems is controllable and absolute errors are small with present choice of (see Tables 1–6 and Figures 1–6). The obtained numerical results justify the advantage of this methodology. We gave transformations to homogenize the boundary conditions for all examples. Additionally, we improved our programme to find numerical results. As shown in Tables 1, 3, and 5 all the numerical results have been found in very short time.
4. Numerical Results
In this section, three numerical examples are provided to show the accuracy of the present method.
Example 10. We first consider the seventh-order nonlinear boundary value problem:
The exact solution of (24) is given as 
After homogenizing the boundary conditions of (24), we obtain where we used the following transformation: Using the RKHSM for this example we obtain Tables 1-2 and Figures 1-2.
Example 11. We now consider the seventh-order linear BVP
The exact solution of (28) is given as 
After homogenizing the boundary conditions of (28), we get where, we used the following transformation Using RKHSM for this example we obtain Tables 3-4 and Figures 3-4.
where we used the following transformation:
Remark 13. Using our method we chose points on . In Tables 1–6, we computed the absolute errors at the points . The RKHSM tested on three problems, one linear and two nonlinear. A comparison with VIM , HPM , ADM , VPM , and HAM  was made and it was seen that the present method yields good results (see Tables 1–6 and Figures 1–6).
In this paper, we introduced an algorithm for solving the seventh-order problem with boundary conditions. For illustration purposes, we chose three examples which were selected to show the computational accuracy. It may be concluded that the RKHSM is very powerful and efficient in finding exact solution for a wide class of boundary value problems. The approximate solution obtained by the present method is uniformly convergent. Clearly, the series solution methodology can be applied to much more complicated nonlinear differential equations and boundary value problems. However, if the problem becomes nonlinear, then the RKHSM does not require discretization or perturbation and it does not make closure approximation. Results of numerical examples show that the present method is an accurate and reliable analytical method for the seventh-order boundary value problem.
Proof of Theorem 3. Let . By Definition 2 we have Through several integrations by parts for (A.1) we have Note that property of the reproducing kernel Now, if then (A.2) implies that when , then and therefore Since we have Since , it follows that From (A.4)–(A.10), the unknown coefficients and can be obtained. This completes the proof.
This paper is a part of the Ph.D. thesis of Ali Akgül.
Conflict of Interests
The authors declare that they do not have any competing or conflict of interests.
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