Abstract and Applied Analysis

Abstract and Applied Analysis / 2014 / Article
Special Issue

Analytical and Numerical Approaches for Complicated Nonlinear Equations

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Research Article | Open Access

Volume 2014 |Article ID 745287 | 9 pages | https://doi.org/10.1155/2014/745287

Numerical Solution of Seventh-Order Boundary Value Problems by a Novel Method

Academic Editor: Hossein Jafari
Received25 Nov 2013
Revised03 Feb 2014
Accepted05 Feb 2014
Published23 Mar 2014

Abstract

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.

1. Introduction

Consider the seventh-order boundary value problem [15]: 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) [628] 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 [2931].

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.

3. Exact and Approximate Solutions of (1)-(2) in

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 .

Proof. From the (17) and uniqueness of solution of (11), we have This completes the proof.

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 .

Proof. From (18) and (20), we obtain Thus In addition Then, is monotonically decreasing in .

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 16 and Figures 16). 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.


Exact solutionApproximate solutionAbsolute errorRelative error

0.0 1.0 1.0 0.0 0.0
0.1 1.1051709180756476248 1.1051709180727538232
0.2 1.2214027581601698339 1.2214027581330885422
0.3 1.3498588075760031040 1.3498588075089984856
0.4 1.4918246976412703178 1.4918246975765762858
0.5 1.6487212707001281468 1.6487212707474912552
0.6 1.8221188003905089749 1.8221188006600701498
0.7 2.0137527074704765216 2.0137527079512967073
0.8 2.2255409284924676046 2.2255409289689961077
0.9 2.4596031111569496638 2.4596031113394646558
1.0 2.7182818284590452354 2.7182818284590452354


Absolute error [2] Absolute error [1] Absolute error [RKHSM]

0.0 0.0 0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0


Exact solutionApproximate solutionAbsolute error

0.0 0.0 0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0


Absolute error [4] Absolute error [3] Absolute error [5] Absolute error [RKHSM]

0.0 0.0 0.0 0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0


Exact solution Approximate solution Absolute error

0.0 1.0 1.0 0.0
0.1 0.81435367623236361584 0.81435367623236697064
0.2 0.65498460246238548694 0.65498460246237032242
0.3 0.51857275447720250625 0.51857275447711724998
0.4 0.40219202762138358044 0.40219202762145521926
0.5 0.30326532985631671180 0.30326532985631655134
0.6 0.21952465443761057305 0.21952465443760682026
0.7 0.14897559113742285441 0.14897559113743255406
0.8 0.089865792823444318286 0.089865792823449635588
0.9 0.040656965974059911188 0.040656965974059021714
1.0 0.0


Absolute error [3] Absolute error [5] Absolute error [RKHSM]

0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0

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 [1]
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 [3]
After homogenizing the boundary conditions of (28), we get where, we used the following transformation