Abstract and Applied Analysis

Volume 2013 (2013), Article ID 578942, 11 pages

http://dx.doi.org/10.1155/2013/578942

Research Article

A Novel Method for Solving KdV Equation Based on Reproducing Kernel Hilbert Space Method

Mustafa Inc,¹ Ali Akgül,² and Adem Kiliçman³

¹Department of Mathematics, Science Faculty, Firat University, 23119 Elaziğ, Turkey

²Department of Mathematics, Education Faculty, Dicle University, 21280 Diyarbakır, Turkey

³Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia (UPM), Selangor, 43400 Serdang, Malaysia

Received 19 September 2012; Revised 17 December 2012; Accepted 23 December 2012

Academic Editor: Lan Xu

Copyright © 2013 Mustafa Inc 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.

We propose a reproducing kernel method for solving the KdV equation with initial condition based on the reproducing kernel theory. The exact solution is represented in the form of series in the reproducing kernel Hilbert space. Some numerical examples have also been studied to demonstrate the accuracy of the present method. Results of numerical examples show that the presented method is effective.

1. Introduction

In this paper, we consider the Korteweg-de Vries (KdV) equation of the form

with initial condition

The constant factor is just a scaling factor to make solutions easier to describe. Most of the authors chose to be one or six. Some mathematicians and physicians investigated the exact solution of the KdV equation without having either initial conditions or boundary conditions [1], while others studied its numerical solution [2, 3].

The numerical solution of KdV equation is of great importance because it is used in the study of nonlinear dispersive waves. This equation is used to describe many important physical phenomena. Some of these studies are the shallow water waves and the ion acoustic plasma waves [4]. It represents the long time evolution of wave phenomena, in which the effect of nonlinear terms is counterbalanced by the dispersion . Thus it has been found to model many wave phenomena such as waves in enharmonic crystals, bubble liquid mixtures, ion acoustic wave, and magnetohydrodynamic waves in a warm plasma as well as shallow water waves [5, 6].

The KdV equation exhibits solutions such as solitary waves, solitons and recurrence [7]. Goda [8] and Vliengenthart [9] used the finite difference method to obtain the numerical solution of KdV equation. Soliman [2] used the collocation solution with septic splines to obtain the solution of the KdV equation. Numerical solutions of KdV equation were obtained by the variational iteration method, finite difference method [3, 10], and by using the meshless based on the collocation with radial basis functions [11]. Wazwaz presented the Adomian decomposition method for KdV equation with different initial conditions [12]. Syam [13] worked the ADM for solving the nonlinear KdV equation with appropriate initial conditions.

In present work, we use the following equation:

by transformation for homogeneous initial condition of (1) and (2), we get the following:

where

In this paper, we solve (1) and (2) by using reproducing kernel method. The nonlinear problem is solved easily and elegantly without linearizing the problem by using RKM. The technique has many advantages over the classical techniques; mainly, it avoids linearization to find analytic and approximate solutions of (1) and (2). It also avoids discretization and provides an efficient numerical solution with high accuracy, minimal calculation, and avoidance of physically unrealistic assumptions. In the next section, we will describe the procedure of this method.

The theory of reproducing kernels was used for the first time at the beginning of the 20th century by Zaremba in his work on boundary value problems for harmonic and biharmonic functions [14]. Reproducing kernel theory has important application in numerical analysis, differential equations, probability and statistics [14, 15]. Recently, using the RKM, some authors discussed fractional differential equation, nonlinear oscillator with discontinuity, singular nonlinear two-point periodic boundary value problems, integral equations, and nonlinear partial differential equations [14, 15].

The efficiency of the method was used by many authors to investigate several scientific applications. Geng and Cui [16] applied the RKHSM to handle the second-order boundary value problems. Yao and Cui [17] and Wang et al. [18] investigated a class of singular boundary value problems by this method and the obtained results were good. Zhou et al. [19] used the RKHSM effectively to solve second-order boundary value problems. In [20], the method was used to solve nonlinear infinite-delay-differential equations. Wang and Chao [21], Li and Cui [22], and Zhou and Cui [23] independently employed the RKHSM to variable-coefficient partial differential equations. Geng and Cui [24] and Du and Cui [25] investigated to the approximate solution of the forced Duffing equation with integral boundary conditions by combining the homotopy perturbation method and the RKHSM. Lv and Cui [26] presented a new algorithm to solve linear fifth-order boundary value problems. In [27, 28], authors developed a new existence proof of solutions for nonlinear boundary value problems. Cui and Du [29] obtained the representation of the exact solution for the nonlinear Volterra-Fredholm integral equations by using the reproducing kernel space. Wu and Li [30] applied iterative reproducing kernel method to obtain the analytical approximate solution of a nonlinear oscillator with discontinuities. Inc et al. [15] used this method for solving Telegraph equation.

The paper is organized as follows. Section 2 introduces several reproducing kernel spaces and a linear operator. The representation in is presented in Section 3. Section 4 provides the main results. The exact and approximate solutions of (1) and (2) and an iterative method are developed for the kind of problems in the reproducing kernel space. We have proved that the approximate solution uniformly converges to the exact solution. Some numerical experiments are illustrated in Section 5. We give some conclusions in Section 6.

2. Preliminaries

2.1. Reproducing Kernel Spaces

In this section, we define some useful reproducing kernel spaces.

Definition 1 (reproducing kernel). Let be a nonempty abstract set. A function is a reproducing kernel of the Hilbert space if and only if

(a)for all ,  ,(b)for all ,  ,  . This is also called “the reproducing property”: the value of the function at the point is reproduced by the inner product of with .

Then we need some notation that we use in the development of the paper. In the next we define several spaces with inner product over those spaces. Thus the space is defined as

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

Similarly, we define the space

The inner product and the norm in are defined, respectively, by

Thus the space is also a reproducing kernel space and its reproducing kernel function can be given by

and the space

where the inner product and and the norm in are defined, respectively, by

The space is a reproducing kernel space, and its reproducing kernel function is given by

Similarly, the space is defined 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

Further we define the space as

and the inner product and the norm in are defined, respectively, by

Now we have the following theorem.

Theorem 2. The space is a complete reproducing kernel space and, its reproducing kernel function can be denoted by

where

Proof. Since

through iterative integrations by parts for (22) we have

Note that property of the reproducing kernel

If

then by (23) we obtain the following equation:

when ,

therefore

Since

we have

From (25)–(31), the unknown coefficients ve    can be obtained. Thus is given by

Theorem 3. The is a reproducing kernel space, and its reproducing kernel function is

such that for any ,

where ,   are the reproducing kernel functions of   and , respectively.

Similarly, the space is defined as

The inner product and the norm in    are defined, respectively, by

Then the space is a reproducing kernel space and its reproducing kernel function is

3. Solution Representation in

On defining the linear operator as

model problem (1) changes to the following problem:

Lemma 4. The operator is a bounded linear operator.

Proof. We have

since

on using the the continuity of , we have

Similarly for ,

and then

Therefore

Now, choose a countable dense subset in and define

where is the adjoint operator of . The orthonormal system of can be derived from the process of Gram-Schmidt orthogonalization of as

Theorem 5. Suppose that is dense in ; then is complete system in and

Proof. We have

Clearly . For each fixed , if

then

Note that is dense in , hence, . It follows that from the existence of . So the proof is complete.

Theorem 6. If is dense in , then the solution of (39) is

Proof. Since is complete system in , we have

Now the approximate solution can be obtained from the -term intercept of the exact solution and

Obviously

4. The Method Implementation

If we write

then (52) can be written as

Now let ; then from the initial conditions of (39), is known. We put and define the -term approximation to by

where

In the sequel, we verify that the approximate solution converges to the exact solution, uniformly. First the following lemma is given.

Lemma 7. If  ,  , and is continuous, then

Proof. Since

From the definition of the reproducing kernel, we have

It follows that

From the convergence of , there exists a constant , such that

At the same time, we can prove

using Theorem 3. Hence

In a similiar way it can be shown that

So

This completes the proof.

Theorem 8. Suppose that is a bounded in (58) and (39) has a unique solution. If   is dense in , then the -term approximate solution derived from the above method converges to the analytical solution of (39) and

where is given by (59).

Proof. First, we prove the convergence of . From (58), we infer that

The orthonormality of   yields that

In terms of (71), it holds that . Due to the condition that is bounded, is convergent and there exists a constant such that

This implies that

If , then

On account of

consequently

The completeness of shows that as . Now, let we prove that is the solution of (39). Taking limits in (58) we get

Note that

Therefore

In view of (71), we have

Since is dense in, for each , there exists a subsequence such that

We know that

Let ; by Lemma 7 and the continuity of , we have

which indicates that satisfy (39). This completes the proof.

Remark 9. In a same manner, it can be proved that

where

where is given by (59).

5. Numerical Results

In this section, two numerical examples are provided to show the accuracy of the present method. All computations are performed by Maple 16. Results obtained by the method are compared with exact solution and the ADM [13] of each example are found to be in good agreement with each others. The RKM does not require discretization of the variables, that is, time and space, it is not effected by computation round off errors and one is not faced with necessity of large computer memory and time. The accuracy of the RKM for the KdV equation is controllable and absolute errors are very small with present choice of and (see Tables 1, 2, 3, and 4 and Figures 1, 2, and 3). The numerical results that we obtained justify the advantage of this methodology.

Table 1: The exact solution of Example 10 for initial condition at , .

0.1
0.2
0.3
0.4
0.5
0.6

0.1
1.830273924
1.269479180
0.718402632
0.36141327
0.17121984
0.07882210

0.2
1.922085966
1.423155525
0.839948683
0.43230491
0.20711674
0.09585068

0.3
1.980132581
1.572895466
0.973834722
0.51486639
0.25001974
0.11644607

0.4
2
1.711277572
1.118110335
0.61003999
0.30105415
0.14130164

0.5
1.980132581
1.830273924
1.269479180
0.71840263
0.36141327
0.17121984

0.6
1.922085966
1.922085966
1.423155525
0.83994868
0.43230491
0.20711674

Table 2: The approximate solution of Example 10 for initial condition at , .

0.1
0.2
0.3
0.4
0.5
0.6

0.1
1.830273864
1.269478141
0.718402628
0.36141327
0.17128272
0.07883011

0.2
1.922085928
1.423155537
0.839948629
0.43230491
0.20711710
0.09585155

0.3
1.980132606
1.572896076
0.973834717
0.51486633
0.25001974
0.11644677

0.4
2.000000027
1.711278098
1.118110380
0.61004008
0.30105468
0.14130128

0.5
1.980133013
1.830274266
1.269479288
0.71840299
0.36141338
0.17122050

0.6
1.922086667
1.922086057
1.423155510
0.83994874
0.43230465
0.20711669

Table 3: The absolute error of Example 11 for initial condition at , .

0.1
0.2
0.3
0.4
0.5
0.6

0.1
1.78 × 10⁻⁶
3.01 × 10⁻⁹
8.55 × 10⁻⁷
3.49 × 10⁻⁷
2.85 × 10⁻⁷
6.28 × 10⁻⁷

0.2
6.38 × 10⁻⁷
6.98 × 10⁻⁷
6.52 × 10⁻⁷
4.51 × 10⁻⁷
8.33 × 10⁻⁶
2.42 × 10⁻⁷

0.3
2.2 × 10⁻⁸
9.09 × 10⁻⁷
6.88 × 10⁻⁶
1.35 × 10⁻⁷
2.97 × 10⁻⁶
1.69 × 10⁻⁷

0.4
1.70 × 10⁻⁷
1.03 × 10⁻⁷
5.38 × 10⁻⁷
1.20 × 10⁻⁶
3.98 × 10⁻⁷
1.68 × 10⁻⁷

0.5
2.26 × 10⁻⁷
1.29 × 10⁻⁷
8.74 × 10⁻⁷
3.13 × 10⁻⁷
4.02 × 10⁻⁷
9.63 × 10⁻⁷

0.6
8.94 × 10⁻⁷
7.83 × 10⁻⁷
4.34 × 10⁻⁷
9.79 × 10⁻⁷
2.77 × 10⁻⁷
1.45 × 10⁻⁸

Table 4: The relative error of Example 11 for initial condition at , .

0.1
0.2
0.3
0.4
0.5
0.6

0.1
9.201 × 10⁻⁷
5.113 × 10⁻¹⁰
5.707 × 10⁻⁷
3.868 × 10⁻⁷
6.06 × 10⁻⁷
2.76 × 10⁻⁶

0.2
3.441 × 10⁻⁷
3.498 × 10⁻⁷
3.968 × 10⁻⁷
4.320 × 10⁻⁷
1.48 × 10⁻⁶
8.84 × 10⁻⁷

0.3
1.255 × 10⁻⁸
4.555 × 10⁻⁷
3.881 × 10⁻⁶
1.132 × 10⁻⁷
4.49 × 10⁻⁶
5.13 × 10⁻⁷

0.4
1.032 × 10⁻⁷
5.264 × 10⁻⁸
2.862 × 10⁻⁷
8.964 × 10⁻⁷
5.13 × 10⁻⁷
4.27 × 10⁻⁷

0.5
1.460 × 10⁻⁷
6.857 × 10⁻⁸
4.469 × 10⁻⁷
2.089 × 10⁻⁷
4.44 × 10⁻⁷
2.04 × 10⁻⁶

0.6
6.079 × 10⁻⁷
4.410 × 10⁻⁷
2.175 × 10⁻⁷
5.958 × 10⁻⁷
2.65 × 10⁻⁷
2.58 × 10⁻⁸

Figure 1: The absolute error for Example 10 at ,  .

Figure 2: The absolute error for Example 11 at ,  .

Figure 3: The relative error for Example 11 at ,  .

Example 10 (see [13]). Consider the following KdV equation with initial condition

with . The exact solution is . If we apply (3) to (86), then the following (87) is obtained

Example 11 (see [13]). We now consider the KdV equation with initial condition

The exact solution is . If we apply (3) to (88), then the following (89) is obtained:

Using our method we choose points on . We replace with for simplicity. In Tables 3 and 4, we compute the absolute errors and the relative errors at the points  .

Remark 12. The problem discussed in this paper has been solved with Adomian method [13] and Homotopy analysis method [31]. In these studies, even though the numerical results give good results for large values of , these methods give away values from the analytical solution for small values of and . However, the method is used in our study for large and small values of and , results are very close to the analytical solutions can be obtained. In doing so, it is possible to refine the result by increasing the intensive points.

6. Conclusion

In this paper, we introduce an algorithm for solving the KdV equation with initial condition. For illustration purposes, we chose two examples which were selected to show the computational accuracy. It may be concluded that the RKM is very powerful and efficient in finding exact solution for wide classes of problem. 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 RKM 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 KdV equation with initial or boundary conditions.

Acknowledgment

A. Kiliçman gratefully acknowledge that this paper was partially supported by the University Putra Malaysia under the ERGS Grant Scheme having project no. 5527068 and Ministry of Science, Technology and Inovation (MOSTI), Malaysia under the Science Fund 06-01-04-SF1050.

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