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A Novel Method for Solving KdV Equation Based on Reproducing Kernel Hilbert Space Method
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.
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 , 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 . 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 . Goda  and Vliengenthart  used the finite difference method to obtain the numerical solution of KdV equation. Soliman  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 . Wazwaz presented the Adomian decomposition method for KdV equation with different initial conditions . Syam  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 . 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  applied the RKHSM to handle the second-order boundary value problems. Yao and Cui  and Wang et al.  investigated a class of singular boundary value problems by this method and the obtained results were good. Zhou et al.  used the RKHSM effectively to solve second-order boundary value problems. In , the method was used to solve nonlinear infinite-delay-differential equations. Wang and Chao , Li and Cui , and Zhou and Cui  independently employed the RKHSM to variable-coefficient partial differential equations. Geng and Cui  and Du and Cui  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  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  obtained the representation of the exact solution for the nonlinear Volterra-Fredholm integral equations by using the reproducing kernel space. Wu and Li  applied iterative reproducing kernel method to obtain the analytical approximate solution of a nonlinear oscillator with discontinuities. Inc et al.  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.
2.1. 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
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  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.
Example 11 (see ). 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  and Homotopy analysis method . 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.
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.
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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