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 Kortewegde Vries (KdV) equation of the form
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 [
The KdV equation exhibits solutions such as solitary waves, solitons and recurrence [
In present work, we use the following equation:
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 [
The efficiency of the method was used by many authors to investigate several scientific applications. Geng and Cui [
The paper is organized as follows. Section
In this section, we define some useful reproducing kernel spaces.
Let
for all
for all
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 space
Since
The
Similarly, the space
On defining the linear operator
The operator
We have
Now, choose a countable dense subset
Suppose that
We have
If
Since
Now the approximate solution
If we write
If
Since
From the definition of the reproducing kernel, we have
Suppose that
First, we prove the convergence of
In a same manner, it can be proved that
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 [
The exact solution of Example

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 
The approximate solution of Example

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 
The absolute error of Example

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} 
The relative error of Example

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} 
The absolute error for Example
The absolute error for Example
The relative error for Example
Consider the following KdV equation with initial condition
We now consider the KdV equation with initial condition
The problem discussed in this paper has been solved with Adomian method [
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 060104SF1050.