Collocation Method for Solving the Generalized KdV Equation

In this work, we have obtained numerical solutions of the generalized Korteweg-de Vries (GKdV) equation by using septic B-spline collocation finite element method. The suggested numerical algorithm is controlled by applying test problems including; single soliton wave. Our numerical algorithm, attributed to a Crank Nicolson approximation in time, is unconditionally stable. To control the performance of the newly applied method, the error norms, 2 L and L∞ and invariants 1 I , 2 I and 3 I have been calculated. Our numerical results are compared with some of those available in the literature.


Introduction
Several physical processes for example dispersion of long waves in shallow water waves under gravity, bubble-liquid mixtures, ion acoustic plasma waves, fluid mechanics, nonlinear optics and wave phenomena in enharmonic crystals can be expressed by the KdV equation which was first introduced by Korteweg

and de
Vries [1]. The equation was solved analytically by Zabusky, Fornberg and Whitham, [2] [3]. Zabusky and Kruskal [4] were first obtained numerical solutions of the equation with finite difference method. Gardner et al. [5] demonstrated existence and uniqueness of solutions of the KdV equation. Several scientists have used various numerical methods including pseudospectral method [4], finite difference method [6] [7], finite element method [8]- [15] and heat balance integral method [15] to solve the equation. Numerical solutions of the KdV equation were obtained using differential quadrature method based on cosine ex-pansion by B. Saka [16]. Like the KdV equation, in recent years, various numerical methods have been improved for the solution of the MKdV equation. Kaya [17] calculated the explicit solutions of the higher order modified Korteweg de-Vries equation by Adomian decomposition method. MKdV equation has been solved by using Galerkins' method with quadratic B-spline finite elements by Biswas et al. [18]. Raslan and Baghdady [19] [20] indicated the accuracy and stability of the difference solution of the MKdV equation and they obtained the numerical aspects of the dynamics of shallow water waves along lakes' shores and beaches modeled by the MKdV equation. A new variety of (3 + 1)-dimensional MKdV equations and multiple soliton solutions for each new equation were established by Wazwaz [21] [22]. Lumped Petrov-Galerkin and Galerkin methods were practiced to the MKdV equation by Ak et al. [23] [24].
GKdV equation has received much less attention, presumably because of its higher nonlinearity for 2 p > . The symmetry group was calculated for the equation and several classes of solutions were obtained in [25]. Liu and Yi [26] developed and analyzed a Hamiltonian preserving DG method for solving the generalized KdV equation. The initial value problem of a kind of GKdV equations is considered by using Sobolev space theory and finite element method by Lai et al. [27]. Alvarado and Omel'yanov [28] create a finite differences scheme to simulate the solution of the Cauchy problem and present some numerical results for the problem of the solitary waves interaction. A class of fully discrete scheme for the generalized Korteweg-de Vries equation in a bounded domain ( ) 0, L has studied by Sepulveda and Villagran [29]. Collocation finite element method based on quintic B-spline functions is applied to the generalized KdV equation by Ak et al. [30]. Solitary wave solution for the GKdV equation by using ADM has been obtained by Ismail et al. [31].
In this article, we will take in consideration for the following GKdV equation with the homogeneous boundary conditions and an initial condition where t is time, x is the space coordinate, ε and µ are positive parameters.
One of the primary mathematical models for describing the theory of water waves in shallow channels is the following Korteweg de-Vries (KdV) equation: The terms x UU and xxx U in the Equation (4) stand for the nonlinear convection and dispersion, respectively. In this paper, we have numerically solved the GKdV equation using collocation method with septic B-spline finite elements. We have investigated the motion of a single soliton wave to show the performance and profiency of the proposed method. Also we have showed the suggested method is unconditionally stable applying the von-Neumann stability analysis.

Septic B-Spline Collocation Method
We think of a mesh as a uniform divide of the solu- ) at the knots m x are given by [32] ( ) The set of septic B-spline functions Using Equation (5) and Equation (6) where the symbols , , ' '' ''' symbolize differentiation according to x, respectively.
Using (5) and (8) in the Equation (1) where ( ) If time parameters i δ and its time derivatives i δ  in Equation (9) are separated by the Crank-Nicolson form and finite difference approach, respectively: and usual finite difference approximation we acquired a repetition relationship between two time levels n and 1 n + relating two unknown parameters 1 ,   T  3  2  1  1  2  3 , , , , , , To acquire a solution of this system, we require six additional restrictions. These are obtained from the boundary conditions (2) and can be used to remove So, by taking account (18) 0  0  1  2  2  1 , , , , , ,

Stability Analysis
To The modulus of the (19) is found 1, hence the linearized algorithm is unconditionally stable.

Test Problems
In this section, we introduce some numerical examples including: motion of single soliton wave whose exact solution is known to test validity of our algorithm for solving GKdV equation. The initial boundary value problem (1)-(2) possesses following conservative quantities; which correspond to the mass, momentum and energy of the shallow water waves, respectively [33] [34]. To compare the numerical solution with the exact solution we use the following error norms:

The Motion of Single Solitary Wave
For this test problem, Equation (1) [30]. All parameters are given in all refarans. For these parameters, the single solitary wave has the amplitude 0.9 and 0.3, respectively. The three invariants 1 2 , I I and 3 I together with the 2 L , L ∞ error norms for the problem are documented and compared in Table   1 for times up to 1 t = . As seen from the table that 2 L and L ∞ error norms are found small enough and the conservation of the invariants can be seen to be almost constant.
Solitary wave profiles are demonstrated at 0, 0.1, 0.2, ,1 t =  in Figure 1 in which the soliton moves to the right at a nearly unchanged speed and amplitude as time increases, as expected.
Method Time For the second set, we choose the parameters 2 p = , 3  I . The calculated values are presented in Table 3. As can be seen in Table 3, the error norms

Conclusion
In this paper, a septic B-spline collocation method has been successfully applied to the GKdV equation to examine the motion of a single solitary wave whose analytical solution is known. To show how good and accurate the numerical solutions of the test problems, we have computed the error norms 2 L and L ∞ and conserved quantities 1 2 , I I and 3 I . According to the tables in the paper, one can have easily seen that our error norms are enough small and the invariants are well conserved. Stability analysis has been done and the linearized numerical scheme has been obtained unconditionally stable. Thus, we can say (a) (b) Figure 3. Motion of single solitary wave for (a)

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.