New Approach for Solving of Extended KdV Equation

The Extended Korteweg de-Vries equation solved by using a finite element algorithm based on Bubnov-Galerkin’s method using quintic B-spline functions. Crank–Nicolson approximation in time has been used for time discretezation.The method can faithfully simulate the physics of the Extended Korteweg de-Vries equation, according to simulations


INTRODUCTION
Early in 1877, Joseph Valentin Boussinesq introduced the KdV equation.Then in 1895, Diederik Korteweg and Gustav de Vries have just been rediscovered and formed by (1) where is a field variable, and are positive constants, and and denote time and space differentiation, respectively.The KdV Eq.( 1) is a third-order one-dimensional nonlinear partial differential equation that is used in nonlinear dispersive wave analysis.The equation was constructed to describe the one-dimensional behaviour of solitary shallow water waves.
There are many forms for KdVequation like Rosenau-KdV [1], extended KdV [2], generalized KdV [3], Rosenau KdV-RLW [4], KdV-Burger [5], the coupled Schrödinger-KdV equation [6], ect..The KdV equation arises as an approximate equation governing weakly nonlinear long waves when terms up to the second order in the (small) wave amplitude are retained and when the weakly nonlinear and weakly dispersive terms are in balance.If effects of higher order are of interest then retention of terms up to the third order in the (small) wave amplitude leads to the extended KdV equation.[7] We chose Extended Kdv equation as one type of KdV equation, which in the form where is a field variable, , and β are positive constants, and and denote time and space differentiation, respectively The extended Korteweg-de Vries equation, which includes terms of third order in wave amplitude, is derived in two ways; the first is an extension of the derivation of Whitham (1974) of the Korteweg-de Vries equation from the water-wave equations and the second is from the Lagrangian for the water-wave equations derived by Luke (1967).Since a Lagrangian for the extended Korteweg-de Vries equation is required to apply modulation theory, the second method of derivation is useful as it leads directly to this Lagrangian.Deriving the modulation equations for the full extended Korteweg-de Vries equation.[2] In this work, we choose the Bubnov-Galerkin finite element method using quintic B-spline are the basis functions.Spline functions have highly desirable characteristics which have made them a powerful mathematical tool for numerical approximations, are employed to set up approximate functions.The quintic B-spline bases together with finite element methods are shown to provide very accurate solutions in solving some partial differential equations.For instance, quintic B-spline finite element method for the numerical solution of the Korteweg-de Vries equation is designed by Gardner.
If we want to talk about KdV applications, we need more than one paper, so we will name some of these aplications.Other features in the Jovian atmosphere, such as the Great Red Spot (GRS), as visible in cloud patterns, are perhaps the most daring use of KdV to date [8], both the cold and the hot plasma mathematically rigorously [9], Plasma physics should be mentioned in any list of KdV applications [10].The soliton features of KdV were first confirmed using ion-acoustic waves [11], [12].
The KdV equation can be solved in many numerical ways like finite difference method [13], finite element method [14], [15], and collocation method [16], ect..In this study, we'll examine one sort of KdV equation.the extended KdV equation [17], which would be solved by finite element method using Bubnov-Galerkin's with quintic b-spline.First, applying finite elemet method; second, studying Crank-Nicholson Approach; third, introduced the initial state; and then apply the algorithm in two experiments.

FINITE ELEMENT METHOD
The finite element method is a very successful application of classical methods such as: the Ritz method, the Galerkin method, the Least Squares method; for approximating the solutions of boundary value problems arising in the theory of elliptic partial differential equations.[18] giving specific applications of the finite element in the three major categories of boundary value problems, namely (1) equilibrium or steady state or time-independent problems, (2) eigenvalue problems, and (3) propagation or transient problems.
The extended KdV equation is solved numerically throughout that is a finite region with boundary conditions.Let a partition of be by the equally spaced knots and let quintic B-splines with knots at the points , are where are a collection of splines.serves as a foundation for functions that are sought within the finite region .The solution approximation to is defined as follows: ∑ where are time-dependent parameters that can be calculated from boundary and condition conditions. (4) The intervals are used to identify finite elements with nodes at and .Each element is thus covered by six splines , , which are represented as a local coordinate system given by where and .The expressions for all of these splines over through the element are as follows [19] (5) Outside the interval , the spline and its fifth derivatives equal zero.When we use Eq.( 4) to formulate equations based on the element parameters., these curves operate as "shape" functions for the element.The variation across the element is provided by ∑ The derivatives at the knots and the nodal value of are expressed in terms of the element parameters as shown below.(7) The dashes imply differentiation in regard to .When the Bubnov-Galerkin method is applied in Eq.(2) using weight functions , then the result is We'll now set up the appropriate element matrices.We have the contribution for the typical element , we obtain which the matrix form is formed by where the dot is the differentiation with respect to the time t, and The element matrices are given by where take only for this element .The matrices and are therefore and , , and are .Instead of , , , we utilise the associated matrix , , and in our algorithm.In our algorithm, we use This is dependent on the variables .The matrices of elements and are determined algebraically from Eq.( 13), which is given by Eq.( 12).The equation below is obtained by assembling the elements Eq. (11).
where the matrices are constructed from the element matrices respectively in the usual way and (17)

CRANK-NICHOLSON APPROACH
The Crank-Nicolson technique is a finite difference method for numerically solving the heat equation and other partial differential equations in numerical analysis [20].Time centre on , where is the time step, then utilise Crank-Nicholson method [21], with Substituting Eq.( 17) into Eq.(15), we obtainthe recurrence relationship (19) and then (20) where the time labels are represented by the superscripts and .The system ( 19) is made up of linear equations with variables.Four more conditions must be met in order to obtain a unique solution to the system.These are derived from the boundary conditions and may be utilised to exclude from the recurrence relationships (19), resulting in an 11 banded matrix equation.At each time step, an inner iteration is performed to verify that the nonlinear term converges.The following is the iteration algorithm [22]: First, is known.The first approximation which is derived from Eq.( 17), is calculated to using .The second approximation is found with , and the third with .We found that 10 rounds are usually enough to get a fair approximation for in this first stage.
To find a first approximation in general to , we use , A second approximation is then found from, and so on.Convergence is normally achieved after two or three iterations [23].
The time evolution of is determemined by a system of the decadiagonal [See Appendix A], and as a result, after the initial vector of the parameters is obtained, can be begun.

THE INITIAL STATE
We using recurrence relationships (19) to begin the time assessment of by determining the vector from the starting condition.From Eq.( 6), if we rewrite the global trial functions as follows denotes unknown parameters that must be determined.must meet the following requirements in order to determine the initial vector [24].At the knots , it agrees with the analytical initial condition; applying Eq.( 6), it leads to conditions.The solution of matrix equations is then used to get the start up vector , (22) ( 21) where After determining the initial vector as the solution of the undecadiagonal matrix Eq.( 19), the system is solved using a Thomas algorithm.
The numerical algorithm developed in Section 3 will be validated by studying test problems concerned with the migration and interaction of solitons.We use the L 2 and L ∞ error norms to measure the difference between the numerical and analytical solutions and hence to show how well the scheme predicts the position and amplitude of the solution as the simulation proceeds.The and error norms of the solution are defined by

Experiment 1
Subject to the boundary conditions [25] (26) The analytical solution of of the extended KdV Eq.( 2) is as follows The analytic solution for the initial condition: Figure 1 comparison between the numerical solution and the exact solution at the same time, which agree with the exact solution.In order to determine the accuracy of the current scheme, we used and complete the simulation up to .
Figure 2 Numerical solution at different time [t=0, 5, 10, 15, 20 seconds], which agree with the exact solution.In order to determine the accuracy of the current scheme, and shows that the wave moves when the time changed.The wave acts like a pulse.
The L2 and L¥ error norms are also recorded and the L2 norm is less than 3*10 25 , while the L∞ norm is less than 6_10 19   The analytical solution of of the extended Extended KdV Eq.( 2) is as follows The analytic solution for the initial condition: Figure3 comparison between the numerical solution at at time t=5 seconds, which agree with the exact solution.In order to determine the accuracy of the current scheme, we used α=β=0.9, in the fisrt term, and α =0.2, β =3 in the second term x 0 =18, and A 1 =1. .Figure 4 Numerical solution at time t=0, 5, 10, 15, 20 seconds, which agree with the exact solution.In order to determine the accuracy of the current scheme, we used α=β=0.9, in the first term, and and complete the simulation up to t=20.0 seconds, shown in the figure that the two solitons merge and dismerge when the time changed .

CONCLUSIONS
The Extended KdV equation is a nonlinear transient dispersive equation, any numerical system that replicates it must accurately reflect all of its properties.To deal with the fifth derivative of the extended KdV equation, Based on Galerkin and quintic B-spline shape and weight functions, we developed a one-dimensional B-spline finite element method.The Crank Nicholson technique is used to create time discretization.This results in a nonlinear equation system with 11 diagonal matrices.Matlab code was used to complete all calculations.To solve the equation in this work, we employed the finite element method with a quintic B-spline.We used several functions to study extended KdV equation and the wave form in each function in diffierent times.To solve the equation in this work, we employed the finite element method with a quintic B-spline.We used several functions to study extended KdV equation and the wave form in each function in diffierent times.From experiment 1 and 2 (one solitary wave and two solitons), the results obtained proved the method to be reliable, accurate and efficient through the calculated error norms.We believe that the technique given here could be applicable in other situations where derivative continuity is required.We can say that our numerical method can be reliably used to obtain the numerical solution of the Extended KdV equation and similar type non-linear equations.

Appendix A: An Undecadiagonal Matrix Algorithm
Consider the problem of solving simulation equations, which can be stated as follows: where is a matrix of known coefficients and and represent unknown and known equations, respectively, and is a vector of unknown and known equations.[26].
[ ] The 11-diagonal matrix is decomposed into two tridiagonal matrices using LU decomposition.When a result, defining the following parameters using forward recursion as needed, and As a result, the lower tringular matrix can be used to deduce the following vector .
Those calculations then use reverse recursion to angender the computation of the unknown vector from the higher triangular matrix.

Fig( 1 )Fig( 2 )
Fig(1): Comparison between numerical solution and the exact solution at time 5 seconds

Table 1 : Error norms for the single solitary wave of the extended KdV equation at
t = 1,2, .. ,5 .