In this paper we consider three methods of approximation for the nonlinear water wave equation. In particular we are interested of KdV equation as a stationary water wave. The first is the method of approximation with a polynomial, the second method is the finite–volumemethod and the third method is Laplace decomposition method (LDM). A comparison between the methods is mentioned in this article.We treat the considered methods comparing the obtained solutions with the exact ones. We give in particular the numerical results compared with the analytical results. We show that the used methods are effective and convenient for solving the water wave equations. We can propose and sure that the method of approximation with a polynomial gives accurate results.

We consider the initial-boundary value problem associated with the nonlinear dispersive and dissipative wave which was formulated by Korteweg, de Vries and Burgers in the form (see [2])

where μ,θ,δ are constant coefficients.

In [1] we see the change form of long waves in a rectangular zone and on a new type of stationary waves.

In particular, we deal with the nonlinear wave equation, named Korteweg de Vries-Bezuci form as follow:

∂U/∂x-δ ∂u/∂x+μu ∂u/∂x-θ (∂^2 u)/(∂x^2 )=0 (1.2)

where, the coefficients μ and θ in Eq. (1.3) represent the damping and the dispersion coefficients, respectively and δ=(p_z (x,y))/S is the wave breaking coefficient.

Numerical methods, numerical results and comparisons are considered in this article. It is well known that many physical phenomena can be described by the Korteweg-de Vries–Bezuci equation. Eq. (1.2) can serve as a nonlinear wave model of a fluid in an elastic tube

Johnson

Numerical methods for solutions of the KdV equation are known since long time. Recall here

In this section, a brief outline of LDM is explained

Lu(x,t)+Ru(x,t)+N(u(x,t))=h(x,t) (2.1)

with the following initial condition:

u(x,0)=f(x) ( 2.2)

where 𝐿 is the first-order differential operator, 𝐿=𝜕/𝜕𝑡, 𝑁(𝑢) presents the nonlinear term, ℎ(𝑥,𝑡) is the source term and. R is the remainder of the linear operator. Thus we get

L(u) = g(t)- R(u)- N(u) (2.3)

The methodology consists of applying Laplace transform first on both sides of (2.1)

L[Lu(x,t)]+L[Ru(x,t)]+L[N(u(x,t))]=L[h(x,t)] (2.4)

We discuss the solution of the KdV-Bezuci equation using LDM. Eq. (1.2) can be written in an operator form:

Lu(x,t)=〖δ u〗_xx-μ uu_x-θ〖 u〗_xx (2.5)

where the differential operator 𝐿 is 𝐿=𝜕/𝜕𝑡 . We will define the solution u(𝑥,t) by the series in the form:

u(x,t)=∑_(n=0)^∞▒ u_n (x,t) (2.6)

and the nonlinear operator 𝑁(𝑢) represented by an infinite series of the so-called Adomian's polynomials:

N(u)=∑_(n=0)^∞▒〖P_n (x) (2.7)

where u_n (x,t), n≥0 are the components of 𝑢(𝑥,𝑡) that will be elegantly determined and P_n are called Legendre's polynomials, which are a system of complete orthogonal system with respect to the weight function v(x) = 1 over the interval −1 ≤ x ≤ 1. That is a polynomial of degree n, such that

∫_(-1)^1▒〖P_m (x) P_n (x) 〗 dx=0 if n≠m

and defined by Rodrigues’s formula

P_n=1/(2^n n!) [d^n/(dλ^n ) (∑_(i=0)^∞▒(n¦k)^2 λ^i u_i )]_(λ=0), n≥0 (2.8)

that represent the nonlinear term _{x} and given by

P_0=u_0x u_0 P_1=u_0x u_(1 )+u_1x u_(0 )

P_2=u_0x u_(2 )+u_1x u_(1 )+u_2x u_(0 )

P_3=u_0x u_(3 )+u_1x u_(2 )+u_2x (2.9)

The inverse operator L^{-}^{1 }is an integral operator defined by

L^(-1)=∫_0^t▒(.) dt (2.10)

U =f_0+L^(-1) (g(t)- R(u)- N(u)) (2.11)

where Is the solution of homogenous equation L(u)=0

The first few components of u_n (x,t) follows as

u_0 (x,t)=f(x)

u_1 (x,t)=L^(-1) (- P_0+θu_0x -δ u_0xx)

u_2 (x,t)=L^(-1) (- P_1+θ u_1x -δ u_1xx)

u_3 (x,t)=L^(-1) (- P_2+θu_2x -δ u_2xx)

(2.12)

The scheme in (2.12) can easily determine the components un (x,t) ,n ≥ 0.So it is possible to calculate more components in the decomposition series to enhance the approximation. Using initial condition and plotting the solution of KdV-Bezuci equation at t = 0.01, 1, 2, 2.5 by using the fundamentals of modified Laplace decomposition method we get,

The MOL (The method of lines) has the merits of both the finite difference method and analytical method, it does not yield spurious modes nor have the problem of relative convergence. This method

The method of lines is regarded as a special finite difference method but more effective with respect to accuracy and computational time than the regular finite difference method. The MOL is generally recognized as a comprehensive and powerful approach to the numerical solution of time-dependent partial differential equations (PDEs).

Let’ s we study our problem inthe rectangle zone a≤x≤≤b ,0≤t≤T.

Consider KdV-Bezuci equation (1.2) with the initial condition

u(x,0)=(25c/μ-100μ^2 δ+(θ^2 )/δ)+12μ^2 δ〖sin〗^2 (μx)-(12 )/5 μθ tan(x) (3.1.1)

and the boundary conditions

u(a,t)=0,98 , u(x,0)=0,02 (3.1.2)

The exact solution is given by

u(x,t)=(25c/μ-100μ^2 δ+(θ^2 )/δ)+12μ^2 δ〖sin〗^2 (μx-ct)-(12 )/5 μθ tan(μx-ct) (3.1.3)

The solution domain is the rectangle a≤x≤≤b ,0≤t≤T.

The solution using a second order finite difference scheme for _{x} and _{xx} is denoted by Method Of Line II.

The derivative u_x in KdV-Bezuci equation (1.2) is computed by finite differences scheme in second order approximations

u_x=(u_(i+1)-u_(i-1))/2h +O(h^2).

The derivative u_xx in KdV-Bezuci equation (1) is computed by finite difference scheme in second order approximations

u_xx=(u_(i-1)-2u_i+ u_(i+1) )/h^2 +O(h^2).

Applying the above finite difference schemes to Eq. (1.2) yields a system of ordinary differential equations for the unknown u_i as functions in t as follows:

(du_i (t) )/dt =f(u_i) , i=1,……,N-1

Thus, we have the system of differential equations of one independent variable t. This system can be easily solved by using fourth order Runge–Kutta scheme.

U^(n+1)=U^n+(∆t(K_1 +K_2 +2K_3 +K_4) )/6 , K_1=F(U^n ),K_2=F(U^n+(∆t )/2) K_1 ,K_3=F(U^n+(∆t )/2) K_2 ,K_4=F(U^n+∆tK_3 )

The results obtained using the method of lines have been compared with the exact solution as a plots of the solution and the absolute error (AE) profiles of the KdV-Bezuci equation where θ and δ are constants at

^{-3},

We obtain the Method of line solutions of KdV–Bezuci equation with high accuracy. The obtained results demonstrate the reliability of the MOL and its wider applicability to nonlinear evolution equations.

In this article, we conclude that the nonlinear KdV– Bezuci equation gives solution, which represents an important application in physical problems. The computations associated here were performed using Maple. We have presented the following tables to describe the absolute errors between the exact and numerical solutions. The tables illustrate the errors for both methods, the Laplace decomposition method and the method of lines compared with the exact solution, at different values of

A review of the decomposition method

A comparison between the numerical MOL and the decomposition methods with those obtained by exact solution are given for 10^{−}^{3}. From the tables, we can observe that the decomposition method is accurate as compared with MOL at small period of time but with increasing the time, the MOL is more accurate when compared with LDM. It is observed that if we increase the number of terms in algorithm, the size of calculation is maximized with no increase in accuracy so the reduction of terms facilities the construction of Legendre polynomials for nonlinear operators and gives the same accuracy. LDM can provide the solution with minimal number of iterations.

From the comparative study between LDM and the MOL we may conclude that the MOL is more accurate than LDM.

From the above tables we can infer that LDM have better convergence at small