A Good Numerical Method for the Solution of Generalized Regularized Long Wave Equation

The generalized regularized long wave equation is very important that can be applied in the field of physics, science and technology. Some authors have put forward many different numerical method, but the precision is not enough high. In this paper, we will illustrate the high-precision numerical method to solve the generalized regularized long wave equation. Three numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by our method indicate new algorithm has the following advantages: small computational work, fast convergence speed and high precision.


Introduction
The nonlinear wave is one of the most important scientific research areas.The generalized regularized long wave (GRLW) equation can be used to describe nonlinear dispersive waves and the phenomena of weak nonlinearity and dispersion waves, including nonlinear transverse waves in shallow water, ion-acoustic and magneto hydrodynamic waves in plasma, phonon packets in nonlinear crystals and a large number of physical phenomena.The GRLW equation can be written as: The regularized long wave (RLW) equation for =1, 0, 0, and 0. is the dispersion effect term that makes the wave form spread, the solitons appear as a result of the balance between this weak nonlinearity and dispersion, t is the time and x is the space coordinate.
It is well known that solving the GRLW equation have many methods.Some of the previous works on the GRLW equation include an implicit second-order accurate and stable energy preserving finite difference method based on the use of central difference equations for the time and space derivatives (Lu-Ming, 2005), a method of lines based on the discretization of the spatial derivatives by means of Fourier pseudo-spectral approximations (Durn, 2002), a Fourier spectral method for the initial value problem of the GRLWequation (Ben-Yu, 1988), and a linearized implicit pseudo-spectral method (Djidjeli, 2003).Moreover numerical techniques such as finite difference methods (Ramos, J. I., 2006), a spectral method (Hammada, 2016), finite element methods based on the least square principle (Fu-Zheng, 2015), finite element methods based on Galerkin and collocation principles (Noureddine, 2013), the Petrov-Galerkin method (Thoudam, 2012), the radial basis function collocation method (Siraj-ul-Islam, 2009), the Sinc-collocation method (Reza, 2010), the collocation method with quintic B-splines (Battal, 2016) and the cubic B-spline finite element method (Khalifa, 2008) have been devised for finding numerical solutions of special kinds of GRLW equations.As we know, the barycentric interpolation collocation method already applied to the integral calculus, linear equation and ordinary differential equation very well, but few people to applied to nonlinear partial differential equation.In the following, we extending the barycentric interpolation collocation method to solve the GRLW equation.
In this study, a new algorithm for solving the GRLW equation in the meshless space is proposed.The advantages of the approach show in the following facts.The paper is organized as follows.Section 2 is recommend the barycentric interpolation collocation method.Through numerical examples to illustrate the advantages in section 3. The last section is a brief conclusion.

The Partial Differential Matrix of Barycentric Interpolation
In this section , we illustrate the partial differential matrix of barycentric interpolation.We consider a regular rectangle region and also divided into N different nodes in the interval [0, T ] : In the region Ω, the nodes generate tensor type nodes, that is: The functional value u(x, t) at nodes (x i , t j ) is defined as: The barycentric interpolation of u(x, t) at nodes (x i , t j ) can be written as: Here, ξ i (x),η j (t) are respectively barycentric interpolation basis function of u(x, t) on the interval [a, b] nodes and the interval [0, T ] nodes.
Use formula (3), the p + q order partial derivative of function u(x, t) at nodes can be expressed as: and the functional values of partial derivative at nodes (x i , t j ) are defined as: T be respectively defineded as the column vectors of nodes at x,t axis, and the matrixes X,T composed of tensor type node coordinates are respectively defined as: Let the matrix X,T by columns stretch into a N × M dimensional column vector x,t, that is: The relationship between components of vector x,t and vector x 0 ,t 0 exist the following : So,formula (8) can be written in following matrix form: In the above formula,D (p,q) = C (p) ⊗ D (q) is the Kronecker product of matrix C (p) and D (q) ,and it is also called p + q order partial differential matrix at nodes p) and D (q) are respectively called p order partial differential matrix and q order partial differential matrix.Denote: where I M and I N are respectively M order unit matrix and N order unit matrix.
The partial differential matrix has the following relations: Use the barycentric interpolation collocation method to solve the partial differential equation, there are several types of nodes to choose.Such as: equidistant nodes, Chebyshev nodes, Legendre nodes and so on.The Chebyshev nodes calculation formula is: This paper that we use the Chebyshev nodes, which can improve the accuracy effectively.

Direct Linearized Iterative Method
Next, we put the nonlinear equations transformed into linear equations and introduce a direct linearized iterative method for solving nonlinear equations.
The nonlinear partial differential equation Du(x, t) = f (x, t) can divided into two parts, which are linear term and nonlinear term: in the formula, L is linear differential operator and N is nonlinear differential operator.Take initial hypothesis function u 0 (x, t) which has been given into formula (11), we obtain: Therefore, the formula( 12)is transformed into a linear differential equation.We can get a new function u 1 (x, t) after solving formula (12), and through the iteration we can obtain: A control precision ε is given in the course of the actual calculation, if | u n (x)−u n−1 (x) | ε, then the iteration stops, the approximate solution of the nonlinear problem is obtained.This process is referred to as the direct linearized iteration method.

The Barycentric Interpolation Collocation Formula of the Generalized Regularized Long Wave Equation
Think about function (1) where α = 1, β 0, δ 0 and µ = 0.So the function can be written as the nonlinear term is βu ∂u ∂x .Take initial hypothesis function u 0 (x, t) which has been given into formula (15),we obtain: Therefore,the formula( 16)is transformed into a linear differential equation.We can get a new function u 1 (x, t) after solving formula ( 16), and we can get a linear iterative scheme A control precision ε is given in the course of the actual calculation, if | u n (x)−u n−1 (x) | ε, then the iteration stops, the approximate solution of the nonlinear problem is obtained.This process is referred to as the direct linearized iteration method.

The Convergence Analysis of the Generalized Regularized Long Wave Equation
Let E be Lu + NU = f , let E n be T n U = Lu + NU n = f .{E n } is a sequence of the linear partial differential equation, then On account of knowledge of the topology , to the continuous deformation E, then Assumption the genuine solution is U * * , it can approximate representation linear equation T U = f .Assumption the solution is U * , then U * * and U * are arbitrarily close.And the continuous linear operator T n converge to the continuous linear operator T .
Assumption the nonlinear operator N is differentiable of Frechet.

Numerical Experiment
In this section, three numerical examples are studied to demonstrate the advantage of the present method.
Example 1 [14−16] Considering following nonlinear RLW equation with initial condition and boundary conditions.
The RLW equation has three conservation quantities corresponding to mass, momentum and energy: The RLW equation has an analytical solution of the form is where v = 1 + βc is the wave velocity, k = 1 2 √ βc δv and c is a constant.This solution represents the motion of a single solitary wave with amplitude 3c and initially centered at x 0 = 0 and α = β = δ = 1.The initial condition is taken as The numerical experiments are done in the region −40 ≤ x ≤ 60, T = 20 with parameters c = 0.1, M = 150, N = 120 that are presented in Table 1.Fig 1 .With the increase of the points, accuracy is higher and highter that are presented in Table 2. Example 2 [17] Considering following nonlinear RLW equation with initial condition and boundary conditions.
The RLW equation has an analytical solution of the form is The initial condition is taken as u(x, 0) = sech 2 ( x 4 ) where We give the numerical solution in the region −50 ≤ x ≤ 50, a = 0.5, x 0 = 10, M = 200, N = 100, T = 1 that are presented in Table 4. Example 3 [6,18] Considering following nonlinear RLW equation with initial condition and boundary conditions.
where the numerical experiments are done in the region 0 ≤ x ≤ 100, T = 10, q = 1, µ = 1.Single solitary wave: the RLW equation has an analytical solution of the form is We consider the initial condition of given by where amplitude (A) and the inverse width (B) of the RLW solution are respectively given by A Interaction of two solitary waves: the RLW equation has an analytical solution of the form is Here,we take the initial condition given by the form where amplitude A j and the inverse width B j of the RLW solution are respectively given byA j = [

Discussion
In order to solve the generalized regularized long wave equation, a new algorithm that the barycentric interpolation collocation method is constructed skillfully.Through the calculation the present method to solve the GRLW equation better.Choose the points in Examples of present method less than the other method, but the accuracy still well.Our method makes it easy to solve the GRLW equation and we can extend the present method to other the partial differential equation.The numerical results demonstrate that the new method is quite accurate and efficient for nonlinear equation.All computations are performed by the Matlab software package.
and δ are nonnegative constants, β and δ are the behavior of an undular bore, the nonlinear term Fig 2. The numerical experiments are done in the region −80 ≤ x ≤ 120, T = 20 with parameters c = 0.03, M = 150, N = 120 that are presented in Table 3.Fig 3.

Table 5 .
Comparison of L 2 and L ∞ for Example 3 (Single solitary wave)