MQ Quasi-Interpolation Method And Its Application In B-Equation

The shallow water theories widely describe the wave equation(system of equation) in weak nonlinear condition. Different completely integrable nonlinear partial differential equations arise at various levels of approximation. Such equations possess soliton solutions. B-family equation is a class important equation of shallow water theory. The D-P and C-H equation are two important equation of b-equation. The C-H equation models the unidirectional propagation of waves at free surface of shallow water. The D-P equation models the propagation of nonlinear dispersive waves. The D-P and C-H equation have a strong similarity formation. They are completely integrable and have Hamiltonian structure and infinity conservative laws. But they are truly different.Radial basis functions method which is a meshless method is an important method of solving partial differential equation. Multi-quadric(MQ) function is an important kernel function of radial basis functions method. It’s convenient for solving PDEs by using MQ quasi-interpolation method which is not need to solve linear system of equations. In this paper, the MQ quasi-interpolation method was used to solve b-equation.First, the background and the properties of b-equation were introduced. The basic theory of the MQ quasi-interpolation method and Gegenbauer reconstruction theory also were introduced.Next the D-P equation was solved. We first reduced the differential order by introducing an auxiliary variable. Coupled with FDM, the MQ quasi-interpolation method was used to approximate the spatial variables. And then the numerical method of D-P equation was obtained by using TVD Runge-Kutta method to integrate the time variable. The examples showed the efficiency of proposed method.And then the C-H equation was solved. We first reduced the differential order by introducing an auxiliary variable. The MQ quasi-interpolation method and TVD Runge-Kutta method were used to solve C-H equation. A numerical method of C-H equation was obtained. The examples showed the efficiency of proposed method.Last, we gave an idea of error analysis. The summary and expectation were given. Although the proposed method was easy and efficient, some drawbacks also exist.