Improved Moving Least Square Approximation Meshless Collocation Method For Two Types Of Nonlinear Evolution Equations

With the development of modern science and technology and computers,numerical calculations have been used in various fields of science and technology and social life.The nonlinear Gilson-Pickering equation and the nonlinear improved Boussinesq equation are two types of nonlinear evolution equations that describe the propagation process of solitary wavelets.The analytical solutions of these two types of equations are not easy to obtain,making them very difficult in mathematical modeling and theoretical analysis.Researchers hope to find efficient numerical calculation methods for solving these two types of equations.The meshless method is one of the hot spots in current computational science research,and the collocation method is a more mature meshless method.At present,there is no report about using the improved moving least squares approximation meshless collocation method to study the nonlinear Gilson-Pickering equation and the nonlinear improved Boussinesq equation.This paper proposes an improved moving least squares approximation meshless collocation method for solving nonlinear Gilson-Pickering equations and nonlinear improved Boussinesq equations.The main contents are arranged as follows.The first part introduces the background and numerical calculation research progress of nonlinear Gilson-Pickering equation and nonlinear improved Boussinesq equation,as well as the research progress of moving least squares approximation and related meshless methods.The second part first discusses the stability of the improved moving least squares approximation,and then gives the calculation formula of the improved moving least squares approximation meshless collocation method for a class of second-order linear partial differential equations,and finally analyzes the meshless collocation method.The theoretical error of the lattice collocation method is given,and corresponding examples are given to verify the theoretical results.The third part presents the improved moving least squares approximation meshless collocation method for the nonlinear Gilson-Pickering equation.Firstly,the difference format is used to discretize the time derivative,then the improved moving least squares is used to approximate the discrete spatial derivative,and finally the collocation technique is used to obtain Discrete nonlinear algebraic equations.Numerical examples show that this method can effectively solve nonlinear Gilson-Pickering equations with third-order partial derivatives and time-dependent variables,and the method has higher accuracy than the finite element method.The fourth part presents the improved moving least squares approximation meshless collocation method for the nonlinear improved Boussinesq equation.By approximating the time derivative,a system of nonlinear discrete algebraic equations is established and solved by an iterative algorithm.The convergence of the iterative algorithm is discussed.This method introduces the displacement basis function and the scale basis function to ensure the convergence and stability of the numerical results.Numerical examples show that the method has fast convergence speed and high calculation accuracy.Finally,the last part summarizes the above research work and prospects for the follow-up research.