Modified MQ Quasi-interpolation And Its Application In Numerical Solutions Of Partial Differential Equations

Due to the nature of the radial basis function itself,more and more scholars have become interested in radial basis functions and apply them to computational geometry,partial differential equations,and so on.The radial basis function interpolation is one of its many applications,but with the increase of the number of interpolation nodes,it becomes very difficult to solve the coefficient matrix corresponding to the interpolation of radial basis function,and sometimes the coefficient matrix is ill-conditioned,which often makes the calculation becomes very unstable.In this case,the radial basis function quasi-interpolation without solving the linear equations appears and some types of quasi-interpolation have polynomial reproduction property,and shape-preserving properties,such as the monotonicity preserving property,convexity(concavity)preserving property.Among them,the most representative is the Multiquadric(MQ)quasi-interpolation method.In this paper,an improved MQ quasi-interpolation method is proposed,and this method is used to solve the numerical solutions of KdV equations.This paper is divided into five chapters.The first chapter is introduction,introducing the back ground of radial basis function,summarizing the research trends of MQ quasi-interpolation,and the research on the application of MQ quasi-interpolation for solving partial differential differential equations.The second chapter two is the basics section,summarizing the relevant knowledge of radial basic function and radial function interpolation and mainly focusing on four types of classical MQ quasi-interpolation operators and their properties.In addition,two improved quasi-interpolation operators are introduced in this paper,one is the nested interpolation operator LRf(x) proposed by Ling,and the other is the MQ quasi-interpolation operator f*(x) constructed by Chen Rong-hua.In chapter three,a new improved MQ quasi-interpolation operator is proposed.The new method improved MQ quasi-interpolation operator Lf*Rf(x) is based on the MQ quasi-interpolation operator f*(x) constructed by Chen Rong-hua.Numerical experiments show that the new method has good approximation property.In chapter four,the improved MQ quasi-interpolation operator is used to solve the KdV equation.the improved quasi-interpolation operator LfR*f(x) is used to solve the KdV equation.The numerical solution shows that this improved method has good approximation accuracy.The fifth chapter is the summary and outlook.The main contents of this paper are summarized,as well as the work to be done in the next stage.