Research On The Quasi-interpolation Scheme Of The Radial Basis Function And Its Application In Numerical Calculation Of Partial Differential Equations

Multi-Quadric(MQ)function is an important radial basis function(RBF).The MQ quasi-interpolant is convenient for solving equations because it does not need to solve any linear equations.RBF quasi-interpolation method is widely used in the field of scientific research,practical production and life,and plays an important role in solving partial differential equations.Therefore,many scholars have paid close attention to the research of RBF quasi-interpolant.More and more mathematicians also have made the research.In this paper,MQ quasi-interpolation scheme and its properties are introduced in detail.The MQ quasi-interpolation scheme is used to solve nonlinear partial differential equations numerically.In this paper,we introduce the significance of nonlinear partial differential equations,the generation and development of RBFs and their applications in numerical solution of nonlinear partial differential equations.Then the basic concept and theory of RBF and MQ quasi-interpolation format and its properties are introduced.Moreover,a new MQ quasi-interpolation scheme is constructed,and numerical experiments are given.The numerical results show that the algorithm is simple,easy to implement and has high accuracy.Then,the construction and properties of the high order MQ quasi-interpolation scheme are introduced in detail,and the approximate function of the high order MQ quasi-interpolation scheme is used to compare it with the traditional MQ quasi-interpolation schem-e.It is found that the high order MQ quasi-interpolation scheme has a high computational accuracy.Thus,the scheme is applied to the numerical solution of the KdV-Burgers(KdVB)equation,and an algorithm based on the higher-order MQ quasi-interpolation scheme is proposed to solve the KdVB equation.Compared with the traditional MQ quasi-interpolation scheme,this method is more accurate and suitable for numerical approximation of KdVB equation.Finally,the research background of Degasperis-Procesi equation is introduced and the DP equation is approximated by using high accuracy MQ quasi-interpolation scheme.Firstly,the equation is reduced to its equivalent form by introducing auxiliary variables,then the time derivative term is discretized by the TVD Runge-Kuttta method,and the spatial derivatives are approximated by the high accuracy MQ quasi-interpolation scheme.Finally,a numerical algorithm for solving the DP equation is presented.Numerical results show that the proposed algorithm can capture the shock waves in the DP equation effectively and has a high accuracy.