Research On The Radial Basis Quasi-interpolation And Its Application

The radial basis function quasi-interpolation has very extensive application in mathematics and in the production and life,therefore,many scholars pay attention to the study of it.The MQ radial basis quasi-interpolation method is ideal in all numerical schemes of scattered data,thus we introduce the MQ quasi-interpolation algorithm and its properties,propose variable parameter MQ quasi-interpolation scheme and apply it to the numerical solution of Burgers equations.The main work includes the following three parts:Firstly,we introduce the commonly used the MQ quasi-interpolation algorithm,and use the _DL method to approximate some functions.Numerical examples show that it has high accuracy.Then,we use the scheme to solve the Burgers equation,by comparing with the exact solution,thus we find that this algorithm can get a good approximation effort.In the calculation,we find that the choice of shape parameter is very critical during the solving process with the MQ quasi-interpolation algorithm,which directly affects the accuracy of the approximate solution.However,the selection of optimal parameter is very difficult.To avoid this problem,a variable parameter MQ quasi-interpolation scheme is proposed,where the shape parameter is not a constant,but variable,so that the problem of the optimal parameter will be changed into the problem of an interval,which greatly simplifies the scheme.Furthermore,three kinds of variablel MQ quasi-interpolation schemes are introduced and applied to approximate some functions.By comparing with the accurate solution,we find that the result of the algorithm is much better than the constant one.Among the three forms of variable parameter,the exponential variable scheme is the best one.Therefore,we apply the exponential variable parameter MQ quasi-interpolation algorithm to the numerical solution of the Burgers equation,and get a good result.Through the comparison of the two methods,we find that the result of our method is obviously better than that of the constant parameter algorithm.Finally,we introduce the radial basis function quasi-interpolation scheme based on the MQ-B spline,which is a high order algorithm.It can satisfy the demand of high precision algorithm in solving differential equations,therefore it deserves our detailed study.