Study On Approximation Capability Of Multiquadric Radial Basis Functions

At present,the functions are common tool that we use to represent or describe things in nature and their rules.However,with the development of science and technology,the simple functions cannot satisfy the changing rules in the real word.The multiquadric(MQ)function is a powerful tool to solve this problem.This function is a kind of radial basis function(RBF),the essence of this function is to apply the univariate function to represent the multivariate function and it's applications are very extensive.In view of this,the research worked in this paper is organized as follows:Firstly,we construct a numerical approximation method for solutions to nonlinear dy-namic systems based on MQ quasi-interpolation functions.We analyze the advantages and disadvantages of this new method and the existing numerical methods theoretically,mean-while,the corresponding numerical example is given,with the error estimations and mean square errors to verify the conclusion is correct.Secondly,four kinds of MQ quasi-interpolation operators were constructed by studying the ideal of the existing MQ quasi-interpolation operators.Various properties of these new operators are discussed,and furthermore,the ideal upper and lower bound estimations on the approximation accuracy are present from the theoretically.We demonstrate two numerical examples with different degrees of complexity that show good agreement with theory.Thirdly,a new format of variable shape parameter is proposed.The properties and error estimations of its corresponding of new MQ quasi-interpolation function with a variable shape parameter are studied.Two numerical examples show that the selection of variable shape parameter is more reasonable.Finally,the main achievements are summarized and some expectations are bring for-ward.