The Construction Of High-precision MQ Quasi-interpolation Operator And Its Application In Partial Differential Equations

In this article,based on the quasi-interpolation operator LR given by Wang Ziqiang,we construct a brand new multi-quadric(MQ)quasi-interpolation operator Lw1 with high approximation accuracy.The first part is the introduction.It introduces the background of radial basis functions and the current research status of MQ quasi-interpolation.At the same time,it also explains the application of MQ quasi-interpolation in partial differential equations and outlines the main work of this article.The second part is the preparatory knowledge part.It introduces the relevant knowl-edge of radial basis functions and MQ functions and their corresponding interpolations.At the same time,two improved quasi-interpolation operators are introduced.One is the quasi-interpolation operator LR proposed by Wang Ziqiang,the other is the MQ quasi-interpolation operator LR constructed by Ling.The third and fourth parts are the main work of this article.In the third part,we propose an improved MQ quasi-interpolation operator Lw1f(x).The construction method of the new operator is introduced,and the approximation accuracy of the new operator is given.At the same time,we prove that the operator has convexity preservation and linear reproducibility.In addition,we use numerical examples to compare the new operator Lw1 with the commonly used quasi-interpolation operators,and the experimental results show that Lw1 has higher approximation accuracy.In the fourth part,we take the Burgers equation,the Degasperis-Procesi(DP)equation and the Newell-whitehead equation as examples,applying the new MQ quasi-interpolation operator proposed in this paper to the solution of partial differential equations.Numerical experiments show that the newly constructed quasi-interpolation operator Lw1 has a good simulation effect.The fifth part is a summary of the full text.This part explains the main work of this article and proposes future work prospects.