Optimal Shape Factor and Fictitious Radius in the MQ-RBF:Solving Ill-Posed Laplacian Problems  

作　　者：Chein-Shan Liu Chung-Lun Kuo Chih-Wen Chang 

机构地区：[1]Center of Excellence for Ocean Engineering,Taiwan Ocean University,Keelung,202301,Taiwan,China [2]Department of Mechanical Engineering,National United University,Miaoli,36063,Taiwan,China

出　　处：《Computer Modeling in Engineering & Sciences》2024年第6期3189-3208,共20页工程与科学中的计算机建模（英文）

基　　金：supported by the the National Science and Technology Council(Grant Number:NSTC 112-2221-E239-022).

摘　　要：To solve the Laplacian problems,we adopt a meshless method with the multiquadric radial basis function(MQRBF)as a basis whose center is distributed inside a circle with a fictitious radius.A maximal projection technique is developed to identify the optimal shape factor and fictitious radius by minimizing a merit function.A sample function is interpolated by theMQ-RBF to provide a trial coefficient vector to compute the merit function.We can quickly determine the optimal values of the parameters within a preferred rage using the golden section search algorithm.The novel method provides the optimal values of parameters and,hence,an optimal MQ-RBF;the performance of the method is validated in numerical examples.Moreover,nonharmonic problems are transformed to the Poisson equation endowed with a homogeneous boundary condition;this can overcome the problem of these problems being ill-posed.The optimal MQ-RBF is extremely accurate.We further propose a novel optimal polynomial method to solve the nonharmonic problems,which achieves high precision up to an order of 10^(−11).

关 键 词：Laplace equation nonharmonic boundary value problem Ill-posed problem maximal projection optimal shape factor and fictitious radius optimal MQ-RBF optimal polynomial method 

分 类 号：O17[理学—数学] TP3[理学—基础数学]