TY  - EJOU
AU  - Liu, Chein-Shan 
AU  - Kuo, Chung-Lun 
AU  - Chang, Chih-Wen 

TI  - Optimal Shape Factor and Fictitious Radius in the MQ-RBF: Solving Ill-Posed Laplacian Problems
T2  - Computer Modeling in Engineering \& Sciences

PY  - 2024
VL  - 139
IS  - 3
SN  - 1526-1506

AB  - To solve the Laplacian problems, we adopt a meshless method with the multiquadric radial basis function (MQ-RBF) 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 the MQ-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<sup>−11</sup>.
KW  - Laplace equation; nonharmonic boundary value problem; Ill-posed problem; maximal projection; optimal shape factor and fictitious radius; optimal MQ-RBF; optimal polynomial method

DO  - 10.32604/cmes.2023.046002