@Article{cmes.2023.046002,
AUTHOR = {Chein-Shan Liu, Chung-Lun Kuo, Chih-Wen Chang},
TITLE = {Optimal Shape Factor and Fictitious Radius in the MQ-RBF: Solving Ill-Posed Laplacian Problems},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {139},
YEAR = {2024},
NUMBER = {3},
PAGES = {3189--3208},
URL = {http://www.techscience.com/CMES/v139n3/55631},
ISSN = {1526-1506},
ABSTRACT = {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>.},
DOI = {10.32604/cmes.2023.046002}
}