KANSA METHOD BASED ON THE HAUSDORFF FRACTAL DISTANCE FOR HAUSDORFF DERIVATIVE POISSON EQUATIONS

Abstract

This study proposes the radial basis function (RBF) based on the Hausdorff fractal distance and then applies it to develop the Kansa method for the solution of the Hausdorff derivative Poisson equations. The Kansa method is a meshless global technique promising for high-dimensional irregular domain problems. It is, however, noted that the shape parameter of the RBFs can have a significant influence on the accuracy and robustness of the numerical solution. Based on the leave-one-out cross-validation algorithm proposed by Rippa, this study presents a new technique to choose the optimal shape parameter of the RBFs with the Hausdorff fractal distance. Numerical experiments show that the Kansa method based on the Hausdorff fractal distance is highly accurate and computationally efficient for the Hausdorff derivative Poisson equations.