Radial Basis Function Methods for Fractional Derivative Applications

Abstract

In recent decades, the theoretical researches and experimental results show that fractional derivative model can be a powerful tool to describe the contaminant transport through complex porous media and the dynamic behaviors of real viscoelastic materials. Consequently, growing attention has been attracted to numerical solution of fractional derivative model. Radial basis function (RBF) meshless technique is one of the most popular and powerful numerical methods, which are mathematically simple, and avoid troublesome mesh generation for high-dimensional problems involving irregular or moving boundary. Recently, RBF-based meshless methods, such as the Boundary Particle Method and the Method of Approximate Particular Solutions, have been successfully applied to fractional derivative problems. The Boundary Particle Method is one of truly boundary-only RBF collocation schemes, which employs both the semi-analytical basis functions to approximate the FDE solutions. Inspired by the boundary collocation RBF techniques, the Method of Approximate Particular Solutions is one of the domain-type RBF collocation schemes with easy-to-use merit, which employs the particular solution RBFs for the solution of FDEs. This study will make a numerical investigation on the abovementioned RBF meshless methods to fractional derivative problems. The convergence rate, numerical accuracy and stability of these schemes will be examined through several benchmark examples.