A meshfree method for the fractional advection-diffusion equation

“Non-local” phenomena are common to problems involving strong heterogeneity, fracticality, or statistical correlations. A variety of temporal and/or spatial fractional partial differential equations have been used in the last two decades to describe different problems such as turbulent flow, contaminant transport in ground water, solute transport in porous media, and viscoelasticity in polymer materials. The study presented herein is focused on the numerical solution of spatial fractional advection-diffusion equations (FADEs) via the reproducing kernel particle method (RKPM), providing a framework for the numerical discretization of spacial FADEs. However, our investigation found that an alternative formula of the Caputo fractional derivative should be used when adopting Gauss quadrature to integrate equations with fractional derivatives. Several one-dimensional examples were devised to demonstrate the effectiveness and accuracy of the RKPM and the alternative formula.