Least-Squares Spectral Method for the solution of a fractional advection–dispersion equation

Fractional derivatives provide a general approach for modeling transport phenomena occurring in diverse fields. This article describes a Least Squares Spectral Method for solving advection–dispersion equations using Caputo or Riemann–Liouville fractional derivatives.A Gauss–Lobatto–Jacobi quadrature is implemented to approximate the singularities in the integrands arising from the fractional derivative definition. Exponential convergence rate of the operator is verified when increasing the order of the approximation.Solutions are calculated for fractional-time and fractional-space differential equations. Comparisons with finite difference schemes are included. A significant reduction in storage space is achieved by lowering the resolution requirements in the time coordinate.