Numerical Solution of Space-time Variable Fractional Order Advection-Dispersion Equation using Jacobi Spectral Collocation Method

This article is aimed at studying computational solution of variable order fractional advection-dispersion equation for one-dimensional and two-dimensional spaces utilizing spectral collocation method. In the considered model, the time derivative is Coimbra fractional derivative and space derivative is a Riemann-Liouville derivative. Jacobi polynomials are applied as basic functions in approximation of the solution. The presented approach is an application of the shifted Jacobi-Gauss collocation (SJ-G-C) and the shifted Jacobi-Gauss-Radau collocation (SJ-GR-C) methods using for discretizing along space and time, respectively. Using the related collocation points, the problem would be changed to an algebraic equation system, which can be tackled applying a computational technique. At the end, several examples in one and two dimensional cases have been solved by introduced approach, it would be shown that the proposed numerical algorithm has considerably higher accuracy in contrast to the existing computational schemes including finite difference approach.