Space-fractional advection–dispersion equations by the Kansa method

The paper makes the first attempt at applying the Kansa method, a radial basis function meshless collocation method, to the space-fractional advection–dispersion equations, which have recently been observed to accurately describe solute transport in a variety of field and lab experiments characterized by occasional large jumps with fewer parameters than the classical models of integer-order derivative. However, because of non-local property of integro-differential operator of space-fractional derivative, numerical solution of these novel models is very challenging and little has been reported in literature. It is stressed that local approximation techniques such as the finite element and finite difference methods lose their sparse discretization matrix due to this non-local property. Thus, the global methods appear to have certain advantages in numerical simulation of these non-local models because of their high accuracy and smaller size resultant matrix equation. Compared with the finite difference method, popular in the solution of fractional equations, the Kansa method is a recent meshless global technique and is promising for high-dimensional irregular domain problems. In this study, the resultant matrix of the Kansa method is accurately calculated by the Gauss–Jacobi quadrature rule. Numerical results show that the Kansa method is highly accurate and computationally efficient for space-fractional advection–dispersion problems.