An implicit numerical method of a new time distributed-order and two-sided space-fractional advection-dispersion equation

Distributed-order differential equations have recently been investigated for complex dynamical systems, which have been used to describe some important physical phenomena. In this paper, a new time distributed-order and two-sided space-fractional advection-dispersion equation is considered. Firstly, we transform the time distributed-order fractional equation into a multi-term time-space fractional partial differential equation by applying numerical integration. Then an implicit numerical method is constructed to solve the multi-term fractional equation. The uniqueness, stability and convergence of the implicit numerical method are proved. Some numerical results are presented to demonstrate the effectiveness of the method. The method and techniques can be extended to other time distributed-order and space-fractional partial differential equations