A novel finite volume method for the Riesz space distributed-order diffusion equation

In recent years, considerable attention has been devoted to distributed-order differential equations mainly because they appear to be more effective for modelling complex processes which obey a mixture of power laws or flexible variations in space. In this paper, we propose a novel finite volume method (FVM) for a distributed-order space-fractional diffusion equation (FDE). Firstly, we use the mid-point quadrature rule to transform the space distributed-order diffusion equation into a multi-term fractional equation. Secondly, the transformed multi-term fractional equation is solved by discretising in space using the finite volume method and then in time using the Crank-Nicolson scheme. Thirdly, we prove that the Crank-Nicolson scheme with FVM is unconditionally stable and convergent with second order accuracy in both time and space. Finally, two numerical examples are presented to show the effectiveness of the numerical method. These methods and techniques can also be used to solve other types of fractional partial differential equations.