A novel finite volume method for the nonlinear two-sided space distributed-order diffusion equation with variable coefficients

Fractional differential equations have been proved to be powerful tools for modelling anomalous diffusion in many fields of science and engineering. However, when comes to the anomalous diffusion characterized by two or more scaling exponents in the mean squared displacement (MSD), or even by logarithmic time dependency of the MSD, distributed-order diffusion equations are shown to be more useful than the general single or multi-term fractional diffusion equations. In this paper, we construct a novel finite volume method for solving a nonlinear two-sided space distributed-order diffusion equation with variable coefficients. Firstly, we apply the modified Gaussian integral formula to approximate the distributed-order integral. Secondly, we propose the finite volume method based on a piecewise-linear polynomial to discretize the problem and establish the Crank–Nicolson scheme. Furthermore, we prove that the proposed method is stable and convergent with second order accuracy in both space and time. Finally, some numerical examples are given to show the efficiency of the proposed method.