Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media

The inherent heterogeneities of many geophysical systems often give rise to fast and slow pathways to water and chemical movement. One approach to model solute transport through such media is by fractional diffusion equations with space–time dependent variable coefficients. Many physical processes appear to exhibit fractional-order behavior that may vary with time, or space, or space and time. The theory of pseudodifferential operators and equations has been used to deal with this situation. In this paper we use a fractional Darcy’s law with variable order Riemann–Liouville fractional derivatives, this leads to a new variable-order fractional diffusion equation with variable coefficients.In this paper we consider a new two-dimensional variable-order fractional percolation equation with variable coefficients. An alternating direct method for the two-dimensional variable-order fractional percolation equation is proposed. Stability and convergence of the implicit alternating direct method are discussed. Finally, some numerical results are given. The numerical results demonstrate the effectiveness of the methods. These techniques can be used to simulate three-dimensional variable-order fractional partial differential equations.