Second-order accurate numerical approximations for the fractional percolation equations

Abstract

First, we examine a practical numerical method which based on the classical Crank-Nicholson (CN) method combined with Richardson extrapolation is used to solve a class of one-dimensional initial-boundary value fractional percolation equation (FPE) with variable coefficients on a finite domain. Secondly, we present ADI-CN method for the two-dimensional fractional percolation equation. Stability and convergence of these methods are proved. Using these methods, we can achieve second-order convergence in time and space. Finally, numerical examples are presented to verify the order of convergence.

Alternating-direction implicit finite difference methods for a new two-dimensional two-sided space-fractional diffusion equation

“…are similar to the definitions of the x direction. As we cannot easily get the explicit analytical solutions of the fractional equations, so many researchers resort to their numerical solutions [4][5][6][7][8][9][10].…”

A Fast Implicit Finite Difference Method for Fractional Advection-Dispersion Equations with Fractional Derivative Boundary Conditions

“…Fractional advectiondispersion equations are used to model many problems in physics, biology, and finance [3][4][5]. Fractional differential equations have attracted fantastic attention of many authors in recent years [6][7][8][9][10][11].…”