Sturm-Liouville problem and numerical method of fractional diffusion equation on fractals

Classical diffusion theory is widely applied in natural science and has made a great achievement. However, the phenomenon of anomalous diffusion in discontinuous media (fractal, porous, etc.) shows that classical diffusion theory is no longer suitable. The differential equations with fractional order have recently been proved to be powerful tools for describing anomalous diffusion. Nevertheless, the analysis methods and numerical methods for fractional differential equations are still in the stage of exploration. In the paper, we consider the Sturm-Liouville problem and the numerical method of a fractional sub-diffusion equation with Dirichlet condition, respectively. We have given the series solution of equation and proved the stability and the convergence of the implicit numerical scheme. It is found that the numerical results are in satisfactory agreement with the analytical solution. Through the robustness analysis, it is also found that the diffusion processes on fractals are more sensitive to the spectral dimension than to the anomalous diffusion exponent.