A center Box method for radially symmetric solution of fractional subdiffusion equation

In this paper, a center Box difference method is considered for the radially symmetric solution of fractional subdiffusion equation. By method of order reduction, the derivative boundary condition is transformed into Dirichlet boundary condition and thus the geometrical singularity is successfully removed from the original problem. As a matter of course, a natural discretization scheme is obtained. To investigate the stability and convergence of the method, we define a new norm with a weight rd-1. Thus, the usual Sobolev inequality is not suitable to the new norm. Therefore, we prove three new Sobolev-like embedding inequalities which can also be applied to the other problems in polar coordinates. Then, the scheme is proved to be unconditionally stable and convergent in maximum norm with the help of the new Sobolev-like embedding inequalities. Some illustrative examples are provided to demonstrate the theoretical results. By some comparisons, it can be seen that the natural discretization scheme is accurate and effective in physical simulations. And it can be used to both long time and short time computation.