A finite difference method for elliptic equations with the variable-order fractional derivative

An efficient finite difference method for the multi-dimensional differential equation with variable-order Riemann-Liouville derivative is studied. Firstly, we construct an efficient discrete approximation for the multi-dimensional variable-order Riemann-Liouville derivative by the generating functions approximation theory. The convergence of the discrete operator in the Barron space is analyzed. Based on it, we present the finite difference method for the elliptic equation with variable-order Riemann-Liouville derivative. The stability and convergence of the method are proven by the maximum principle. Moreover, a fast solver is presented in the computation based on the fast Fourier transform and the multigrid algorithm in order to reduce the storage and speed up the BiCGSTAB method, respectively. We extend this method to time-dependent problems and several numerical examples show that the proposed schemes and the fast solver are efficient.