Solving mixed classical and fractional partial differential equations using short–memory principle and approximate inverses

The efficient numerical solution of the large linear systems of fractional differential equations is considered here. The key tool used is the short–memory principle. The latter ensures the decay of the entries of the inverse of the discretized operator, whose inverses are approximated here by a sequence of sparse matrices. On this ground, we propose to solve the underlying linear systems by these approximations or by iterative solvers using sequence of preconditioners based on the above mentioned inverses.