Abstract: Traditional second-order diffusion PDEs model Fickian diffusion processes, in which the particles follow Brownian motion. However, many diffusion processes were found to exhibit anomalous diffusion behavior, in which the probability density functions of the underlying particle motions are characterized by an algebraically decaying tail and so cannot be modeled properly by second-order diffusion PDEs. Fractional PDEs provide a powerful tool for modeling these problems, as the probability density functions of anomalous diffusion processes satisfy these equations.

Fractional PDEs present new difficulties that were not encountered in the context of integer-order PDEs. Computationally, the numerical methods for space-fractional PDEs generate dense matrices. Direct solvers were traditionally used, which require $O(N^3)$ computations per time step and $O(N^2)$ memory, where $N$ is the number of unknowns.

Mathematical difficulties include the loss of coercivity of the Galerkin formulation for variable-coefficient problems, non-existence of the weak solution to inhomogeneous Dirichlet boundary-boundary value problems, and low regularity (the solution to homogeneous Dirichlet boundary-value problem of a one-dimensional fractional PDE with constant coefficient and source term is not in the Sobolev space $H^1$).