  For describing the probability distribution of the positions and times of
particles performing anomalous motion, fractional PDEs are derived from the
continuous time random walk models with waiting time distribution having
divergent first order moment and/or jump length distribution which has
divergent second order moment. It can be noted that the fractional PDEs are
essentially dealing with the multiscale issues. Generally the regularity of the
solutions for fractional PDEs is weak at the areas close to boundary and
initial time. This paper focuses on developing the applications of wavelet
bases to numerically solving fractional PDEs and digging out the potential
benefits of wavelet methods comparing with other numerical methods, especially
in the aspects of realizing preconditioning, adaptivity, and keeping the
Toeplitz structure. More specifically, the contributions of this paper are as
follows: 1. the techniques of efficiently generating stiffness matrix with
computational cost $\mathcal{O}(2^J)$ are provided for first, second, and any
order bases; 2. theoretically and numerically discuss the effective
preconditioner for time-independent equation and multigrid method for
time-dependent equation, respectively; 3. the wavelet adaptivity is detailedly
discussed and numerically applied to solving the time-dependent (independent)
equations. In fact, having reliable, simple, and local regularity indicators is
the striking benefit of the wavelet in adaptively solving fractional PDEs (it
seems hard to give a local posteriori error estimate for the adaptive finite
element method because of the global property of the operator).
