  The functional distributions of particle trajectories have wide applications,
including the occupation time in half-space, the first passage time, and the
maximal displacement, etc. The models discussed in this paper are for
characterizing the distribution of the functionals of the paths of anomalous
diffusion described by time-space fractional diffusion equation. This paper
focuses on providing effective computation methods for the models. Two kinds of
time stepping schemes are proposed for the fractional substantial derivative.
The multiresolution Galerkin method with wavelet B-spline is used for space
approximation. Compared with the finite element or spectral polynomial bases,
the wavelet B-spline bases have the advantage of keeping the Toeplitz structure
of the stiffness matrix, and being easy to generate the matrix elements and to
perform preconditioning. The unconditional stability and convergence of the
provided schemes are theoretically proved and numerically verified. Finally, we
also discuss the efficient implementations and some extensions of the schemes,
such as the wavelet preconditioning and the non-uniform time discretization.
