  This paper establishes explicit solutions for fractional diffusion problems
on bounded domains. It also gives stochastic solutions, in terms of Markov
processes time-changed by an inverse stable subordinator whose index equals the
order of the fractional time derivative. Some applications are given, to
demonstrate how to specify a well-posed Dirichlet problem for space-time
fractional diffusions in one or several variables. This solves an open problem
in numerical analysis.
