Title: Well-posedness of the Space-Time
Fractional Diffusion Problems and Its Numerical Solutions


The universality of anomalous diffusion phenomenon in physical and
biological experiments has led to an intensive investigation on the
fractional differential equations in recent years. 
The physical background includes modeling turbulent flow, chaotic
dynamics charge transport in amorphous semiconductors,
NMR diffusometry in disordered materials, and dynamics of
a bead in polymer network, and so on.

In this talk, we consider initial boundary value problems of the
space-time fractional diffusion equation (STFDE) and its numerical
solutions.  STFDE is of interest not only in its own right, but also
in that it constitutes the principal part in solving many other FPDEs.
Two definitions, i.e. Riemann--Liouville definition and Caputo one, of
the fractional derivative are considered in parallel.  In both cases,
we establish the well-posedness of the weak solution.  Moveover, based
on the proposed weak formulation, we construct an efficient spectral
method for numerical approximations of the weak solution. The main
results are as follows: First, a theoretical framework for the
variational solution of the space-time fractional diffusion equation
is developed. We find suitable functional spaces and norms in which
the space-time fractional diffusion problem can be formulated into an
elliptic weak problem, and the existence and uniqueness of the weak
solution is then proved by using existing theory for elliptic
problems. Secondly, we show that in the case of Riemann--Liouville
definition, the well-posedness of the space-time fractional diffusion
equation does not require any initial conditions. This contrasts with
the case of Caputo definition, in which the initial condition has to
be integrated into the weak formulation in order to establish the
well-posedness. Finally, thanks to the weak formulation, we are able
to construct some numerical methods for efficiently solving the
space-time fractional diffusion problem.