We discuss the time cone method by Cahn for nucleation and growth mechanisms.
The model by Cahn is an integral equation and so the computations for
the forward problem is
rather heavy.  First we show the reduction of the integral model
equation to hyperbolic systems.
Thanks to the hyperbolic system, we can create a dramatically fast
forward solver even in the
three dimensional case.  Next on the basis of the hyperbolic systems,
we argue several inverse problems of determining the nucleation rate
or the growth rate of e.g., ferrite grain to establish the uniqueness
and stability.
This is a joint work with Mr. Yikan Liu (The University of Tokyo).