We discuss Cahn's time cone method which models the growth process in phase transformation kinetics. Cahn's model is an integral equation, preventing us from smooth arguments for both forward and inverse problems. To this end, we derive a class of multiple hyperbolic systems from the original model. Especially, the integral terms in odd dimensional cases are completely eliminated. Thanks to this reduction, dramatically fast forward solver is developed in practical spatial dimensions. Next, on basis of the hyperbolic systems, we study several inverse problems of determining the nucleation rate or the growth rate to establish the uniqueness and stability