  This paper presents the syntax and semantics of a novel type of hybrid
automaton (HA) with partial differential equation (PDE) dynamic, partial
differential hybrid automata (PDHA). In PDHA, we add a spatial domain $X$ and
harness a mathematic conception, partition, to help us formally define the
spatial relations. While classically the dynamics of HA are described by
ordinary differential equations (ODEs) and differential inclusions, PDHA is
capable of describing the behavior of cyber-physical systems (CPS) with
continuous dynamics that cannot be modelled using the canonical hybrid systems'
framework. For the purposes of analyzing PDHA, we propose another model called
the discrete space partial differential hybrid automata (DSPDHA) which handles
discrete spatial domains using finite difference methods (FDM) and this simple
and intuitive approach reduces the PDHA into HA with ODE systems. We conclude
with two illustrative examples in order to exhibit the nature of PDHA and
DSPDHA.
