  The differential form of the Maxwell's equations was first derived based on
an assumption that the media are stationary, which is the foundation for
describing the electro-magnetic coupling behavior of a system. For a general
case in which the medium has a time-dependent volume, shape and boundary and
may move at an arbitrary velocity field v(r,t) and along a general trajectory,
we derived the Maxwell's equations for a mechano-driven slow-moving media
system directly starting from the integral forms of four physics laws, which
should be accurate enough for describing the coupling among
mechano-electro-magnetic interactions of a general system in practice although
it may not be Lorentz covarance. Our key point is directly from the four
physics laws by describing all of the fields, the space and the time in the
frame where the observation is done. The equations should be applicable to not
only moving charged solid and soft media that has acceleration, but also
charged fluid/liquid media, e.g., fluid electrodynamics. This is a step toward
the electrodynamics in non-inertia frame of references. General strategies for
solving the Maxwell's equations for mechano-driven slowing moving medium are
presented using the perturbation theory both in time and frequency spaces.
Finally, approaches for the electrodynamics of moving media are compared, and
related discussions are given about a few interesting questions.
