  The aim of this paper is to establish the framework of the enclosure method
for some class of inverse problems whose governing equations are given by
parabolic equations with discontinuous coefficients.
  The framework is given by considering a concrete inverse initial boundary
value problem for a parabolic equation with discontinuous coefficients. The
problem is to extract information about the location and shape of unknown
inclusions embedded in a known isotropic heat conductive body from a set of the
input heat flux across the boundary of the body and output temperature on the
same boundary. In the framework the original inverse problem is reduced to an
inverse problem whose governing equation has a large parameter. A list of
requirements which enables one to apply the enclosure method to the reduced
inverse problem is given.
  Two new results which can be considered as the application of the framework
are given. In the first result the background conductive body is assumed to be
homogeneous and a family of explicit complex exponential solutions are
employed. Second an application of the framework to inclusions in an isotropic
inhomogeneous heat conductive body is given. The main problem is the
construction of the special solution of the governing equation with a large
parameter for the background inhomogeneous body required by the framework. It
is shown that, introducing another parameter which is called the virtual
slowness and making it sufficiently large, one can construct the required
solution which yields an extraction formula of the convex hull of unknown
inclusions in a known isotropic inhomogeneous conductive body.
