Abstract
A homogeneous boundary condition is constructed for the parabolic equation in an arbitrary cylindrical domain
(
being a bounded domain,
and
being the identity operator and the Laplacian) which generates an initial-boundary value problem with an explicit formula of the solution
. In the paper, the result is obtained not just for the operator
, but also for an arbitrary parabolic differential operator
, where
is an elliptic operator in
of an even order with constant coefficients. As an application, the usual Cauchy-Dirichlet boundary value problem for the homogeneous equation
in
is reduced to an integral equation in a thin lateral boundary layer. An approximate solution to the integral equation generates a rather simple numerical algorithm called boundary layer element method which solves the 3D Cauchy-Dirichlet problem (with three spatial variables).