On explicit and numerical solvability of parabolic initial-boundary value problems

A homogeneous boundary condition is constructed for the parabolic equation (partial derivative(t) + I-Delta) u = f in an arbitrary cylindrical domain Omega x R (Omega subset of R-n being a bounded domain, I and. being the identity operator and the Laplacian) which generates an initial-boundary value problem with an explicit formula of the solution u. In the paper, the result is obtained not just for the operator partial derivative(t) + I - Delta, but also for an arbitrary parabolic differential operator partial derivative(t) + A, where A is an elliptic operator in R-n of an even order with constant coefficients. As an application, the usual Cauchy-Dirichlet boundary value problem for the homogeneous equation (partial derivative(t) + I - Delta) u = 0 in Omega x R 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).