On Explicit and Numerical Solvability of Parabolic Initial-boundary Value Problems

A homogeneous boundary condition is constructed for the parabolic equation (∂t + I − Δ)u = f in an arbitrary cylindrical domain Ω×R (Ω ⊂ Rn being a bounded domain, I and Δ 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 ∂t + I −Δ, but also for an arbitrary parabolic differential operator ∂t +A, where A is an elliptic operator in Rn of an even order with constant coefficients. As an application, the usual Cauchy-Dirichlet boundary value problem for the homogeneous equation (∂t + I −Δ)u = 0 in Ω×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).