On explicit and numerical solvability of parabolic initial-boundaryvalue problems

A homogeneous boundary condition is constructed for the parabolic equation (∂_t + I - Δ)u = f in an arbitrary cylindrical domain Ω × Rdbl; (Ω ⊂ ℝⁿ being a bounded domain, I and Δ being the identity operator and the Laplacian) which generates aninitial-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).