Boundary Value Problems 
Volume 2006 (2006), Article ID 75458, 12 pages
doi:10.1155/BVP/2006/75458

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

Alexander Kozhevnikov1 and Olga Lepsky2

 1Department of Mathematics, University of Haifa, Haifa 31905, Israel
2Department of Mathematics, Natural Sciences Programs, Lesley Collage, Lesley University, 29 Everett Street, Cambridge 39762, MA, USA
Received 26 June 2005; Revised 15 January 2006; Accepted 22 March 2006

Abstract

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).