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

Alexander Kozhevnikov,Olga Lepsky

doi:10.1155/bvp/2006/75458

Alexander Kozhevnikov, Olga Lepsky

Open Access

https://doi.org/10.1155/bvp/2006/75458

Copy DOI

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).

Highlights

It is well known that the initial-boundary value problem with the Dirichlet/Neumann boundary condition for the parabolic equation (∂t + I − Δ)u = f can be solved using the Green function
In the case of an arbitrary domain Ω, there is no explicit formula for the solution
We show that the equation (∂t + I − Δ)u±(x, t) = f±(x, t)((x, t) ∈ Ω± × R+) under the homogeneous boundary condition ∂nu± − Ψ∓u± = 0 on ∂Ω × R+ has a unique solution belonging to an anisotropic Sobolev space

Summary

Introduction

It is well known that the initial-boundary value problem with the Dirichlet/Neumann boundary condition for the parabolic equation (∂t + I − Δ)u = f can be solved using the Green function. The Green function can be found explicitly just for a few very specific domains Ω such as balls and half-spaces. How can one define boundary conditions for an arbitrary domain Ω in order to obtain an explicitly solvable initial boundary value problem? An answer is obtained not just for the operator ∂t + I − Δ, and for a rather general parabolic differential operator of the form ∂t + A, where A is an elliptic differential operator of even order with constant coefficients. Similar questions for elliptic boundary value problems have been investigated in [4].

On explicit and numerical solvability

Theorem on explicit solvability