- Research Article
- Open Access
On explicit and numerical solvability of parabolic initial-boundary value problems
- Alexander Kozhevnikov1Email author and
- Olga Lepsky2
https://doi.org/10.1155/BVP/2006/75458
© A. Kozhevnikov and O. Lepsky 2006
Received: 26 June 2005
Accepted: 22 March 2006
Published: 31 May 2006
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).
Keywords
Authors’ Affiliations
References
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