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On explicit and numerical solvability of parabolic initial-boundary value problems
Boundary Value Problems volume 2006, Article number: 75458 (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).
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Keywords
- Boundary Layer
- Integral Equation
- Bounded Domain
- Parabolic Equation
- Explicit Formula
