Boundary Value Problems

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

Boundary Value Problems20062006:75458

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).

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Authors’ Affiliations

(1)
Department of Mathematics, University of Haifa
(2)
Department of Mathematics, Natural Sciences Programs, Lesley Collage, Lesley University

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Copyright

© A. Kozhevnikov and O. Lepsky 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.