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

Boundary Value Problems volume 2006, Article number: 75458 (2006) Cite this article

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 and Affiliations

  1. Department of Mathematics, University of Haifa, Haifa, 31905, Israel

    Alexander Kozhevnikov

  2. Department of Mathematics, Natural Sciences Programs, Lesley Collage, Lesley University, 29 Everett Street, Cambridge, MA, 39762, USA

    Olga Lepsky

Authors
  1. Alexander Kozhevnikov
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  2. Olga Lepsky
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Correspondence to Alexander Kozhevnikov.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Kozhevnikov, A., Lepsky, O. On explicit and numerical solvability of parabolic initial-boundary value problems. Bound Value Probl 2006, 75458 (2006). https://doi.org/10.1155/BVP/2006/75458

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  • DOI: https://doi.org/10.1155/BVP/2006/75458

Keywords

  • Boundary Layer
  • Integral Equation
  • Bounded Domain
  • Parabolic Equation
  • Explicit Formula