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Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations

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Abstract

A numerical method for the Dirichlet initial boundary value problem for the heat equation in the exterior and unbounded region of a smooth closed simply connected 3-dimensional domain is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and an integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the parabolic problem to a sequence of stationary elliptic problems which are solved by a boundary layer approach giving a sequence of boundary integral equations of the first kind to solve. Under the assumption that the boundary surface of the solution domain has a one-to-one mapping onto the unit sphere, these integral equations are transformed and rewritten over this sphere. The numerical discretisation and solution are obtained by a discrete projection method involving spherical harmonic functions. Numerical results are included.

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Acknowledgments

This research was carried out while the first author was visiting the University of Linz (Austria) on a Stipendium Lemberg OeAD-GmbH. The hospitality and the support are gratefully acknowledged.

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Authors and Affiliations

  1. Faculty of Applied Mathematics and Informatics, Ivan Franko National University of Lviv, Lviv, 79000, Ukraine

    Roman Chapko

  2. School of Mathematics, Aston University, Birmingham, B4 7ET, UK

    B. Tomas Johansson

Authors
  1. Roman Chapko
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  2. B. Tomas Johansson
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Correspondence to B. Tomas Johansson.

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Chapko, R., Johansson, B.T. Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations. J Eng Math 103, 23–37 (2017). https://doi.org/10.1007/s10665-016-9858-6

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  • DOI: https://doi.org/10.1007/s10665-016-9858-6

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