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

Roman Chapko, B. Tomas Johansson*

*Corresponding author for this work

Research output: Contribution to journalArticle

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.

Original languageEnglish
Number of pages15
JournalJournal of Engineering Mathematics
VolumeEarly online
Early online date4 Apr 2016
DOIs
Publication statusE-pub ahead of print - 4 Apr 2016

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Heat Equation
Initial-boundary-value Problem
Dirichlet Problem
Boundary value problems
Integral equations
Integral Equations
Numerical Solution
Harmonic functions
Spherical Functions
Surjection
Boundary integral equations
Spherical Harmonics
Boundary Integral Equations
Unit Sphere
Parabolic Problems
Projection Method
Harmonic Functions
Elliptic Problems
Boundary Layer
Numerical methods

Bibliographical note

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Keywords

  • boundary integral equations of the first kind
  • discrete projection method
  • exterior 3-dimensional domain
  • heat equation
  • Laguerre transformation

Cite this

Chapko, Roman ; Johansson, B. Tomas. / Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations. In: Journal of Engineering Mathematics. 2016 ; Vol. Early online.
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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. / Chapko, Roman; Johansson, B. Tomas.

In: Journal of Engineering Mathematics, Vol. Early online, 04.04.2016.

Research output: Contribution to journalArticle

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