### 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 language | English |
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Number of pages | 15 |

Journal | Journal of Engineering Mathematics |

Volume | Early online |

Early online date | 4 Apr 2016 |

DOIs | |

Publication status | E-pub ahead of print - 4 Apr 2016 |

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### Bibliographical note

-### Keywords

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

### Cite this

*Journal of Engineering Mathematics*,

*Early online*. https://doi.org/10.1007/s10665-016-9858-6

}

*Journal of Engineering Mathematics*, vol. Early online. https://doi.org/10.1007/s10665-016-9858-6

**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.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations

AU - Chapko, Roman

AU - Johansson, B. Tomas

N1 - -

PY - 2016/4/4

Y1 - 2016/4/4

N2 - 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.

AB - 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.

KW - boundary integral equations of the first kind

KW - discrete projection method

KW - exterior 3-dimensional domain

KW - heat equation

KW - Laguerre transformation

UR - http://www.scopus.com/inward/record.url?scp=84962163342&partnerID=8YFLogxK

UR - https://link.springer.com/article/10.1007%2Fs10665-016-9858-6

U2 - 10.1007/s10665-016-9858-6

DO - 10.1007/s10665-016-9858-6

M3 - Article

AN - SCOPUS:84962163342

VL - Early online

ER -