Modelling with measures: Approximation of a mass-emitting object by a point source
We consider a linear diffusion equation on $\Omega:=\mathbb{R}^2\setminus\overline{\Omega_\mathcal{o}}$, where $\Omega_\mathcal{o}$ is a bounded domain. The time-dependent flux on the boundary $\Gamma:=∂\Omega_\mathcal{o}$ is prescribed. The aim of the paper is to approximate the dynamics by the sol...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2014-11-01
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| Series: | Mathematical Biosciences and Engineering |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2015.12.357 |
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| Summary: | We consider a linear diffusion equation on $\Omega:=\mathbb{R}^2\setminus\overline{\Omega_\mathcal{o}}$, where $\Omega_\mathcal{o}$ is a bounded domain. The time-dependent flux on the boundary $\Gamma:=∂\Omega_\mathcal{o}$ is prescribed. The aim of the paper is to approximate the dynamics by the solution of the diffusion equation on the whole of $\mathbb{R}^2$ with a measure-valued point source in the origin and provide estimates for the quality of approximation. For all time $t$, we derive an $L^2([0,t];L^2(\Gamma))$-bound on the difference in flux on the boundary. Moreover, we derive for all $t>0$ an $L^2(\Omega)$-bound and an $L^2([0,t];H^1(\Omega))$-bound for the difference of the solutions to the two models. |
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| ISSN: | 1551-0018 |