Mathematical Biosciences and Engineering, 2015, 12(2): 357-373. doi: 10.3934/mbe.2015.12.357.

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Modelling with measures: Approximation of a mass-emitting object by a point source

1. Institute for Complex Molecular Systems & Centre for Analysis, Scientific computing and Applications, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven
2. Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden

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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Keywords Point source; modelling with measures.; model reduction; diffusion; quantitative flux estimates; boundary exchange

Citation: Joep H.M. Evers, Sander C. Hille, Adrian Muntean. Modelling with measures: Approximation of a mass-emitting object by a point source. Mathematical Biosciences and Engineering, 2015, 12(2): 357-373. doi: 10.3934/mbe.2015.12.357

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Copyright Info: 2015, Joep H.M. Evers, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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