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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| Language: | English |
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AIMS Press
2014-11-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2015.12.357 |
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| author | Joep H.M. Evers Sander C. Hille Adrian Muntean |
| author_facet | Joep H.M. Evers Sander C. Hille Adrian Muntean |
| author_sort | Joep H.M. Evers |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-a4e3904a522a46b78df4bf0477168186 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2014-11-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-a4e3904a522a46b78df4bf04771681862025-01-24T02:31:45ZengAIMS PressMathematical Biosciences and Engineering1551-00182014-11-0112235737310.3934/mbe.2015.12.357Modelling with measures: Approximation of a mass-emitting object by a point sourceJoep H.M. Evers0Sander C. Hille1Adrian Muntean2Institute for Complex Molecular Systems & Centre for Analysis, Scientific computing and Applications, Eindhoven University of Technology, P.O. Box 513, 5600 MB EindhovenMathematical Institute, Leiden University, P.O. Box 9512, 2300 RA LeidenInstitute for Complex Molecular Systems & Centre for Analysis, Scientific computing and Applications, Eindhoven University of Technology, P.O. Box 513, 5600 MB EindhovenWe 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.https://www.aimspress.com/article/doi/10.3934/mbe.2015.12.357point sourcemodelling with measures.model reductiondiffusionquantitative flux estimatesboundary exchange |
| spellingShingle | 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 point source modelling with measures. model reduction diffusion quantitative flux estimates boundary exchange |
| title | Modelling with measures: Approximation of a mass-emitting object by a point source |
| title_full | Modelling with measures: Approximation of a mass-emitting object by a point source |
| title_fullStr | Modelling with measures: Approximation of a mass-emitting object by a point source |
| title_full_unstemmed | Modelling with measures: Approximation of a mass-emitting object by a point source |
| title_short | Modelling with measures: Approximation of a mass-emitting object by a point source |
| title_sort | modelling with measures approximation of a mass emitting object by a point source |
| topic | point source modelling with measures. model reduction diffusion quantitative flux estimates boundary exchange |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2015.12.357 |
| work_keys_str_mv | AT joephmevers modellingwithmeasuresapproximationofamassemittingobjectbyapointsource AT sanderchille modellingwithmeasuresapproximationofamassemittingobjectbyapointsource AT adrianmuntean modellingwithmeasuresapproximationofamassemittingobjectbyapointsource |