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Mathematical Biosciences and Engineering

2015, Volume 12, Issue 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
  • Received: 01 April 2014 Accepted: 29 June 2018 Published: 01 December 2014

  • MSC : Primary: 35K05, 35A35; Secondary: 35B45.

  • 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.

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

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  • 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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  • This article has been cited by:

    1. Joep H. M. Evers, Sander C. Hille, Adrian Muntean, Measure-Valued Mass Evolution Problems with Flux Boundary Conditions and Solution-Dependent Velocities, 2016, 48, 0036-1410, 1929, 10.1137/15M1031655
    2. Qiyao Peng, Sander C. Hille, Quality of approximating a mass-emitting object by a point source in a diffusion model, 2023, 151, 08981221, 491, 10.1016/j.camwa.2023.10.034
    3. Xiao Yang, Qiyao Peng, Sander C. Hille, Approximation of a compound-exchanging cell by a Dirac point, 2025, 59, 24058963, 73, 10.1016/j.ifacol.2025.03.014
    4. Qiyao Peng, Sander C. Hille, 2025, Chapter 25, 978-3-031-86168-0, 243, 10.1007/978-3-031-86169-7_25
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