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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

    Related Papers:

  • 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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    [1] 2nd edition, Academic Press, 2003.
    [2] Trends in Cell Biology, 13 (2003), 282-285.
    [3] Development, 140 (2013), 2253-2268.
    [4] J. Funct. Anal., 147 (1997), 237-258.
    [5] Journal of Experimental Botany, 63 (2012), 4213-4218.
    [6] Mathematical Models and Methods in Applied Sciences, 23 (2013), 1795-1859.
    [7] Nonlinear Analysis: Real World Applications, 11 (2010), 4524-4532.
    [8] Mem. Am. Math. Soc., 166 (2003), Viii+114 pp.
    [9] Math. Z., 257 (2007), 193-224.
    [10] Springer-Verlag, New York, 2000.
    [11] 2nd edition, Wiley, New York, 1999.
    [12] 3rd edition, Pearson Education, 2008.
    [13] Journal of Statistical Mechanics: Theory and Experiment, 2013 (2013), p04025.
    [14] Journal of Evolution Equations, 8 (2008), 423-448.
    [15] Integr. Equ. Oper. Theory, 63 (2009), 351-371.
    [16] Neuron, 68 (2010), 610-638.
    [17] Second edition, John Wiley and Sons, New York-London-Sydney, 1975.
    [18] Science, 255 (1982), 1523-1531.
    [19] Journal of Experimental Botany, 59 (2008), 45-53.
    [20] Appl. Math. Optim., 6 (1980), 287-333.
    [21] Springer Verlag, 1972.
    [22] Ann. Rev. Neurosci., 20 (1997), 125-156.
    [23] Trends in Plant Science, 12 (2007), 384-390.
    [24] Eur. J. Appl. Math, 25 (2014), 749-781.
    [25] Kluwer Academic Publishers, Dordrecht, 1991.
    [26] Springer-Verlang, New York, 1983.
    [27] Journal of Experimental Botany, 64 (2013), 1-9.
    [28] Math. Zeit., 27 (1928), 218-244.
    [29] Eleventh Copper Mountain Conference on Multigrid Methods, 2003.
    [30] Numer. Math., 122 (2012), 709-723.
    [31] Trudy Mat. Fust. Steklov, 83 (1965), 3-163 (Russian). Engl. Transl.: Proc. Steklov Inst. Math., 83 (1965), 1-184.
    [32] Princeton University Press, Princeton, New Jersey, 1970.
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  • © 2015 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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