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On the Homogenization of the Renewal Equation with Heterogeneous External Constraints

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Abstract

We study the homogenization limit of the renewal equation with heterogeneous external constraints by means of the two-scale convergence theory. We prove that the homogenized limit satisfies an equation involving non-local terms, which are the consequence of the oscillations in the birth and death terms. We have moreover shown that the numerical approximation of the homogenized equation via the two-scale limit gives an alternative way for the numerical study of the solution of the limiting problem.

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Acknowledgements

This article has been written under the auspices of the Italian National Institute of Higher Mathematics (INdAM), GNFM group. The authors are grateful to the referee for his/her comments which helped us to improve the manuscript.

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Authors and Affiliations

  1. CERMICS, École des Ponts ParisTech, 6 et 8 avenue Blaise Pascal Cité Descartes - Champs sur Marne, Marne la Vallée Cedex 2, 77455, France

    Étienne Bernard

  2. Léonard de Vinci Pôle Universitaire, 92916, Paris La Défense, France

    Francesco Salvarani

  3. Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, Via Ferrata 1, 27100, Pavia, Italy

    Francesco Salvarani

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  1. Étienne Bernard
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Bernard, É., Salvarani, F. On the Homogenization of the Renewal Equation with Heterogeneous External Constraints. Acta Appl Math 187, 4 (2023). https://doi.org/10.1007/s10440-023-00594-2

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