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Approximations of kinetic equations of swarm formation: Convergence and exact solutions

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  • Lachowicz, Mirosław
  • Leszczyński, Henryk
  • Topolski, Krzysztof A.

Abstract

In the present paper we study Euler–type approximations along characteristics for a class of kinetic equations that describe swarm formations in the case when the interactions rate is variable. The proposed numerical schemes preserve essential properties of the kinetic equations and in particular preserve the probabilistic measure and are able to approximate the solution almost to the appearance of blow-ups. The blow–ups are referred here to the self–organization swarm behavior. Moreover we define a class of exact solutions — traveling wave–type equilibrium solutions that we called TWES.

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  • Lachowicz, Mirosław & Leszczyński, Henryk & Topolski, Krzysztof A., 2022. "Approximations of kinetic equations of swarm formation: Convergence and exact solutions," Applied Mathematics and Computation, Elsevier, vol. 417(C).
    • Handle: RePEc:eee:apmaco:v:417:y:2022:i:c:s0096300321008602
    DOI: 10.1016/j.amc.2021.126778
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    References listed on IDEAS

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    1. Lachowicz, Mirosław & Leszczyński, Henryk & Topolski, Krzysztof A., 2019. "Self-organization with small range interactions: Equilibria and creation of bipolarity," Applied Mathematics and Computation, Elsevier, vol. 343(C), pages 156-166.
    2. Pareschi, Lorenzo & Toscani, Giuseppe, 2013. "Interacting Multiagent Systems: Kinetic equations and Monte Carlo methods," OUP Catalogue, Oxford University Press, number 9780199655465, Decembrie.
    3. Mirosław Lachowicz & Henryk Leszczyński, 2020. "Modeling Asymmetric Interactions in Economy," Mathematics, MDPI, vol. 8(4), pages 1-14, April.
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    1. Lachowicz, Mirosław & Matusik, Milena & Topolski, Krzysztof A., 2024. "Population of entities with three individual states and asymmetric interactions," Applied Mathematics and Computation, Elsevier, vol. 464(C).

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