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Transport through a network of capillaries from ultrametric diffusion equation with quadratic nonlinearity
UNAM, Mexico.
Linnaeus University, Faculty of Technology, Department of Mathematics. (Int Ctr Math Modeling Phys & Cognit Sci)ORCID iD: 0000-0002-9857-0938
2017 (English)In: Russian journal of mathematical physics, ISSN 1061-9208, E-ISSN 1555-6638, Vol. 24, no 4, p. 505-516Article in journal (Refereed) Published
Abstract [en]

This paper is about a novel mathematical framework to model transport (of, e.g., fluid or gas) through networks of capillaries. This framework takes into account the tree structure of the networks of capillaries. (Roughly speaking, we use the tree-like system of coordinates.) As is well known, tree-geometry can be topologically described as the geometry of an ultrametric space, i.e., a metric space in which the metric satisfies the strong triangle inequality: in each triangle, the third side is less than or equal to the maximum of two other sides. Thus transport (e.g., of oil or emulsion of oil and water in porous media, or blood and air in biological organisms) through networks of capillaries can be mathematically modelled as ultrametric diffusion. Such modelling was performed in a series of recently published papers of the authors. However, the process of transport through capillaries can be only approximately described by the linear diffusion, because the concentration of, e.g., oil droplets, in a capillary can essentially modify the dynamics. Therefore nonlinear dynamical equations provide a more adequate model of transport in a network of capillaries. We consider a nonlinear ultrametric diffusion equation with quadratic nonlinearity - to model transport in such a network. Here, as in the linear case, we apply the theory of ultrametric wavelets. The paper also contains a simple introduction to theory of ultrametric spaces and analysis on them.

Place, publisher, year, edition, pages
MAIK NAUKA/INTERPERIODICA/SPRINGER , 2017. Vol. 24, no 4, p. 505-516
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
URN: urn:nbn:se:lnu:diva-69754DOI: 10.1134/S1061920817040094ISI: 000417289900009OAI: oai:DiVA.org:lnu-69754DiVA, id: diva2:1173371
Available from: 2018-01-12 Created: 2018-01-12 Last updated: 2018-01-12Bibliographically approved

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Khrennikov, Andrei

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