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Transport through diffusive and nondiffusive regions, embedded objects, and clear layers. (English) Zbl 1020.45004
The diffusion approximation of transport equations with discontinuities of interfaces between different media was devepoped by G. Bal and L. Ryzhik [SIAM J. Appl. Math. 60, 1887-1912 (2000; Zbl 0976.45008)]. The present paper is devoted to the study of the diffusion approximation of transport equations in diffusion domains with nondiffusion inclusions, where classical diffusion is not valid.
Two classes of inclusions are considered; the first one concerns large nodiffusive objects and the second one extends regions of relatively small volume such as clear layers. Generalized diffusion equations with suitable nonlocal interface conditions at the boundary of inclusions are derived. The asymptotic convergence of the transport solution to a solution of diffusion equation for two such inclusions is established.
A numerical treatment, given in the case of straight clear layers, shows that the solution to the diffusion equation is an accurate approximation of the solution to the considered transport equation. Applications to medical imaging for monitoring certain properties of human tissues are indicated.

MSC:
45K05 Integro-partial differential equations
82C70 Transport processes in time-dependent statistical mechanics
92C55 Biomedical imaging and signal processing
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