Spectrum of a stationary uniform-speed transport equation with reflection-type boundary conditions
type: Item
instance of: scholarly article
title: Spectrum of a stationary uniform-speed transport equation with reflection-type boundary conditions
zbMATH Open document ID: 0706.45011
DOI: 10.1007/BF01077714
author: R. S. Makin
publication date: 1990-01-01T00:00:00Z
review text: The following stationary single-velocity transport equation with boundary conditions of reflection type is considered: \[ (1)\quad v^{-1}u_ t(x,\Omega,t)=-\Omega \cdot \nabla u(x,\Omega,t)-\sigma (x,\Omega,t)+(\gamma /4\pi)\int_{\Gamma}d\Omega 'u(x,\Omega ',t), \] \[ (2)\quad u(s,\Omega,t)=\alpha u(s,\Omega -2(n(s)\cdot \Omega)n(s),t),\quad s\in \partial V,\quad \Omega \cdot n(s)\geq 0. \] Here: \(x\in V\), V is a bounded, convex, open domain with sufficiently smooth boundary \(\partial V\); \(t>0\); \(\Gamma\) is the surface of the unity sphere of \(R^ 3\); \(\Omega\) is the unity vector defined on \(\Gamma\) which gives the sense of movement of the particles with constant velocity \(v>0\); n(s) is the unity vector of the interior normal at \(\partial V\); \(\sigma\), \(\gamma\) are positive constants; \(\alpha\) is a parameter, \(\alpha\in [0,1].\) Some spectral properties for the stationary problem (1), (2) are studied.
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 45C05
Mathematics Subject Classification ID: 82C70
zbMATH DE Number: 4159385
zbMATH Keywords: spectrum
zbMATH Keywords: stationary single-velocity transport equation
zbMATH Keywords: boundary conditions of reflection type
reviewed by: Ion Onciulescu
MaRDI profile type: Publication
cites work: On the spectrum of an unsymmetric operator arising in the transport theory of neutrons
published in: Functional Analysis and Its Applications
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description: scientific article; zbMATH DE number 4159385