A stability estimate in the problem of determining the dispersion index and relaxation for the transport equation
type: Item
instance of: scholarly article
title: A stability estimate in the problem of determining the dispersion index and relaxation for the transport equation
zbMATH Open document ID: 0878.45004
DOI: 10.1007/BF02104875
published in: Siberian Mathematical Journal
publication date: 1997-12-15T00:00:00Z
review text: The transport equation \[ L(\sigma,K)V \equiv\nabla U\cdot V(\theta)+ \sigma U+SU=0, \quad (x,\theta)\in G\equiv D\times [0,2\pi], \] where \(\sigma= \sigma(x)\), \(\nabla= ({\partial\over \partial x_1}, {\partial\over \partial x_2})\), \(V(\theta)= (\cos\theta, \sin \theta)\), \(\nabla U\cdot V\) is the inner product of the vectors \(\nabla U\) and \(V(\theta)\), and \(SU\) is the operator: \[ SU\equiv \int^{2\pi}_0 K\bigl(x,V (\theta)\cdot V(\theta')\bigr) \cdot U(x, \theta') d\theta' \] describing the scattering has been the object of serious study for long. The present author has contributed already much to this field of studies. The inverse problem for the transport equation, consisting in finding the relaxation \(\sigma(x)\), \(x\in\mathbb{R}^2\) and the dispersion index \(K(x,V(\theta), V(\theta'))\) is considered here. After presenting the statement of the problem and the main results, two theorems and ten lemmas are given in four sections. Details of most of these are also written down alongwith certain stability estimates as well.
Mathematics Subject Classification ID: 45K05
zbMATH DE Number: 1023969
zbMATH Keywords: transport equation
zbMATH Keywords: inverse problem
zbMATH Keywords: relaxation
zbMATH Keywords: dispersion index
zbMATH Keywords: stability estimates
author: Vladimir G. Romanov
MaRDI profile type: Publication
cites work: Investigation of scattering and absorbing media by the methods of X-ray tomography
cites work: Q4150304
cites work: Conditional stability estimates for the two-dimensional problem of restoring the right-hand side and absorption in the transport equation
cites work: Q4003938
full work available at URL: https://doi.org/10.1007/bf02104875
OpenAlex ID: W2019274245
description: scientific article; zbMATH DE number 1023969