Identification of nonlinear coefficient in a transport equation
type: Item
instance of: scholarly article
title: Identification of nonlinear coefficient in a transport equation
zbMATH Open document ID: 1038.65089
author: A. V. Lapin
author: Serguei Lapin
published in: Lobachevskii Journal of Mathematics
publication date: 2004-02-10T00:00:00Z
full work available at URL: https://eudml.org/doc/229796
full work available at URL: lpl.html
review text: Convective transport of a chemical substance subject to a sorption process through a porous medium may be modeled by the nonlinear initial boundary-value problem \[ \frac{\partial c}{\partial t} + \frac{\partial c}{\partial x}+ \frac{\partial a(c)}{\partial t}\;=\; 0\quad x\in(0,1)\quad t\in(0,T], \] \[ c(0, t) = 1\quad c(x, 0) = 0. \] The function \(\;c\;\) represents the concentration of the dissolved chemical and must then satisfy the additional constraint \[ \;0\;\leq\;c\;\leq\;1. \] Further \(\;a(c)\;\) stands for the so-called sorption isotherm. A parameter identification problem is considered for this model, namely, the function \(\;a(c)\;\) is unknown and its determination is searched with the known observations for the values of \(\;\phi(x)\), related to it through \[ \;a(c(x, T))\; =\; \phi(x)\;. \] This turns out to be a highly ill-posed problem which is treated through the standard optimal control approach. Use of a finite-difference scheme for the transport equation brings it down to a finite-dimension optimization framework. It is then shown that the corresponding problem is always solvable and computer implementation of the resulting algorithm exhibits quite dependable results.
Mathematics Subject Classification ID: 65M32
Mathematics Subject Classification ID: 80A30
Mathematics Subject Classification ID: 80M20
Mathematics Subject Classification ID: 65M30
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 76S05
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 76M20
zbMATH DE Number: 2039393
zbMATH Keywords: nonlinear coefficient identification
zbMATH Keywords: transport equation
zbMATH Keywords: final observation
zbMATH Keywords: finite difference scheme
zbMATH Keywords: multilevel algorithm
zbMATH Keywords: numerical examples
zbMATH Keywords: inverse problem
zbMATH Keywords: convective transport
zbMATH Keywords: chemical substance
zbMATH Keywords: parameter identification
zbMATH Keywords: ill-posed problem
zbMATH Keywords: optimal control
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2039393