Lapin, Alexander; Lapin, Serguei Identification of nonlinear coefficient in a transport equation. (English) Zbl 1038.65089 Lobachevskii J. Math. 14, 69-84 (2004). 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. Reviewer: Carlos A. De Moura (Flamengo) Cited in 1 Document MSC: 65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs 80A30 Chemical kinetics in thermodynamics and heat transfer 80M20 Finite difference methods applied to problems in thermodynamics and heat transfer 65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 76S05 Flows in porous media; filtration; seepage 35L60 First-order nonlinear hyperbolic equations 35R30 Inverse problems for PDEs 76M20 Finite difference methods applied to problems in fluid mechanics Keywords:nonlinear coefficient identification; transport equation; final observation; finite difference scheme; multilevel algorithm; numerical examples; inverse problem; convective transport; chemical substance; parameter identification; ill-posed problem; optimal control PDFBibTeX XMLCite \textit{A. Lapin} and \textit{S. Lapin}, Lobachevskii J. Math. 14, 69--84 (2004; Zbl 1038.65089) Full Text: EuDML EMIS