Identification of nonlinear coefficient in a transport equation
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.
scientific article; zbMATH DE number 2039393