Identification of nonlinear heat conduction laws

Abstract

We consider the identification of nonlinear heat conduction laws in stationary and instationary heat transfer problems. Only a single additional measurement of the temperature on a curve on the boundary is required to determine the unknown parameter function on the range of observed temperatures. We first present a new proof of Cannon's uniqueness result for the stationary case, then derive a corresponding stability estimate, and finally extend our argument to instationary problems.

Parameter identification in a semilinear hyperbolic system

“…In the coefficient inverse problem under investigation, the distance to the approximating linearization could be chosen freely by a proper experimental setup. A similar argument was already used in [12] for the identification of a nonlinear diffusion coefficient in a quasi-linear heat equation. The general strategy might however be useful in a more general context and for many other applications.…”

“…This shows the Hölder stability of the inverse problem (6.5) for stationary experiments. As a next step, we will now extend these results to the instationary case by a perturbation argument as proposed in [12] for a related inverse heat conduction problem.…”

Inverse Problem for an Equation of the Reaction-Diffusion-Advection Type with Data on the Position of a Reaction Front: Features of the Solution in the Case of a Nonlinear Integral Equation in a Reduced Statement

“…Since the system ( 15) contains k max − k min + 1 equations, we can determine from it no more than k max − k min + 1 unknowns, namely only q n for n = k min , k max . The system (15) can be rewritten in the following form, using only the specified grid values q n :…”

“…Problems for nonlinear singularly perturbed reaction-diffusion-advection equations arise in gas dynamics [1], combustion theory [2], chemical kinetics [3][4][5][6][7][8][9][10], nonlinear wave theory [11], biophysics [12][13][14][15][16], medicine [17][18][19][20], ecology [21][22][23][24][25], finance [26] and other fields of science [27]. A specific feature of problems of this type is the presence of processes of different scales.…”

Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay

“…This paper discusses the inverse problem of numerical recovering of the initial condition for a nonlinear singularly perturbed reaction-diffusion-advection equation with data on the position of a reaction front, measured in an experiment with a delay relative to the initial time. Problems for equations of this type arise in gas dynamics [1], chemical kinetics [2][3][4][5][6], nonlinear wave theory [7], biophysics [8][9][10][11][12], medicine [13][14][15][16], ecology [17][18][19] and other fields of science [20]. A feature of this type of problem is the presence of multiscale processes.…”