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Identification of a heat transfer coefficient depending on pressure and temperature. (English) Zbl 1251.65111

The authors consider the following heat transfer problem: \[ \begin{cases} T'(t)=H(T(t),P(t),)(T^e-T(t))+\alpha P'(t)T(t), \quad t\in [t_0,t_f], \\ T(t_0)=T_0, \end{cases} \] where the heat transfer coefficient \(H\) is to be identified (note that there is no temperature gradient in the model). The pressure \(P\) is given as a function of the time \(t\). \(H\) is only identified if \(T\) is separated from the (constant) temperature \(T^e\) of the environment by a certain threshold \(\mu>0\). It is assumed that \(H\) depends either on the pressure \(P\) or the temperature \(T\) alone.
In the first case, the differential equation has a closed solution and there is an explicit solution for \(H\) if \(T\) is known for all times. If only a finite number of \(T\)-measurements is available, also possibly containing errors, an algorithm is given that mainly relies on interpolation of the set of measurements.
In the second case, where \(H\) is only a function of \(T\) it is assumed that the pressure is constant, simplifying the state equation. Here, too, \(H\) can be easily explicitely calculated if \(T\) and \(T'\) are known. If only a finite number of \(T\)-measurements is available, also possibly containing errors, the inverse problem \[ \int_{t_0}^t H(T(s))\,ds = \int_{t_0}^t \frac{T'(s)}{T^e-T(s)}\,ds=- \ln \left ( \frac{T^e-T(t)}{T^e-T_0} \right) \] is taken for determinig \(H\). Various known techniques, such as Tikhonov regularization, discrepancy principle of Morozow, Landweber’s iterative method, are considered for its solution. Also an iterative algorithm based on interpolating the measurements is presented. For both cases numerical experiments are provided.

MSC:

65L09 Numerical solution of inverse problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34A55 Inverse problems involving ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
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