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Initial temperature reconstruction for nonlinear heat equation: application to a coupled radiative–conductive heat transfer problem

In a cooling process described by a nonlinear heat equation, we are interested to recover the initial temperature from the temperature measurements that are available on a part of the boundary for some time. Up to now, even for the linear heat equation, this problem usually has been studied as a nonlinear ill-posed operator equation, and regularization methods involving Fréchet derivatives have been applied. We propose a fast derivative-free iterative method. Numerical results are presented for the glass cooling process where the nonlinearity appears due to radiation.