Determination of time-dependent coefficients for a weakly degenerate heat equation

In this paper, we consider solving numerically for the first time inverse problems of determining the time-dependent thermal diffusivity coefficient for a weakly degenerate heat equation, which vanishes at the initial moment of time, and/or the convection coefficient along with the temperature for a one-dimensional parabolic equation, from some additional information about the process (the so-called over-determination conditions). Although uniquely solvable these inverse problems are still ill-posed since small changes in the input data can result in enormous changes in the output solution. The finite difference method with the Crank-Nicolson scheme combined with the nonlinear Tikhonov regularization are employed. The resulting minimization problem is computationally solved using the MATLAB toolbox routine lsqnonlin. For both exact and noisy input data, accurate and stable numerical results are obtained.