Determination of time-dependent coefficients and multiple free boundaries

A difficult inverse problem consisting of determining the time-dependent coefficients and multiple free boundaries, together with the temperature in the heat equation with Stefan condition and several-orders heat moment measurements is, for the first time, numerically solved. The time-dependent missing information matches up quantitatively with the time-dependent additional information that is supplied. Although the inverse problem has a unique local solution, this problem is still ill-posed since small errors in input data cause large errors in the output solution. For the numerical realization, the finite difference method with the Crank-Nicolson scheme combined with the Tikhonov regularization are employed in order to obtain an accurate and stable numerical solution. The resulting nonlinear minimization problem is computationally solved using the MATLAB toolbox routine lsqnonlin. A couple of numerical examples are presented and discussed to verify the accuracy and stability of the approximate solutions.