Determination of time-dependent coefficients in moving boundary problems under nonlocal and heat moment observations

This paper investigates the reconstruction of time-dependent coefficients in the transient heat equation in a moving boundary domain with unknown free boundaries. This problem is considered under Stefan/heat moments overdetermination conditions also dependent of time. This inverse problem is nonlinear. Moreover, although local existence and uniqueness of solution hold, the problem is still ill-posed since small errors into the input data lead to large errors in the reconstructed coefficients. In order to obtain a stable solution, the nonlinear Tikhonov regularization method is employed. This recasts as minimizing a regularization functional subject to simple bounds on variables. Numerically, this is accomplished using the Matlab toolbox optimization routine lsqnonlin. Numerical results illustrate that stable and accurate solutions are obtained.