Reconstructing the Time-Dependent Thermal Coefficient in 2D Free Boundary Problems

The inverse problem of reconstructing the time-dependent thermal conductivity and free boundary coefficients along with the temperature in a two-dimensional parabolic equation with initial and boundary conditions and additional measurements is, for the first time, numerically investigated. This inverse problem appears extensively in the modelling of various phenomena in engineering and physics. For instance, steel annealing, vacuum-arc welding, fusion welding, continuous casting, metallurgy, aircraft, oil and gas production during drilling and operation of wells. From literature we already know that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data. For the numerical realization, we apply the alternating direction explicit method along with the Tikhonov regularization to find a stable and accurate numerical solution of finite differences. The root mean square error (rmse) values for various noise levels p for both smooth and non-smooth continuous time-dependent coefficients Examples are compared. The resulting nonlinear minimization problem is solved numerically using the MATLAB subroutine lsqnonlin. Both exact and numerically simulated noisy input data are inverted. Numerical results presented for two examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data.