Stabilizing of an ill-posed inverse problem by using smoothing splines and hyperbolic heat equation

A smoothing splines method and a hyperbolic heat conduction model is applied to regularize the recovery of the initial profile from a parabolic heat conduction model. The recovery of initial profile with parabolic model is extremely ill-posed and it is desirable to develop a method to recover some useful information about the initial profile. It is demonstrated in this article that the recovery of the initial profile by a parabolic model is impossible in case of corrupted data, however, it is possible to recover the initial profile in a stable way by reformulating the problem by a hyperbolic model together with a smoothing splines method. One example is used for comparison of parabolic model and the proposed model for different modes and time displacements. A graphical comparison is given between the proposed model and the parabolic model.