Inverse unsteady heat conduction problem of second kind

The problem of determining the time-dependent heat flux imposed on the boundary of a solid slab from the temperature distribution at the final time is solved by the conjugate gradient (CG) and the truncated singular value decomposition (TSVD) methods. The Tikhonov regularization is used to regularize the solution when the given data contain random errors. The recovering of the exact boundary condition is shown to depend on the total time of the heating/cooling process. It is found that the exact boundary heat flux can be recovered for about one tenth of the diffusion time, beyond which we obtain only the time-averaged heat flux. However, by using a modified conjugate gradient method, we may reconstruct the boundary heat flux for much larger times if its initial value is known. We also show that these methods can be effectively used to solve the control problem.