Stability of solutions of the overdetermined inverse heat conduction problems when discretized with respect to time

The inverse heat conduction problem with many intend responses is considere d After discretization with respect to time, the problem is described by a system of the Helmholtz equations in a recurrent form An approximate soluti on of a heat conduction problem in an integral form is shown Then, an appro ximate solution of an inverse heat conduction problem in a flat slab is pre sented for many internal temperature responses. Stability of the solution w ith respect to the internal response errors is investigated for hco casts: when the integrals ave calculated with the use of the average value theorem and when they ave calculated numerically. Analysts of the norm of a matrix that is essential for the solution stability shows that in the case of thr ee sensors the norm slightly changes with change of the middle sensor locat ion. If more than three sensors are taken into consideration, thee results practically uda not change comparing to the case of three sensors The inter nal temperature response errors ave suppressed if the time step is greater than a certain critical value.