The inverse heat conduction problem with many internal responses is considered. After discretization with respect to time, the problem is described by a system of the Helmholtz equations in a recurrent form. An approximate solution of a heat conduction problem in an integral form is shown. Then, an approximate solution of an inverse heat conduction problem in a flat slab is presented for many internal temperature responses. Stability of the solution with respect to the internal response errors is investigated for two cases: when the integrals are calculated with the use of the average value theorem and when they are calculated numerically. Analysis of the norm of a matrix that is essential for the solution stability shows that in the case of three sensors the norm slightly changes with change of the middle sensor location. If more than three sensors are taken into consideration, the results practically will not change comparing to the case of three sensors. The internal temperature response errors are suppressed if the time step is greater than a certain critical value.
Maciag, A. and Jehad Al‐Khatib, M. (2000), "Stability of solutions of the overdetermined inverse heat conduction problems when discretized with respect to time", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 10 No. 2, pp. 228-245. https://doi.org/10.1108/09615530010312554
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