The two-dimensional retrospective heat conduction problem

A retrospective heat conduction problem is to estimate the temperature distribution in the previous moment from measured temperature histories at fixed locations inside the body. This is an important problem because an accurate retrospective solution can be used to detect the cause of something abnormal in a manufacturing process or to set up an initial state for a thermal process with required characteristics. In this research, a reverse matrix method is used to estimate the previous state in a two-dimensional heat conduction problem. Two kinds of retrospective problems are studied in this paper: one is to estimate a previous state at the time when the measurement starts and the other at the time before the measurement starts. Comparisons between the exact state and the estimated ones are made in order to confirm the validity of the proposed method. The close agreement between the exact solutions and the estimated results shows the potential of the proposed method for finding the accurate value of the retrospective solution in the heat conduction problem.