An inverse problem utilizing the conjugate gradient method of minimization with adjoint problem is used to estimate the timewise variation of the inlet temperature of a thermally developing, hydrodynamically developed laminar flow between parallel plates by utilizing transient temperature measurements from a single thermocouple located downstream of the entrance. It is assumed that there is no prior information available on the functional form of time dependence of the inlet temperature other than that which can be inferred from the measured downstream temperatures. The effects of functional form of the inlet temperature, sensor position, magnitude of measurement error and data sampling rate on the accuracy of estimates are examined. In order to examine the accuracy of the method under most strict conditions, timewise variations in the form of step changes were also studied. The inverse analysis considered here could predict the timewise variation of inlet temperature even under such strict conditions. The estimates are notably more accurate when the thermocouple is placed near the entrance.