In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent heat flux at the inner surface of a functionally graded hollow circular cylinder from the knowledge of temperature measurements taken within the cylinder. Subsequently, the distributions of temperature and thermal stresses in the cylinder can be determined as well. It is assumed that no prior information is available on the functional form of the unknown heat flux; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements, and the effect of the errors in these measurements upon the precision of the estimated results is also considered. Results show that an excellent estimation on the time-dependent heat flux, temperature distributions, and thermal stresses can be obtained for the test case considered in this study.