This paper presents a joint state and input estimation algorithm for the one-dimensional heat-conduction problem. A computationally efficient method is proposed in this work to solve the inverse heat-conduction problem (IHCP) using orthogonal collocation method (OCM). A Kalman filter (KF) algorithm is used in conjunction with a recursive-weighted least-square (RWLS)-based method to simultaneously estimate the input boundary condition and the temperature field over the heat-conducting element. A comparison study of the algorithm is shown with explicit finite-difference method (FDM) of approximation and analytical solution of the forward problem, which clearly reveals the high accuracy with lower-dimensional modeling. The estimation results show that the performance of the estimator is robust to noise sensitivity up to a certain level, which is practically acceptable.