In this paper, early lumping estimation of space-time varying diffusion coefficient and source term for a nonhomogeneous linear parabolic partial differential equation (PDE) describing tokamak plasma heat transport is considered. The analysis of this PDE is achieved in a finite-dimensional framework using the cubic b-splines finite element method with the Galerkin formulation. This leads to a finite-dimensional linear time-varying state-space model with unknown parameters and inputs. The extended Kalman filter with unknown inputs without direct feedthrough (EKF-UI-WDF) is applied to simultaneously estimate the unknown parameters and inputs and an adaptive fading memory coefficient is introduced in the EKF-UI-WDF, to deal with time varying parameters. Conditions under which the direct problem is well posed and the reduced order model converges to the initial one are established. In silico and real data simulations are provided to evaluate the performances of the proposed technique.