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This paper considers the inverse problems of identifying either the unknown space-dependent heat sourceor the unknown time-dependent heat sourceof the variable coefficient advection–diffusion equation with separable sources of the formfrom supplementary time-dependent temperature measurement at the right boundary of the domain. We proved that solutions of both inverse source problems can be identified uniquely under some regularity assumptions. Two non-iterative inversion algorithms were developed and numerically implemented for the identification of the unknown space-wise and time-wise dependent sources. The robustness and limitations of the two algorithms are investigated through numerical examples related to the reconstruction of continuous and discontinuous sources. [ABSTRACT FROM PUBLISHER]