ABSTRACT This paper investigates the simultaneous inversion of the unknown source term and initial value for a complete parabolic equation in the transport process from a mathematical perspective. We derive the exact solutions using the Fourier transform technique and discover that they are severely ill‐posed. Furthermore, the fractional Landweber iterative regularization method is introduced to address this ill‐posedness. By leveraging the a priori and the a posteriori regularization parameter choice rules, we obtain excellent convergence error estimates. Finally, several numerical experiments validate the effectiveness of this strategy.
Yang et al. (2026) studied this question.