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Abstract This paper investigates the problem of recovering source terms in abstract initial value problems (IVP) commonly used to model various scientific phenomena in physics, chemistry, economics, and other fields. We consider source terms of the form$$F=h+\eta $$ $F=h+\eta$, where$$\eta $$ $\eta$is a Lipschitz continuous background source. The primary objective is to estimate the unknown parameters of non-instantaneous sources$$h(t)=\sum \limits _{j=0}^M h_je^{-\rho _j(t-t_j)}\chi _{[t_j,\infty )}(t)$$ $h\left(t\right)=\sum _{j=0}^M{h}_j{e}^{-{\rho }_j(t-t_j)}{\chi }_{[t_j,\infty )}\left(t\right)$, such as the decay rates, initial intensities and activation times. We present two novel recovery algorithms that employ distinct sampling methods of the solution of the IVP. Algorithm 1 combines discrete and weighted average measurements, whereas Algorithm 2 uses a different variant of weighted average measurements. We analyze the performance of these algorithms, providing upper bounds on the recovery errors of the model parameters. Our focus is on the structure of the dynamical samples used by the algorithms and on the error guarantees they yield.