  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 $F=h+\eta$, where $\eta$ is a Lipschitz continuous
background source. The primary objective is to estimate the unknown parameters
of non-instantaneous sources $h(t)=\sum\limits_{j=0}^M
h_je^{-\rho_j(t-t_j)}\chi_{[t_j,\infty)}(t)$, 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.
