Looking for solution to decomposing function $f(t)$ into stretched exponential functions all with same meta-exponent and decay constants, but with different amplitudes $p$ (known) and offsets $\tau$ (unknown).

$$f(t)=\int_{-\infty}^{t}dt' \:p(t')e^{-\left(\frac{\left( t-\tau(t')\right)}{\alpha}\right)^\beta}$$

where $f(t)>0$ and $p(t)>0$ for all $t$, and both $\alpha>0$ and $\beta>0$. Want a closed-form solution for the delay function $\tau(t)$

Also interested in solving for $\tau(t')$ where the argument of the exponent contains an additional constant positive offset $t_0>0$ as well: $e^{-\left(\frac{\left(t+t_0-\tau(t')\right)}{\alpha}\right)^\beta}$


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