How to solve a series of integral equation [closed]

I am trying to solve the following integral equation analytically:

$$\sum_{n \geq 1} \left( \int_0^\tau e^{-n^2(\tau-s)} f_n(s) \, ds \right) = g(\tau), \quad \tau \in [0, T],$$where $(f_n(\tau))_n$ is the unknown functions to be determined and $g(\tau)$ is a given function in $L^2(0,T)$. I am interested in the existence of the solution for $g(\tau)$ in $L^2(0,T)$.

Could someone provide guidance on references or techniques that could be useful for handling this problem?

Thank you in advance!