I am trying to solve the following equation for the function $f$.
$$t^{-\alpha} \exp{ \left(- \beta x^2 t^{-2 \alpha} \right)} = \int_0^t \frac{f\left(x, s\right)}{t - s}ds$$ where $\alpha$ and $\beta$ are some non negative constants.
I have tried to take Laplace and Fourier transforms of the left hand side, but I believe those integrals do not converge. Is there any other strategy to obtain $f$ ? I am stuck here, any help would be greatly appreciated. Thanks