I've got the following problem I'm working on which is related to some of my research.

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 $0 \le \alpha \le 1$ and $\beta=\alpha \Gamma \left(\alpha+\frac{1}{2}\right)^2$ are some non negative constants.

I have tried to take Laplace and Fourier transforms of the left hand side, but to no avail. Is there any other strategy to obtain $f$ ? I am stuck here, any help would be greatly appreciated. Thanks.