I have an integral equation as follows:
$f(t)=\int_0^t \sqrt{\tau } g(t-\tau ) \, d\tau$
For a prescribed continuous function $f(t)$, How can I determine an analytical solution for $g(t)$? It should be noted that the derivative of $\sqrt{\tau }$ is discontinuous at $t=0$.
If you take $t\geq 0$, then we have by Laplace transform properties that
$$F(s) = \frac{\sqrt{\pi}}{2}s^{-\frac{3}{2}}G(s)$$
Then we can find $g$ on $[0,\infty)$ by claiming
$$g(t) = \mathcal{L}^{-1}\left\{\frac{2}{\sqrt{\pi}}s^{\frac{3}{2}}F(s)\right\}$$