Optimization Problem Involving an Integral Equation

Question

I have an integral equation for a probability current $j(t)$ given by:
$$ \large\frac{1}{\sqrt{t}} e^{-\frac{\bar{x}(t)^2}{4 t}} = \int_0^t \frac{j(t')}{\sqrt{t-t'}} e^{-\frac{\left[\bar{x}(t)-\bar{x}(t') \right]^2}{4 (t-t')}}dt' $$
which I don't think can be solved exactly for general $\bar{x}(t)$ (if it can then that would be excellent!) The problem I would like to be able to solve is to find the function $\bar{x}(t)$ that minimizes something like $$\int_0^{\tau} (j(t')+\bar{x}(t')^2)dt'.$$ I am wondering if anyone knows whether there are numerical techniques for doing this kind of thing, and if so could they point me in the direction of any relevant literature?