# Optimization Problem Involving an Integral Equation

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?

• $\bar x(t)^2$ in the integral looks a bit asymmetric. Should it be squared? – A.Γ. Nov 9 '15 at 15:11
• Sorry, that is a mistake. Now corrected. – Henry Nov 9 '15 at 16:56
• Do you have a explicit equation for $j(t)$? Does $j(t)$ depends on $\bar{x}(t)$? Is $\bar{x}(t)$ the decision variable/function? I'm not aware of the context of your problem, could you clarify? – Marco Aguiar Nov 10 '15 at 12:45
• No the problem is that I don't have an explicit equation for $j(t)$, only the integral equation above. This integral equation should in principle fully specify $j(t)$ in terms of $\bar{x}(t)$ (which is not a function that I am putting in, it is the function I want to optimize with respect to). – Henry Nov 10 '15 at 15:13
• If you could find a differential equation for $dj(t)/dt$ you could use transform in a calculus of variations problem (and solve using Euler-Lagrange equation) or a optimal control problem (ans solve using one of the many methods available). – Marco Aguiar Nov 11 '15 at 13:22