I'm solving initial value problem with non-local boundary condition $u(0,t)=\int_{0}^{l}\beta(s)u(s,t)ds = \gamma(t)$. I have already found function u for to cases $x<t$ and $x>t$. But I have some troubles with finding general solution to function $\gamma(t)$. I need to solve this integral equation of Volterra type of second kind:
$\gamma(t)=\int_{0}^{t}\beta(s)e^{-\int_{0}^{s}\sigma(\tau)d\tau}\gamma(t-s)ds+H(t)$
I'm trying to find resolvent or just use naive itterating method:
$y_n=H(t)+\int_{0}^{t}K(s,t)y_{n-1}(s)ds$
But it gave me only a long recursive expression without any simplifications, that I need.
So maybe I need to use some other method and etc, I need some advertising. Thanks.