I have a system of three coupled integral equations for three unknowns $j(t), \bar{x}(t)$ and $\lambda(t)$ to be solved between $t=0$ and $t=T$ (b.c. are $\bar{x}(0)=x_0$ and $\bar{x}'(T)=0$):
(1): $$ \frac{d^2 x(t)}{dt^2} = \frac{\lambda(t)}{2 \alpha} \left( \int_0^t \frac{(x(t) - x(t'))}{2(t-t')^{3/2}} j(t') e^{-\frac{(x(t)-x(t'))^2}{4(t-t')}}dt' - \\ j(t) \int_t^T \frac{(x(t') - x(t))}{2(t'-t)^{3/2}} e^{-\frac{(x(t')-x(t))^2}{4(t'-t)}}dt' - \frac{x(t)}{2 t^{3/2}} e^{-\frac{x(t)^2}{4t}}\right)$$ (2): $$\frac{1}{\lambda(t)} = \int_t^T \frac{e^{-\frac{(x(t')-x(t))^2}{4(t'-t)}}}{\sqrt{t'-t}}dt'$$ (3): $$\frac{e^{-\frac{x(t)^2}{4 t}}}{\sqrt{t}} = \int_0^t\frac{j(t')}{\sqrt{t-t'}}e^{-\frac{(x(t)-x(t'))^2}{4(t-t')}}dt'$$
I am really interested in finding techniques to solve these numerically, but I'm not even sure how to classify the first equation (and hence what to be searching for in the literature). Equations (2) and (3) are volterra equations of the first kind with weakly singular kernels, but I'm not really sure what to call equation (1). Does anyone know of any numerical techniques which could be applied to these equations?