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The gist of the problem I have is I want to be able to find a numerical solution to these three coupled, rather unpleasant looking integro-differential equations

(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): $$\frac{e^{-\frac{x(t)^2}{4 t}}}{\sqrt{t}} = \int_0^t\frac{j(t')}{\sqrt{t-t'}}e^{-\frac{(x(t)-x(t'))^2}{4(t-t')}}dt'$$

for $x(t), j(t)$ and $\lambda(t)$. The only boundary conditions are that $x(0)=x_0$ is known, and $x'(T)=0$. $\alpha$ will be a known constant. I don't have much experience with numerically solving integral equations but they don't look nice and I've not been able to find much in the literature (in an admittedly short search) that could be applied to this.

To give you some context, the actual problem I'm trying to solve is to minimize the functional: $$ \int_0^T (j(t) + \alpha x'(t)^2)dt$$ but with the state $j(t)$ governed by the integral equation (3) above for a given $x(t)$. The way I've approached this is to consider $\widetilde{j}=j+\epsilon_1 \Delta j$ and $\widetilde{x}=x+\epsilon_2 \Delta x$ and consider the functional: $$K[\epsilon_1, \epsilon_2] = \int_0^T \left[ \widetilde{j}(t) + \alpha \widetilde{x}'(t)^2 + \lambda(t) \left(\frac{e^{-\frac{x(t)^2}{4 t}}}{\sqrt{t}} - \int_0^t\frac{j(t')}{\sqrt{t-t'}}e^{-\frac{(x(t)-x(t'))^2}{4(t-t')}}dt' \right) \right]$$ where $\lambda(t)$ is a Lagrange multiplier which enforces the integral equation (3) holds as a constraint for all time. I'm not sure this is the best way to solve the problem, but when I follow through taking the functional derivative I get the 3 equations above which I guess are in principle soluble. Has anyone come across any good literature which could be applied to solving these equations (or alternatively a better way to approach the initial problem)?

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