I have an equation of the type $$f-\frac{\sigma^2} 2 \frac{d^2 f}{dx^2}-\frac{df}{dx} = \int_0^\infty \left(\frac{df}{dx}\right)^2\exp(ax+f(x)) \, dx.$$
It is an integro-differential equation but the integral is on the fixed domain $[0;\infty[$ independent of $x$. I know we use the Laplace transform for integro-differential equations in some cases. Here I'm trying to rewrite the equation in a simple way. If that fails I would be happy with a numerical method that can deal with it. Usual ODE solvers I tried could not solve it numerically because of the unknown appearing in the integral.
Best Answer
Replace the right side by constant $c$. The d.e. $f - \frac{\sigma^2}{2} f'' - f' = c$ can be solved explicitly. Then you have an equation to solve: $c = \int_0^\infty f'(x)^2 \exp(ax + f(x))\; dx$.
EDIT: With a nonlinear term, there is the real possibility that the solution will blow up at some finite $t$; even without it, the integral may diverge. But if you can identify a region where everything converges nicely, you can use a numerical equation-solver where the integral is evaluated numerically using a numerical solution of the differential equation.