I have a nonlinear ODE I wish to convert into a Fredholm integral equation. This site shows how to convert an ODE with Dirichlet BC into an integral equation, however I cannot figure out how to do it with Neumann BC.
My equation is of the form
$u''(x)+f(x,u)u=0$
With BC
$u'(0)=u'(L)=0$
I've tried following the linked site. Begin by integrating
$\int_0^x u''(x)dx=\int_0^xf(x,u(x))u(x)dx$
$u'(x)-u'(0)=\int_0^xf(x,u(x))u(x)dx$
Here the derivation deviates from the linked site with $u'(0)=0$. Integrate again
$u(x)-u(0)=\iint_0^xf(x,u(x))u(x)dx$
$u(x)-u(0)=\int_0^x(L-t)f(t,u(t))u(t)dt$
Where the rule for reducing multiple integrals was used (by the way, does anyone know if that rule has a name?).
From here I just get stuck. I'm not sure where to go from here or if this method is even the correct approach for neumann BC.
EDIT: After more thinking it seems if I go back to this step
$u'(x)-u'(0)=\int_0^xf(x,u(x))u(x)dx$
And then evaluate the integral out to L
$u'(L)=\int_0^Lf(x,u(x))u(x)dx$
The derivative at L is also zero, and hence I get
$0=\int_0^Lf(x,u(x))u(x)dx$
Does this look correct?