Is it possible to solve a differential equation \begin{equation} f^{''}(x) = g(x) \end{equation} using finite difference method when the boundary condition is zero curvature (second derivative)?
I don't think it's possible since there would be more unknowns than the number of equations, but I'd like to know whether there is a way to keep the slope (first derivative) unchanged (i.e., zero curvature) at the boundary when solving the equation.
Thanks!
The boundary value problem on interval $(a,b)$, $$ f''(x)=g(x),\quad f''(a)=0, \ f''(b)=0 \tag{1}$$ is not well posed. There is no solution if $g$ does not vanish at endpoints. There are infinitely many solutions when it does: one can add any linear function to $f$ without changing the derivative.
The finite difference version of (1) will have the same problems. Generally, it is advisable to fix the well-posedness issues at the level of ODE (1) before going to the difference scheme, which will introduces issues of its own.