Solving ODE with Neumann Boundary: Finite Differences Method

[dinaharchery]

I am new to differential equations, any help would be great.

I have a ODE of the second order u''x = e^x over the domain [1, 1] where u'(0) = 0 is a Neumann boundary on the ODE. I am trying to approximate the solution using the finite differences method, I can do Dirichlet boundaries with finite differences with no problem however the Neumann boundaries are a problem.

The second-order finite difference is
(e^(x - h) - 2*e^(x) + e^(x + h)) / h^2

where h is the computed interval (change in x) across the domain.

How can you model the approximation so that the first derivative at u'(0) = 0 is taken into account. The values I am getting are nothing like the exact solution that I have computed. I am looking to learn this procedure so can anyone point me to the algorithm for this?

Thank you.

 

[dinaharchery]

Maybe my question was not properly worded.

I just want to know how to apply a Neumann boundary on the first derivative (e.g., U'(x) = alpha) with a second-order ODE using finite differences - e.g. U''(x) = f(x)

Is this even possible?

Thanks again