Solving ODE with Neumann Boundary: Finite Differences Method

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To solve a second-order ordinary differential equation (ODE) with Neumann boundary conditions using the finite differences method, it is essential to correctly implement the boundary condition for the first derivative. The finite difference approximation for the second derivative is given, but the challenge lies in applying the Neumann condition, specifically u'(0) = 0. This can be achieved by using a forward difference approximation at the boundary, which allows for the incorporation of the Neumann condition into the finite difference scheme. The user seeks guidance on modeling this approximation effectively to align the computed values with the exact solution. Understanding the proper algorithm for integrating Neumann boundaries in finite difference methods is crucial for accurate results.
dinaharchery
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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.
 
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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
 

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