Solving ODE with Neumann Boundary: Finite Differences Method

In summary, the conversation is about using the finite differences method to approximate the solution of a second-order ODE with a Neumann boundary condition. The speaker is having trouble with the Neumann boundary and is looking for an algorithm or procedure to properly apply it. They also mention wanting to incorporate a first derivative boundary condition into the approximation.
  • #1
dinaharchery
24
0
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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  • #2
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
 

1. What is the Neumann Boundary Condition?

The Neumann Boundary Condition is a type of boundary condition used in solving Ordinary Differential Equations (ODEs). It specifies the derivative of the solution at the boundary, rather than the value of the solution itself.

2. What is the Finite Differences Method?

The Finite Differences Method is a numerical method used to solve ODEs by approximating the derivatives in the equation with finite differences. This method divides the problem domain into a discrete grid and calculates the solution at each grid point using the values at neighboring points.

3. How is the Neumann Boundary Condition incorporated into the Finite Differences Method?

The Neumann Boundary Condition is incorporated into the Finite Differences Method by using a backward difference approximation for the derivative at the boundary point. This means that the derivative at the boundary is calculated using the values at the boundary point and the previous grid point.

4. What are the advantages of using the Finite Differences Method for solving ODEs with Neumann Boundary?

One advantage of the Finite Differences Method is its simplicity and ease of implementation. It also allows for a large variety of boundary conditions to be incorporated, including the Neumann Boundary Condition. Additionally, this method is computationally efficient and can handle complex geometries.

5. Are there any limitations to using the Finite Differences Method for solving ODEs with Neumann Boundary?

One limitation of the Finite Differences Method is that it can only be applied to regular grid structures, which may not accurately represent the problem domain. Additionally, this method may produce less accurate results compared to other numerical methods, such as the Finite Element Method.

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