Solving ODE with Neumann Boundary: Finite Differences Method

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SUMMARY

The discussion centers on solving a second-order ordinary differential equation (ODE) of the form u''(x) = e^x using the finite differences method with Neumann boundary conditions. The user is specifically struggling with implementing the Neumann boundary condition u'(0) = 0, which requires a different approach than Dirichlet boundaries. The finite difference approximation provided is (e^(x - h) - 2*e^(x) + e^(x + h)) / h^2, but the user reports discrepancies between their computed values and the exact solution. The need for a clear algorithm to incorporate Neumann boundaries in finite difference methods is emphasized.

PREREQUISITES
  • Understanding of second-order ordinary differential equations (ODEs)
  • Familiarity with finite differences method for numerical approximation
  • Knowledge of boundary conditions, specifically Neumann boundary conditions
  • Basic calculus, particularly derivatives and exponential functions
NEXT STEPS
  • Research the implementation of Neumann boundary conditions in finite difference methods
  • Study the derivation of finite difference formulas for first and second derivatives
  • Explore numerical methods for solving ODEs, focusing on stability and accuracy
  • Learn about alternative numerical techniques such as the shooting method or finite element method
USEFUL FOR

This discussion is beneficial for students and practitioners in applied mathematics, numerical analysis, and engineering, particularly those working with differential equations and numerical methods for boundary value problems.

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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