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ODE with Neumann (FDM)

  • #1

Homework Statement


Use finite difference central method to approximate the second-order Ordinary Differential Equation U''(x) = e^x over domain [0, 1]
where:
u(1) = 0 (Dirichlet Bound)
U'(0) = 0 (Neumann Bound)

Homework Equations


let 'h' be the change in x direction


The Attempt at a Solution



I am able to get the Dirichlet bound but have problems with the corresponding Neumann bound on the first derivative using central finite difference method.
I get:

(e^(h) - e^(-h)) / 2*h

for the first derivative, setting it equal to zero. I have the boundary approximated but the resulting rest of the equation I am unsure of,
i.e. , how does the finite difference approximation:

(e^(x + h) - 2*e^(x) + e^(x - h)) / h^2

relate the above Neumann boundary condition?

Thanks for any assistance
 

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