ODE with Neumann (FDM)

[dinaharchery]

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

 

[mighty2000]

in this matter.



Thank you for your post. To approximate the second-order ordinary differential equation U''(x) = e^x using the finite difference central method, we can use the following steps:

1. Discretize the domain [0,1] into N equally spaced points with a step size of h = 1/N. This means that we will have N+1 points, including the boundary points x0 = 0 and xN = 1.

2. Approximate the second derivative using the central finite difference method:

U''(x) ≈ (U(x+h) - 2U(x) + U(x-h)) / h^2

3. Substituting this approximation into the original equation, we get:

(U(x+h) - 2U(x) + U(x-h)) / h^2 = e^x

4. Rearrange the equation to solve for U(x+h):

U(x+h) = h^2 * e^x + 2U(x) - U(x-h)

5. Now, let's focus on the Neumann boundary condition at x = 0. We can approximate the first derivative using the central finite difference method as follows:

U'(x) ≈ (U(x+h) - U(x-h)) / 2h

6. Substituting this approximation into the Neumann boundary condition U'(0) = 0, we get:

(U(h) - U(-h)) / 2h = 0

7. Since we have already solved for U(x+h) in step 4, we can substitute it into the above equation and solve for U(-h):

(U(h) - U(-h)) / 2h = 0
=> U(-h) = U(h)

8. Now, we can use this result to approximate the Neumann boundary condition at x = h in step 4:

U(h) = h^2 * e^h + 2U(0) - U(-h)

9. Finally, we can use the Dirichlet boundary condition U(1) = 0 to solve for U(0):

U(0) = U(h) - h^2 * e^h

10. Using these boundary conditions and the approximations for U(x+h) and U(x-h) in the equation from step 4, we can solve for U(x) at each point in