Green's Function with Neumann Boundary Conditions

[Mattbringssoda]

Homework Statement

[/B]
Determine the Green's functions for the two-point boundary value problem u''(x) = f(x) on 0 < x < 1 with a Neumann boundary condition at x = 0 and a Dirichlet condition at x = 1, i.e, find the function G(x; x) solving

u''(x) = delta(x - xbar) (the Dirac delta function); u'(0) = 0; u(1) = 0

and the functions G_0 (x) solving

u''(x) = 0; u'(0) = 1; u(1) = 0

and G_1(x) solving

u''(x) = 0; u'(0) = 0; u(1) = 1:

Homework Equations

The Attempt at a Solution

Right now I'm most concerned with the first part of the problem, the main Green's function G(x,x) solving: u''(x) = delta(x - xbar) (the dirac delta function); u'(0) = 0; u(1) = 0.

I'm stuck because we're using these functions to make discreet approximations to boundary value problems via matrices, so I assume the functions have to be linear.

However, u'(0) = 0 would have to be of some form u'(x) = x, so the solution would have to be a non-linear term, along the lines of u(x) = x^c. That doesn't seem right in light of what we've been learning...so I don't know what to do here...

Any help is much appreciated!

 

[Orodruin]

However, u'(0) = 0 would have to be of some form u'(x) = x, so the solution would have to be a non-linear term, along the lines of u(x) = x^c. That doesn't seem right in light of what we've been learning...so I don't know what to do here...

It is not clear what you are trying to do here. Your problem is a relatively simple differential equation with given boundary conditions and so should have a rather straightforward solution. Constructing the general solution to $G'' = \delta(x-\bar x)$, you should obtain a solution with two undetermined constants that you will have to fix to satisfy the boundary conditions. How are you used to construct the Green's function to ODEs of this type?

 

[Mattbringssoda]

Well, we're only doing a very surface introduction to Greens functions in class, and were using them to construct inverse matrices in which to numerically solve boundary value problems and to look at the stability of those methods.

The only other Greens function we encountered was for one that satisfied G" = δ(x-x), x(0) = 1 and x(1) = 0. The answer was just given and then shown to be the answer. The answers were linear.

In my posted problem, I guess I'm not sure how to get started with the u' as the boundary conditions. I'm guessing something like u^c, then u'(0) will give us 0... but I was under the impression that the terms here has to be linear since were using them in matrices to numerically solve problems... I'm clearly not understanding something...

 

[Orodruin]

Start by considering the differential equation for $x < \bar x$. How does the solution look in this region?

 

[Mattbringssoda]

Hmm, well that's the area I thought would be of form x^c for x < x_bar

(PS Thank you for your responses thus far..)

 

[Orodruin]

Why? How does the differential equation look in that region?

 

[Mattbringssoda]

Well, the G" equals a Dirac function with its jump at xBar,

So below xBar: G" = 0,

Meaning G' = a constant ,

Meaning G is generally some form of Cx, where C is a constant...?

 

[Orodruin]

Meaning G is generally some form of Cx, where C is a constant...?

No, you forgot one integration constant. You also need to adapt the solution to the boundary condition at $x = 0$.

 

[Mattbringssoda]

I think I have it now; u" = 0 means that u MUST beinn linear. u' = some constant, and go from there.

Thanks SO much for your help...