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Boundary conditions, Sturm-Liouville, & Gauss Divergence

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the_dialogue
#1
Jun8-10, 11:41 AM
P: 79
1. The problem statement, all variables and given/known data

I'm getting through a paper and have a few things I can't wrap my head around.

1. In defining the boundary conditions for a membrane (a function of vector 'r'), the author claims that for a small displacement (u) and a boundary movement (f), the boundary condition can be defined as:

[tex]\newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } }

\pd{u(\mathbf{r},t)}{n}{}+\alpha(\mathbf{r})u(\mathbf{r},t)=f(\mathbf{r },t) [/tex]

where alpha>0, n is the normal, and f is some function defined on the boundary.

2. The author presents a series expansion using some eigenfunction [tex]\phi_m(\mathbf{r})[/tex] with eigenvalues 'lambda'. He states that the eigenfunction is derived from the solution to the Sturm-Liouville problem:

[tex](\nabla^2 + \lambda_m^2)\phi_m(\mathbf{r})=0[/tex], on the domain of the membrane

&

[tex]\newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } }

\pd{\phi_m(\mathbf{r},t)}{n}{}+\alpha(\mathbf{r})\phi(\mathbf{r},t)=0[/tex], on the boundary of the membrane

3. The author presents the Gauss divergence theorem in a way I'm not too familiar with.

[tex]
\newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } }

\int\int_{D}^{}(\phi_m\nabla^2u-u\nabla^2\phi_m)dS=\int\int_{B}^{}(\phi_m\pd{u}{n}{}-u\pd{\phi_m}{n}{})dC[/tex] where S is some surface of the domain D, and C is a line element along the boundary B. 'n' defines the normal.

2. Relevant equations

See above.

3. The attempt at a solution
For [1], I believe I understand the first term as representing the geometric relationship between a small displacement u and a small boundary movement f. However I do not see the relevance of the second term (alpha*....)

For [2], I see little resemblance of these equations and what I've seen in introductory texts on the Sturm-Liouville problem. Can someone perhaps point me in the right direction at understanding the meaning of these formulations?

For [3]: I have seen the Gauss divergence theorem relating the Volume integrals to the Surface integrals, but never the "Domain" integral to the "Boundary" integral. I suppose this is just a reduction in dimensions -- is that right? More importantly, I don't understand the meaning of the subtractive terms on each side. This I have not seen in intro texts on the Gauss Divergence theorem.

Any help would be greatly appreciated!
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Coto
#2
Jun8-10, 03:36 PM
P: 308
Quote Quote by the_dialogue View Post
3. The attempt at a solution
For [1], I believe I understand the first term as representing the geometric relationship between a small displacement u and a small boundary movement f. However I do not see the relevance of the second term (alpha*....)
Not sure, but if he is deriving a boundary condition it is possible that the physics of the situation plays a role in determining how he derived it. For this reason it would help if we knew which paper you were talking about, or could provide more insight into the problem.

Quote Quote by the_dialogue View Post
For [2], I see little resemblance of these equations and what I've seen in introductory texts on the Sturm-Liouville problem. Can someone perhaps point me in the right direction at understanding the meaning of these formulations?
In order to expand the function u(x), it makes sense to use some orthonormal functions, which the Sturm-Liouville equation ensures. Substituting in the expansion into the original PDE for u(x) and exploiting the orthogonality of the eigenfunctions, one finds a set of simpler PDEs that you can hope to solve and obtain solutions for the [tex]\phi_m[/tex]'s.

Quote Quote by the_dialogue View Post
For [3]: I have seen the Gauss divergence theorem relating the Volume integrals to the Surface integrals, but never the "Domain" integral to the "Boundary" integral. I suppose this is just a reduction in dimensions -- is that right? More importantly, I don't understand the meaning of the subtractive terms on each side. This I have not seen in intro texts on the Gauss Divergence theorem.
http://en.wikipedia.org/wiki/Green%27s_theorem
the_dialogue
#3
Jun8-10, 05:01 PM
P: 79
Quote Quote by Coto View Post
Not sure, but if he is deriving a boundary condition it is possible that the physics of the situation plays a role in determining how he derived it. For this reason it would help if we knew which paper you were talking about, or could provide more insight into the problem.
Sure thing. We're dealing with a membrane defined by a domain D and boundary B. No other constraints are made until this point. On part of the boundary the membrane may be free and somewhere else it could be forced at this external displacement (f).

Could it be that the author is saying that the boundary motion (f) is equal to 2 terms (in general): [1] using the small angle approx, the component of the membrane movement + [2] some multiple of the membrane movement (alpha)? This seems kind of weak.

Quote Quote by Coto View Post
In order to expand the function u(x), it makes sense to use some orthonormal functions, which the Sturm-Liouville equation ensures. Substituting in the expansion into the original PDE for u(x) and exploiting the orthogonality of the eigenfunctions, one finds a set of simpler PDEs that you can hope to solve and obtain solutions for the [tex]\phi_m[/tex]'s.
I'm not so sure what you mean. Where are these Sturm-Liouville equations taken from and how are they relevant to the problem? Perhaps a simpler example (somewhere online) would help explain this?

Coto
#4
Jun10-10, 12:34 AM
P: 308
Boundary conditions, Sturm-Liouville, & Gauss Divergence

Quote Quote by the_dialogue View Post
Could it be that the author is saying that the boundary motion (f) is equal to 2 terms (in general): [1] using the small angle approx, the component of the membrane movement + [2] some multiple of the membrane movement (alpha)? This seems kind of weak.
Hmm. It's a tough one. I personally don't see how the alpha is coming in either. What's the name of the paper? Is it accessible through web of science?

Quote Quote by the_dialogue View Post
I'm not so sure what you mean. Where are these Sturm-Liouville equations taken from and how are they relevant to the problem? Perhaps a simpler example (somewhere online) would help explain this?
I would say don't get too caught up in where they are coming from. It is more important to understand why they are useful. My guess is the author is deriving a set of bases functions to be used in an expansion of [tex]u(x)[/tex]. The Sturm-Liouville equations can deliver a set of functions which can serve this purpose. The nice properties that these bases functions display make them a good choice when you're expanding the unknown u(x).

In particular take a look at (http://en.wikipedia.org/wiki/Asymptotic_expansion) and (http://en.wikipedia.org/wiki/Perturbation_methods).
the_dialogue
#5
Jun10-10, 09:59 AM
P: 79
Quote Quote by Coto View Post
Hmm. It's a tough one. I personally don't see how the alpha is coming in either. What's the name of the paper? Is it accessible through web of science?
Here's the link to the article: link

I would say don't get too caught up in where they are coming from. It is more important to understand why they are useful. My guess is the author is deriving a set of bases functions to be used in an expansion of [tex]u(x)[/tex]. The Sturm-Liouville equations can deliver a set of functions which can serve this purpose. The nice properties that these bases functions display make them a good choice when you're expanding the unknown u(x).
The thing is, what are the Sturm Liouville "equations" anyway? What I understand, S-L is a differential equation. What are these equations?
Coto
#6
Jun10-10, 10:44 AM
P: 308
Here is the answer to both of your questions:

http://en.wikipedia.org/wiki/Wave_eq...ace_dimensions

As it applies to you, here is the idea:
Although I'm not sure how the boundary condition is derived for the wave equation, it seems it is a well known result. You'll notice that the problem on the Wikipedia page assumes no forcing and that is why the f(r,t) is absent on the RHS.

Notice that the article also states that "This problem may be solved by expanding f and g in the eigenfunctions of the Laplacian in D, which satisfy the boundary conditions."

In other words applying separation of variables to the wave equation, you'll end up with a form of the Sturm-Liouville equation (for the positional, r, part) whose eigenfunctions can be used to expand u(x) since those eigenfunctions automatically satisfy the BCs.

Lastly, the Sturm-Liouville are a particular set of equations that kept arising when solving some fundamental PDEs by separation of variables. If you recall, separation of variables generally leads to a solution which is an infinite sum. One can consider that these solutions are vectors in function spaces where the bases of the space is determined by the eigenfunctions being used. The Sturm-Liouville equation is a general equation (which encompasses many of the ODE equations found through sep. of var.) which has special properties to its solutions including orthogonality and completeness.

You can read more about it here: http://en.wikipedia.org/wiki/Sturm_liouville
the_dialogue
#7
Jun20-10, 04:36 PM
P: 79
More or less I see the general approach. However there are certain equations that I can't entirely explain.

1. When we say that he solves for another solution 'v' (there are 2 v's floating around which makes this notation confusing) that satisfies the boundary condition, is it correct to say that 'v' satisfies a membrane with no acceleration term (i.e. just the boundary term)? That is what equation 17 seems to suggest.

2. Most importantly I'm stuck with equation 19. He seems to be doing a lot of subtraction that I can't justify. How is he "rewriting" equation 13 using equation 18? I see that equation 18 is essentially = 12, but that's about all I can recognize.

3. This subtraction is again seen in equation 20. Basically from 19 onwards I don't see what he's after. If it were me I would just say "new solution = u + v".

Thanks for the help again!


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