# Boundary conditions, Sturm-Liouville, & Gauss Divergence

 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: $$\newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } } \pd{u(\mathbf{r},t)}{n}{}+\alpha(\mathbf{r})u(\mathbf{r},t)=f(\mathbf{r },t)$$ 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 $$\phi_m(\mathbf{r})$$ with eigenvalues 'lambda'. He states that the eigenfunction is derived from the solution to the Sturm-Liouville problem: $$(\nabla^2 + \lambda_m^2)\phi_m(\mathbf{r})=0$$, on the domain of the membrane & $$\newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } } \pd{\phi_m(\mathbf{r},t)}{n}{}+\alpha(\mathbf{r})\phi(\mathbf{r},t)=0$$, on the boundary of the membrane 3. The author presents the Gauss divergence theorem in a way I'm not too familiar with. $$\newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } } \int\int_{D}^{}(\phi_m\nabla^2u-u\nabla^2\phi_m)dS=\int\int_{B}^{}(\phi_m\pd{u}{n}{}-u\pd{\phi_m}{n}{})dC$$ 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!
P: 308
 Quote by the_dialogue 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 by the_dialogue 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 $$\phi_m$$'s.

 Quote by the_dialogue 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
P: 79
 Quote by Coto 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 by Coto 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 $$\phi_m$$'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?

P: 308
Boundary conditions, Sturm-Liouville, & Gauss Divergence

 Quote by the_dialogue 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 by the_dialogue 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 $$u(x)$$. 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).
P: 79
 Quote by Coto 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?
 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 $$u(x)$$. 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).