(adsbygoogle = window.adsbygoogle || []).push({}); 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 begreatlyappreciated!

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# Homework Help: Boundary conditions, Sturm-Liouville, & Gauss Divergence

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