Boundary conditions, Sturm-Liouville, & Gauss Divergence

In summary: MAT-INF2360/v09/undervisningsmateriale/chapter4.pdf)In summary, the author discusses defining boundary conditions for a membrane using a displacement u, boundary movement f, and a normal n. They also present a series expansion using eigenfunctions and eigenvalues from the Sturm-Liouville problem. Additionally, they introduce the Gauss divergence theorem in relation to domain and boundary integrals.
  • #1
the_dialogue
79
0

Homework Statement



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.

Homework Equations



See above.

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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  • #2
the_dialogue said:

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.

the_dialogue said:
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.

the_dialogue said:
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's_theorem
 
  • #3
Coto said:
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.

Coto said:
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?
 
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  • #4
the_dialogue said:
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?

the_dialogue said:
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).
 
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  • #5
Coto said:
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: http://scholar.google.ca/scholar?cluster=12974751515305388156&hl=en&as_sdt=2000"

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 http://mathworld.wolfram.com/Sturm-LiouvilleEquation.html" [Broken]. What are these equations?
 
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  • #6
Here is the answer to both of your questions:

http://en.wikipedia.org/wiki/Wave_equation#Several_space_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
 
  • #7
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!
 

1. What are boundary conditions in scientific equations?

Boundary conditions are constraints placed on the behavior of a physical system at its boundaries or edges. In scientific equations, they specify the values or properties of the system at its boundaries, which are necessary for solving the equations and obtaining a unique solution.

2. What is the Sturm-Liouville equation used for?

The Sturm-Liouville equation is a type of differential equation that is often used in mathematical physics and engineering to model physical phenomena such as heat transfer, fluid flow, and quantum mechanics. It is particularly useful for solving problems involving boundary value conditions.

3. How does the Gauss Divergence theorem relate to boundary conditions?

The Gauss Divergence theorem, also known as the Gauss-Ostrogradsky theorem, is a fundamental theorem in vector calculus that relates the surface integral of a vector field over a closed surface to the volume integral of the divergence of the vector field within the enclosed volume. It can be used in conjunction with boundary conditions to solve problems involving physical systems with varying properties or boundaries.

4. What is the difference between Dirichlet and Neumann boundary conditions?

Dirichlet and Neumann boundary conditions are two types of boundary conditions commonly used in scientific equations. Dirichlet boundary conditions specify the value of the solution at the boundary, while Neumann boundary conditions specify the derivative of the solution at the boundary. In other words, Dirichlet conditions specify the function, while Neumann conditions specify the slope or gradient of the function at the boundary.

5. How are boundary conditions and initial conditions different?

Boundary conditions and initial conditions are both sets of constraints used in solving scientific equations, but they differ in when and where they are applied. Boundary conditions are specified at the edges or boundaries of a physical system, while initial conditions are specified at a specific starting point or time. In other words, boundary conditions dictate the behavior of the system at its boundaries, while initial conditions dictate the behavior of the system at the beginning of the problem.

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