Need an equation....

[brooksmoses]

Here's a math puzzle that I'd like a solution to -- having an answer to it would make a fairly key part of my thesis work much easier.

I need an function z=f(x, y) for a surface in three-space with the following characteristics:

• The surface is, essentially, a bump; it has a finite maximum at (x,y) = (0,0), and decreases monotonically from there down to zero as the radius increases.
  
• The surface is defined and continuous everywhere.
  
• The first-order derivatives of the surface are defined and continuous everywhere, specifically including the origin.
  
• The surface is radially symmetric; it can be expressed exactly as z=g(x²+y²).
  
• The surface has compact support; there exists some radius R such that, for x²+y²>R, z=0.
  

That, as it stands, is the easy part; it suffices to explain the general character of what I'm looking for. I have one final requirement, which makes this a far more difficult problem:

• The integral of this function f over any given rectangle in xy-space must have a relatively simple closed-form solution. Integrals involving special functions (e.g., Bessel functions) are acceptable if there exists a simple and accurate numerical algorithm for computing them.
  

Suggestions welcomed. Needless to say, a proof that a solution does not exist will also be useful.

Comments

[larksdream]

What are you working on?

[brooksmoses]

Ah, the difficult question. I've been working on this project for five years; you'd would think I'd have found a good way to explain it by then, but apparently not....

The short version -- i.e., just this little bit of the project -- is that I'm working on programming some code that applies forces in a fluid-flow simulation. Instead of applying them all at a single point, I want to have the force spread out over an area that includes several grid points of the simulation. That's where the "bump" comes in; it's the shape of the force distribution. I want to be able to easily integrate it, because if I can do that, I can make a bump with a total integral equal to the amount of force I want to apply, and then I can assign each grid point to a rectangle surrounding it (in such a way that all the rectangles are next to each other and cover the whole area once), and give the point a force corresponding to the integral of over it's little rectangle, and then the sum of all of the bits of forces that I put on the grid points will exactly equal the total amount of force I want to apply. There are other ways to get similar results, but they don't seem nearly as elegant.

[kiya]

*absconds with that apostrophe*

[brooksmoses]

While you're editing my explanation, could you edit in a "the bump" after the "integral of" just before the misplaced apostrophe? I seem to have left it out.

[kiya]

*peers* Indeed.

At least it wasn't a parenthesis. That would require drastic measures.

[(Anonymous)]

It's been ages since I've done any real math, so I'm probably missing
something, but wouldn't this work?

f(x,y) = 1/4 + (x² + y²)²/4 - (x² + y²)/2 for x² + y² <= 0
f(x,y) = 0 for x² + y² >= 0

[brooksmoses]

Hmm. After poking at that in Mathematica a bit to evaluate the integrals, I can confirm that, yes, that does indeed seem to work. (Well, ok, it doesn't work as written because the limit should be (x²+y²)=1 rather than (x²+y²)=0, but close enough.)

Writing the code to do all the special cases for the (x²+y²)=1 arc intersecting the rectangle isn't going to be easy, I suppose, but then it wasn't ever going to be -- and it is at least quite possible.

Thanks muchly, masked commenter, whomever you are!

[(Anonymous)]

Oops, typo. I was thinking "0 outside the circle" and wrote 0 instead of 1.

I'm a friend of Aga's.

[larksdream]

Dangit, another nameless friend. I'm sorry, Brooks, I promise to start assigning them names as soon as I think of some. ;-)

[(Anonymous)]

It's Dwight. Who else would get the hard part right and then
write "for x² + y² <= 0"?

[larksdream]

But... but you're not getting comments emailed to you. Please tell me you're not sitting there reloading this page every five minutes? *LOL*

[larksdream]

Actually I thought it might be you, because 1) it sounded like you, and 2) you just emailed me, and 3) you've either read my journal recently, or installed tiny X10 cameras I haven't yet found. ;-)

(I need to do something about this bad habit I have of hijacking other people's posts. *g*)

[kiya]

(I need to do something about this bad habit I have of hijacking other people's posts. *g*)

Hey, the hijack of my feeesheee post wasn't entirely you.

It wasn't even mostly you!

[larksdream]

This is true. I can only plead the extenuating circumstance of actually working on my research today. ;-)

It's so cozy, having everyone to talk to at 3 in the morning. (Well, Brooks doesn't get the virtu points because it's only midnight for him, but still.)

[kiya]

Well, I'm working. Right now, even.

And I'm not cleaning the bathtub.

[brooksmoses]

One imagines that it might be rather difficult to post to LJ while cleaning the bathtub. This also seems rather unfortunate, as if it were possible it might make cleaning the bathtub seem a less unappealing chore.

[kiya]

There are no paper towels, which makes cleaning the bathtub even less an appealing chore than it might otherwise be.

Section 104 is perturbingly exact in its four hundred wordsness.

[brooksmoses]

I can verify that there do exist paper towels, although I suppose that having some here is not particularly useful.

Last time I cleaned a bathtub, though, I mostly used an old sponge. Also salad oil and dish soap, but that's a different story -- a story of gooey soap scum that resisted all commercial cleansers I had to hand, to be specific, but the salad oil got it off reasonably nicely, and then the dish soap and hot water did quite well at getting the salad oil off.

[kiya]

There are no accessible paper towels.

There's enough crud (much of it my hair) in/on the tub that I would really like to be able to scrape it off and throw it away without having to think about it.

*ponders at 105 and whether she's awake enough to write it*

[brooksmoses]

Yeah, I can see the advantage of being able to have disposable scrapers -- i.e., paper towels -- in that sort of situation.

I think I'm going to sleep pretty soon, though, as I need to be up reasonably early tomorrow....

[kiya]

I think this headache is pretty much confirming that whole "sleep" notion.

[brooksmoses]

Yeah; goodnight.... Hope the headache goes away by morning.

[kiya]

Metoo.

Agh.

[larksdream]

I think the key to bathtubs is doing it often enough that the scum says oppressed. A friend of mine greatly favors that daily shower spray thing, but at this point, when I consider trying to remember to spray the bathtub every day, all I can think of is, Oh gods, the stress!! Which is probably not a very good sign. *g*

[brooksmoses]

Well, you were not only working on your research, you were somehow enticing Dwight to work on my research. For which I do muchly thank you.

[larksdream]

*bows* I am always pleased to be able to offer the labor of others. ;)

[brooksmoses]

Many of us, I suspect. Me, for one. :)