The End 2016 Mathematics A To Z: Boundary Value Problems

I went to a grad school, Rensselaer Polytechnic Institute. The joke at the school is that the mathematics department has two tracks, “Applied Mathematics” and “More Applied Mathematics”. So I got to know the subject of today’s A To Z very well. It’s worth your knowing too.

Boundary Value Problems.

I’ve talked about differential equations before. I’ll talk about them again. They’re important. They might be the most directly useful sort of higher mathematics. They turn up naturally whenever you have a system whose changes depend on the current state of things.

There are many kinds of differential equations problems. The ones that come first to mind, and that students first learn, are “initial value problems”. In these, you’re given some system, told how it changes in time, and told what things are at a start. There’s good reasons to do that. It’s conceptually easy. It describes all sorts of systems where something moves. Think of your classic physics problems of a ball being tossed in the air, or a weight being put on a spring, or a planet orbiting a sun. These are classic initial value problems. They almost look like natural experiments. Set a thing up and watch what happens.

They’re not everything. There’s another class of problems at least as important. Maybe more important. In these we’re given how the parts of a system affect one another. And we’re told some information about the edges of the system. The boundaries, that is. And these are “boundary value problems”.

Mathematics majors learn them after getting thoroughly trained in and sick of initial value problems. There’s reasons for that. First is that they almost need to be about problems with multiple variables. You can set one up for, like, a ball tossed in the air. But they’re rarer. Differential equations for multiple variables are harder than differential equations for a single variable, because of course. We have to learn the tools of “partial differential equations”. In these we work out how the system changes if we pretend all but one of the variables is fixed. We combine information about all those changes for each individual changing variable. Lots more, and lots stranger, stuff can happen.

The partial differential equation describes some region. It involves maybe some space, maybe some time, maybe both. There’s a region, called the “domain”, for which the differential equation is true.

For example, maybe we’re interested in the amount of heat in a metal bar as it’s warmed on one end and cooled on another. The domain here is the length of the bar and the time it’s subjected to the heat and cool. Or maybe we’re interested in the amount of water flowing through a section of a river bed. The domain here is the length and width and depth of the river, if we suppose the river isn’t swelling or shrinking or changing much. Maybe we’re intersted in the electric field created by putting a bit of charge on a metal ball. Then the domain is the entire universe except the metal ball and the space inside it. We’re comfortable with boundlessly large domains.

But what makes this a boundary value problem is that we know something about the boundary looks like. Once again a mathematics term is less baffling than you might figure. The boundary is just what it sounds like: the edge of the domain, the part that divides the domain from not-the-domain. The metal bar being heated up has boundaries on either end. The river bed has boundaries at the surface of the water, the banks of the river, and the start and the end of wherever we’re observing. The metal ball has boundaries of the ball’s surface and … uh … the limits of space and time, somewhere off infinitely far away.

There’s all kinds of information we might get about a boundary. What we actually get is one of four kinds. The first kind is “we get told what values the solution should be at the boundary”. Mathematics majors love this because it lets us know we at least have the boundary’s values right. It’s certainly what we learn on first. And it might be most common. If we’re measuring, say, temperature or fluid speed or something like that we feel like we can know what these are. If we need a name we call this “Dirichlet Boundary Conditions”. That’s named for Peter Gustav Lejune Dirichlet. He’s one of those people mathematics majors keep running across. We get stuff named for him in mathematical physics, in probability, in heat, in Fourier series.

The second kind is “we get told what the derivative of the solution should be at the boundary”. Mathematics majors hate this because we’re having a hard enough time solving this already and you want us to worry about the derivative of the solution on the boundary? Give us something we can check, please. But this sort of boundary condition keeps turning up. It comes up, for instance, in the electric field around a conductive metal box, or ball, or plate. The electric field will be, near the metal plate, perpendicular to the conductive metal. Goodness knows what the electric field’s value is, but we know something about how it changes. If we need a name we call this “Neumann Boundary Conditions”. This is not named for the applied mathematician/computer scientist/physicist John von Neumann. Nobody remembers the Neumann it is named for, who was Carl Neumann.

The third kind is called “Robin boundary conditions” if someone remembers the name for it. It’s slightly named for Victor Gustave Robin. In these we don’t necessarily know the value the solution should have on the boundary. And we don’t know what the derivative of the solution on the boundary should be. But we do know some linear combination of them. That is, we know some number times the original value plus some (possibly other) number times the derivative. Mathematics majors loathe this one because the Neumann boundary conditions were hard enough and now we have this? They turn up in heat and diffusion problems, when there’s something limiting the flow of whatever you’re studying into and out of the region.

And the last kind is called “mixed boundary conditions” as, I don’t know, nobody seems to have got their name attached to it. In this we break up the boundary. For some of it we get, say, Dirichlet boundary conditions. For some of the boundary we get, say, Neumann boundary conditions. Or maybe we have Robin boundary conditions for some of the edge and Dirichlet for others. Whatever. This mathematics majors get once or twice, as punishment for their sinful natures, and then we try never to think of them again because of the pain. Sometimes it’s the only approach that fits the problem. Still hurts.

We see boundary value problems when we do things like blow a soap bubble using weird wireframes and ponder the shape. Or when we mix hot coffee and cold milk in a travel mug and ponder how the temperatures mix. Or when we see a pipe squeezing into narrower channels and wonder how this affects the speed of water flowing into and out of it. Often these will be problems about how stuff over a region, maybe of space and maybe of time, will settle down to some predictable, steady pattern. This is why it turns up all over applied mathematics problems, and why in grad school we got to know them so very well.

The Set Tour, Part 11: Doughnuts And Lots Of Them

I’ve been slow getting back to my tour of commonly-used domains for several reasons. It’s been a busy season. It’s so much easier to plan out writing something than it is to write something. The usual. But one of my excuses this time is that I’m not sure the set I want to talk about is that common. But I like it, and I imagine a lot of people will like it. So that’s enough.

T and Tⁿ

T stands for the torus. Or the toroid, if you prefer. It’s a fun name. You know the shape. It’s a doughnut. Take a cylindrical tube and curl it around back on itself. Don’t rip it or fold it. That’s hard to do with paper or a sheet of clay or other real-world stuff. But we can imagine it easily enough. I suppose we can make a computer animation of it, if by ‘we’ we mean ‘you’.

We don’t use the whole doughnut shape for T. And no, we don’t use the hole either. What we use is the surface of the doughnut, the part that could get glazed. We ignore the inside, just the same way we had S represent the surface of a sphere (or the edge of a circle, or the boundary of a hypersphere). If there is a common symbol for the torus including the interior I don’t know it. I’d be glad to hear if someone had.

What good is the surface of a torus, though? Well, it’s a neat shape. Slice it in one direction, the way you’d cut a bagel in half, and at the slice you get the shape of a washer, the kind you fit around a nut and bolt. (An annulus, to use the trade term.) Slice it perpendicular to that, the way you’d cut it if you’re one of those people who eats half doughnuts to the amazement of the rest of us, and at the slice you get two detached circles. If you start from any point on the torus shape you can go in one direction and make a circle that loops around the doughnut’s central hole. You can go the perpendicular direction and make a circle that brushes up against but doesn’t go around the central hole. There’s some neat topology in it.

There’s also video games in it. The topology of this is just like old-fashioned video games where if you go off the edge of the screen to the right you come back around on the left, and if you go off the top you come back from the bottom. (And if you go off to the left you come back around the right, and off the bottom you come back to the top.) To go from the flat screen to the surface of a doughnut requires imagining some stretching and scrunching up of the surface, but that’s all right. (OK, in an old video game it was a kind-of flat screen.) We can imagine a nice flexible screen that just behaves.

This is a common trick to deal with boundaries. (I first wrote “to avoid having to deal with boundaries”. But this is dealing with them, by a method that often makes sense.) You just make each boundary match up with a logical other boundary. It’s not just useful in video games. Often we’ll want to study some phenomenon where the current state of things depends on the immediate neighborhood, but it’s hard to say what a logical boundary ought to be. This particularly comes up if we want to model an infinitely large surface without dealing with infinitely large things. The trick will turn up a lot in numerical simulations for that reason. (In that case, we’re in truth working with a numerical approximation of T, but that’ll be close enough.)

Tⁿ, meanwhile, is a vector of things, each of which is a point on a torus. It’s akin to Rⁿ or S^{2 x n}. They’re ordered sets of things that are themselves things. There can be as many as you like. n, here, is whatever positive whole number you need.

You might wonder how big the doughnut is. When we talked about the surface of the sphere, S², or the surface and interior, B³, we figured on a sphere with radius of 1 unless we heard otherwise. Toruses would seem to have two parameters. There’s how big the outer diameter is and how big the inner diameter is. Which do we pick?

We don’t actually care. It’s much the way we can talk about a point on the surface of a planet by the latitude and longitude of the point, and never care about how big the planet is. We can describe a point on the surface of the torus without needing to refer to how big the whole shape is or how big the hole in the middle is. A popular scheme to describe points is one that looks a lot like latitude and longitude.

Imagine the torus sitting as flat as it gets on the table. Pick a point that you find interesting.

We use some reference point that’s as good as an equator and a prime meridian. One coordinate is the angle you make going horizontally, possibly around the hole in the middle, from the reference point to the point we’re interested in. The other coordinate is the angle you make vertically, going in a loop that doesn’t go around the hole in the middle, from the reference point to the point we’re interested in. The reference point has coordinates 0, 0, as it must. If this sounds confusing it’s because I’m not using a picture. I thought making some pictures would be too much work. I’m a fool. But if you think of real torus-shaped objects it’ll come to you.

In this scheme the coordinates are both angles. Normal people would measure that in degrees, from 0 to 360, or maybe from -180 to 180. Mathematicians would measure as radians, from 0 to 2π, or from -π to +π. Whatever it is, it’s the same as the coordinates of a point on the edge of the circle, what we called S¹ a few essays back. So it’s fair to say you can think of T as S¹ x S¹, an ordered set of points on circles.

I’ve written of these toruses as three-dimensional things. Well, two dimensional-surfaces wrapped up to suggest three-dimensional objects. You don’t have to stick with these dimensions if you don’t want or if your problem needs something else. You can make a torus that’s a three-dimensional shape in four dimensions. For me that’s easiest to imagine as a cube where the left edge and the right edge loop back and meet up, the lower and the upper edges meet up, and the front and the back edges meet up. This works well to model an infinitely large space with a nice and small block.

I like to think I can imagine a four-dimensional doughnut where every cross-section is a sphere. I may be kidding myself. There could also be a five-dimensional torus and you’re on your own working that out, or working out what to do with it.

I’m not sure there is a common standard notation for that, though. Probably the mathematician wanting to make clear she’s working with a torus in four dimensions just says so in text, and trusts that the context of her mathematics makes it clear this is no ordinary torus.

I’ve also written of these toruses as circular, as rounded shapes. That’s the most familiar torus. It’s a doughnut shape, or an O-ring shape, or an inner tube’s shape. It’s the shape you produce by taking a circle and looping it around an axis not on the ring. That’s common and that’s usually all we need.

But if you need some other torus, produced by rotating some other shape around an axis not inside it, go ahead. You’ll need to make clear what that original shape, the generator, is. You’ve seen examples of this in, for example, the washers that fit around nuts and bolts. They’re typically rectangles in cross-section. Or you might have seen that image of someone who fit together a couple dozen iMac boxes to make a giant wheel. I don’t know why you would need this, but it’s your problem, not mine. If these shapes are useful for your work, by all means, use them.

I’m not sure there is a standard notation for that sort of shape. My hunch is to say you’d define your generating shape and give it a name such as A or D. Then name the torus based on that as T(A) or T(D). But I would recommend spelling it out in text before you start using symbols like this.

The Set Tour, Part 9: Balls, Only The Insides

Last week in the tour of often-used domains I talked about Sⁿ, the surfaces of spheres. These correspond naturally to stuff like the surfaces of planets, or the edges of surfaces. They are also natural fits if you have a quantity that’s made up of a couple of components, and some total amount of the quantity is fixed. More physical systems do that than you might have guessed.

But this is all the surfaces. The great interior of a planet is by definition left out of Sⁿ. This gives away the heart of what this week’s entry in the set tour is.

Bⁿ

Bⁿ is the domain that’s the interior of a sphere. That is, B³ would be all the points in a three-dimensional space that are less than a particular radius from the origin, from the center of space. If we don’t say what the particular radius is, then we mean “1”. That’s just as with the Sⁿ we meant the radius to be “1” unless someone specifically says otherwise. In practice, I don’t remember anyone ever saying otherwise when I was in grad school. I suppose they might if we were doing a numerical simulation of something like the interior of a planet. You know, something where it could make a difference what the radius is.

It may have struck you that B³ is just the points that are inside S². Alternatively, it might have struck you that S² is the points that are on the edge of B³. Either way is right. Bⁿ and S^n-1, for any positive whole number n, are tied together, one the edge and the other the interior.

Bⁿ we tend to call the “ball” or the “n-ball”. Probably we hope that suggests bouncing balls and baseballs and other objects that are solid throughout. Sⁿ we tend to call the “sphere” or the “n-sphere”, though I admit that doesn’t make a strong case for ruling out the inside of the sphere. Maybe we should think of it as the surface. We don’t even have to change the letter representing it.

As the “n” suggests, there are balls for as many dimensions of space as you like. B² is a circle, filled in. B¹ is just a line segment, stretching out from -1 to 1. B³ is what’s inside a planet or an orange or an amusement park’s glass light fixture. B⁴ is more work than I want to do today.

So here’s a natural question: does Bⁿ include S^n-1? That is, when we talk about a ball in three dimensions, do we mean the surface and everything inside it? Or do we just mean the interior, stopping ever so short of the surface? This is a division very much like dividing the real numbers into negative and positive; do you include zero among other set?

Typically, I think, mathematicians don’t. If a mathematician speaks of B³ without saying otherwise, she probably means the interior of a three-dimensional ball. She’s not saying anything one way or the other about the surface. This we name the “open ball”, and if she wants to avoid any ambiguity she will say “the open ball Bⁿ”.

“Open” here means the same thing it does when speaking of an “open set”. That may not communicate well to people who don’t remember their set theory. It means that the edges aren’t included. (Warning! Not actual set theory! Do not attempt to use that at your thesis defense. That description was only a reference to what’s important about this property in this particular context.)

If a mathematician wants to talk about the ball and the surface, she might say “the closed ball Bⁿ”. This means to take the surface and the interior together. “Closed”, again, here means what it does in set theory. It pretty much means “include the edges”. (Warning! See above warning.)

Balls work well as domains for functions that have to describe the interiors of things. They also work if we want to talk about a constraint that’s made up of a couple of components, and that can be up to some size but not larger. For example, suppose you may put up to a certain budget cap into (say) six different projects, but you aren’t required to use the entire budget. We could model your budgeting as finding the point in B⁶ that gets the best result. How you measure the best is a problem for your operations research people. All I’m telling you is how we might represent the study of the thing you’re doing.