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.