Last week my sister-in-law's daughter (does that make her my niece?) sent out an emergency broadcast looking for help on solving one of the questions on her take home final. She had been through three Indian tutors (online from India) and was waiting on a reply from a fourth, and she still had no clue on how to deal with this problem.
I took calculus in college and I would have gone on to take differential equations, but I ran into some BS that I wasn't going to deal with. The last bit of calculus was supposed to be an introduction to differential equations. For this we were supposed to memorize a bunch of really complicated equations, with no explanation of what they were for or how they worked. I just don't work like that. I want concepts and understanding. This "memorize or die approach" turned me off, so that was the end my training in mathematics.
Anyway, I heard my niece's cry for help, and while I probably would not be able to help her, I might know someone who could, and in any case I wanted to see just what this intractable problem looked like. Here it is:
Consider a rectangular plate, of size d in the y direction and size l in the x direction. At y=d the temperature is held fixed at f(x). On all other sides the plates temperature is held at 0. The plate is initially 0, everywhere. Solve du/dt = k(delta*u), 0<x<l, 0<y<d. u(0,y,t)=0, u(l,y,t)=0, u(x,0,t)=0, u(x,d,t)=f(x)=-2sin(PI*x/l)+sin(3*PI*x/l), u(x,y,0)=0.
This is how I interpret the problem (O is for an Original statement, M is for My interpretation):
O: Consider a rectangular plate, of size d in the y direction and size l in the x direction.
M: We have a thin, rectangular metal plate.
O: At y=d the temperature is held fixed at f(x).
M: The temperature along the top edge is held at a temperature that is determined by some mysterious device that depends on how far along you are from the top left hand corner. We don't know what the temperature is exactly, any kind of mysterious device may be used. It may heat the edge evenly, or at constantly increasing rate, or it may heat it more in the middle and less on the ends, or it may have apply heat randomly. In any case, once you have picked your mysterious device, it continues to apply the same amount of heat to the same locations.
O: On all other sides the plates temperature is held at 0.
M: The other three edges of our plate are in contact with refrigerated heat sinks that suck all the heat out of those three edges of the plate, holding their temperature to zero.
O: The plate is initially 0, everywhere.
M: When we start, the temperature of the entire plate is zero.
O: Solve du/dt = k(delta*u),
M: This is the crux of the problem. I think what they are asking for is a formula to determine the rate of change in temperature. du/dt is shorthand for change in temperature over change in time, i.e. how fast the temperature is rising. Delta is similar in that it represents change. So it looks like they are asking if the temperature changes by a certain factor, they want to know how fast it is changing when that happens, which doesn't really make a lot of sense, but that's probably because I don't really understand what they are talking about.
O: 0<x<l, 0<y<d.
M: These are the ranges for which we need to find solutions, i.e. we are only interested in the temperature of the various parts of the plate, we are not interested in anything outside the plate.
O: u(0,y,t)=0, u(l,y,t)=0, u(x,0,t)=0,
M: This just says that the temperature of the other three edges are held to zero, which we already know. We have a function u that takes three parameters: the X and Y coordinates of a position on the plate, and the Time. It doesn't matter what time it is, as long as the coordinates place you along one of the three refrigerated edges, the temperature will be zero.
M: This is the formula that models our "mysterious device". This tells us what the temperature is along the top edge of the plate.
M: This just says that the initial temperature of the plate is zero, something else we already knew.
My brother Andy, world wide web crawler that he is, found an MIT website that also has some good math stuff
. Here's a link to a 15 page PDF of Solutions to Problems for 2D & 3D Heat and Wave Equations
, which look kind of similar. I asked my friends and three of them admitted to studying differential equations in college, but none were able to recall enough to be helpful. Stu recognized it as being a Sturm-Liousville problem, whatever that is.
My niece eventually figured it out with some help from her friends:
This problem is actually more advanced than just differential equations. In fact, I don't even think I learned the way to solve it in my math class this semester, but there is a trick that one needs to know and it makes the whole problem easier. The trick is that u(x,y,0)=0 implies that it is a steady state. This means du/dt = 0. So then you only need to solve one side of the problem, cause the other side goes to zero. My teacher always put things like this on the tests, cause he likes these kinds of tricks. Anyway, I hope you can learn more about all this stuff. It's complicated, but it's awesome.
For those of you who waded through all this, here is graph of a temperature gradient from a similar problem in the PDF from MIT.
I've been looking over some of this, and I think the thing that bothers me about it the most is that there doesn't seem to be much correlation between the concepts being dealt with and the symbols being written. You take algebra or geometry and you have equations that can be easily converted into instructions for a calculator or a computer program, or can even be worked by hand. The handwritten expressions of integrals and differentials are not so easily translated. Even just translating them into English can be a formidable challenge.
P.S. You wouldn't believe how much trouble I had putting in less than signs, or I don't know, maybe you would. Blogger is so stupid sometimes. I may have to move to Wordpress or something if I am going to keep this up.
Update October 2016 replaced missing graph.