Engineers have to make decisions during the design process. Nearly all of these decisions are practical ones, and they are the result of certain constraints imposed upon the project by various outside requirements. That’s basically what the bulk of engineering is: solving problems in the presence of practical constraints. I like to refer to these constraints as ‘boundary conditions’ — a term borrowed from differential calculus, but surprisingly apt.
Differential equations are simple and elegant things. They describe the behavior of phenomena in the most ideal form. In that ideal form they are easy to understand and solve, once you comprehend the language used to express them. But for solving practical problems they are, in and of themselves, a non-starter. In order to make them useful to us, we have to apply boundary conditions to describe the particulars of the situation we wish to know more about.
The heat equation is a great example. It describes the distribution of thermal energy in some region over time, and it does a really good job of that. But what if we don’t want to just know the general rule for heat distribution in the universe — we want to know how heat will be distributed along a motorcycle exhaust pipe from manifold to tip. Specifically, we want to know what the temperature of the pipe will be at the point where it crosses the rider’s leg, 10 minutes after the engine is started. Enter boundary conditions. The boundary conditions describe the shape and contours of the pipe, and the thermal properties of the steel and chrome used to make it. We then take this mathematical description of the pipe and stick it into the heat equation to it to get the answer to our question.
Except it’s not as simple as just substituting one function into another and solving for X. There is no single X. The whole damn thing is X. The heat equation is a differential equation, which means the value at one point is based on the values at all the other points over time, so you have to solve the whole thing at once. It gets very involved very quickly. I won’t get in to the nuts and bolts of how this is done — basically you start out solving for a handful of points, and then keep splitting things into smaller and smaller pieces and solving those. It’s a conceptually simple, but arithmetically intense process — a lot of grunt work. (now easily performed by computers, but still).continue reading