Since my studies currently involve a lot of partial differential equations (PDEs) I thought a brief introduction might be in order. I’ll start by introducing ordinary differential equations (ODEs) first, since they are in many ways very similar.
When you solve an ordinary equation like \( 2x^2 – 3x = 5 \) you want to find the specific values of \(x\) that satisfies the equation. With an ordinary differential equation, the solution is not a specific value for \(x\) but rather a function, say \(u(x)\) which satisfies the equation for all values of \(x\) (or for a specific range). The equation \(u(x)\) has to satisfy relates the derivative(s) of \(u(x)\) to some other function (and possibly itself), hence the name “differential equation”.
An example may help to illustrate the idea. Let’s look at a very simple population growth model. Lets say that in a given year, the ratio of people who reproduce during that year is \(r\). The whole population has \(N(t)\) individuals for a given year t, so the increase in population for year \(t\) is \(rN(t)\). Now let’s assume that the process is continous (after all, people make babies all the time). Then we have that the rate of change in the population at any given time \(t\) is \(rN(t)\). The rate of change in the population is the derivative of \(N(t)\). Using Leibniz’s notation we get the following differential equation: \[\frac{dN(t)}{dt} = rN(t).\] This is just another way of expressing the above. If we solve this differential equation, we find that \(N(t) = N_0 e^{rt}\). Here \(e\) is Euler’s number, and \(N_0\) is the population count at \(t = 0\). So if we say that \(r = 0.1\), and we start with \(N_0 = 10000\) individuals, we find that after 10 years the population count is \(N(10) = 10000*e^{0.1*10} \approx 27183\).
When we solved the above equation, \(N_0\) “magically” appeared in the solution. So where did it come from? Well, if I tell you that my collection of fluffy animals grows steadily with one animal per month and ask if you can figure out when I’ll rearch 500 fluffy animals, there’s no way you can realistically answer that without also knowing something else (like how many fluffy animals I got right now). You need some extra information to “pin down the answer”. This information is usually provided as initial conditions (such as \(N_0\) above) or boundary conditions.
At first glance, a partial differential equation isn’t that much different. Instead of “regular” derivatives, it involves partial derivatives. In the above example, \(N(t)\) is a function of \(t\) alone. However lets say you leave a frozen steak on the kitchen table to defrost, then the temperature inside the steak depends not only on position \(x\) (where you measure it) but on time \(t\) (when you measure it). In math terms the temperature is given by \(T(x, t)\). Now you can measure the change in temperature in several different ways. You could for instance measure the change in temperature at different depths. This would be the partial derivative of the temperature with respect to position \[\pdx{T(x,t)},\]where the \(\partial\) symbol indicates that it’s a partial derivative we’re dealing with instead of a “regular” derivative.
By solving the heat equation one can find the temperature distribution throughout the steak at some arbitrary time after it was put on the table. The heat equation relates the partial derivative of the temperature with respect to time to the second partial derivative of the temperature with respect to position (how much the change in temperature is changing with position):\[\pdt{T(x,t)} = k\pddx{T(x,t)}.\] Again you’ll need some extra information to get some meaningful results. Typically you’ll need the initial temperature of the steak, the temperature in the room (initial conditions) and how the surface of the steak loses heat to the environment (boundary condition). If you’re a true mathematician you’ll just assume the steak is shaped like a perfect cylinder, otherwise you’ll also need to know the shape of the steak.
Depending on the shape of the steak it can be very difficult (or impossible) to find an expression for \(T(x, t)\), which is why the mathematician assumes a simple shape. One way of dealing with this is to try to find an approximate solution, typically using a computer. Though I think I’ll leave the details of that for another post 🙂