Partial differential equations. Part I: Waves

PDE’s play a big part in mathematics, physics, and even machine learning and AI

Aside from linear algebra and analysis of all flavors, math students need to learn how to solve, think about, and interpret differential equations. Differential equations describe the world around us, and they make use of the fact that even if don’t know what the original function looks like, we can understand it through its derivatives.

Derivatives are very useful. Let’s look at acceleration. By definition, it is the rate of change of the velocity of some object with respect to time. We can write it down in the following way:

We may not know the original velocity of an object v or its position x, but if we understand that acceleration is the first derivative of velocity with respect to time, and the second derivative of position with respect to time, we can reverse engineer the problem, integrate and restore this information. Pretty cool.

Partial differential equations or PDE’s are a little trickier than that, but because they are tricky, they are very powerful. A PDE is an identity that relates the partial derivatives of a function (let’s call it u), and the function u itself and the independent variables. If in ODE’s (ordinary differential equations) the function depends only on a single variable, PDE’s depend on multiple variables.

There are multiple examples of PDE’s, but the most famous ones are wave equation, heat equation, and Schrödinger equation. We’ll look at all of them, in order.

First, wave equation. It was formulated in 1740’s by a French Jean-Baptiste le Rond d’Alembert, and it touches the problem of a so-called vibrating string (think guitar string). But wave equation is useful for studying waves of all sorts and kinds, not just vibrating strings: water waves, sound waves, seismic waves, light waves. It is usually written in one the following ways:

Here c is a constant describing the propagation speed (it should be greater than zero). The ‘indices’ (tt and xx) describe the fact that the second derivative with respect to time on the left and with respect to x on the right is taken. And of course, since we’re talking about partial differential equations, u is a function of two variables, both x and t.

Just writing down the equation (relations between functions, derivatives, etc.) is insufficient if we are talking about finding a unique solution. We need something called initial condition and boundary condition. We say that we need to solve a wave equation (first line) subject to (second and third line):

Don’t ask me why these functions of x are called Phi and Psi, it is just common to name them this way. The key observation made by d’Alembert, who actually found a solution formula (named after him, of course), is that we are actually dealing with a superposition of traveling waves — to the right (x+t) and to the left (x-t). Here is how it looks:

Since the functions Phi and Psi are given as part of initial value problems, solving wave equation essentially boils down to computing an integral after plugging in the values into the d’Alembert’s formula. Here’s an example. Assume that u(x, 0)=cos(x) and its first derivative with respect to time is equal to 0:

Then we can just plug in cos(x) for Phi and and zero for Psi, and solve:

One more example. Assume the same initial value problem, but this time Phi is 0 and Psi is cos(x):

Then solution is given by:

These are the basics of the wave equation and how it can be solved using d’Alembert’s formula. If the functions Phi and Psi are not too complicated, solution to this PDE is fairly straightforward and simple. One little note: we already know that to compute the function u it is enough to know the values of Phi and Psi on an interval (x-t, x+t). And if Phi and Psi vanish on some interval, say (a, b), then the function u(x, t) also vanishes for x on an interval (a+t, b-t). This is called causality principle.

Next time we’ll talk about more complicated PDE’s, such as heat equation and Schrödinger equation.