PDEs: Explain it like I'm 5

 This post is for Mum who asked for an r/explainlikeim5 style post on what I study. My PhD is very closely related to my Masters project (which will get its own post soon) but I want to talk about the broader field of which this is a part.

There are quantities which depend on one thing: the easiest example is a population depending on time. A differential equation is a relationship between something's growth rate and how its size. In a population that has some reproduction rate, the sentence 'the larger the population, the faster it grows' encodes a differential equation.  Another example is radioactive decay: if I have a block of uranium then after some amount of time, half of it will have decayed. Then after another half-life, another half will decay. This is the same as the population example, but now 'the larger the size, the faster it decays'. These are the two simplest differential equations we can describe, both written as y' = ky.

There are myriad other examples where we measure something depending on one other thing, usually depending on time. For example a pendulum swings and the angle it makes depends on time. Or there's a rocket flying away from earth which burns fuel to propel itself and we can talk about its acceleration changing as its mass decreases. Another example is if I punch a hole in the bottom of a vat and in I pour in squash at a fixed rate at the top: what is the concentration of the squash as time evolves?

In all of these examples the quantity we're measuring (population, distance, concentration, angle, mass) depends on only one number (in every case, time). For this reason they are all examples of ordinary differential equations. But these are all quite simple systems and in general things depend on more than one thing. A relationship between a quantity and it's dependence on multiple different things is called a partial differential equation. Not because it's any easier to solve, that's some unfortunate nomenclature, but because the derivatives are only partial, i.e., they only take into account change with respect to one variable. My best example of this is the height of a string or the temperature along a rod. Both of these numbers depend on both my position along the rod/string and the time that I'm there.

The beauty of PDEs is the breadth of situations they encapsulate. The physical scale is astounding: Schrödinger's equation describes the time evolution of quantum states of subatomic particles. Einstein's equations describe the relationship between energy, momentum and the curvature of space time on a cosmological scale. The Navier-Stokes equations (which I've now written 3 posts on) can be adapted both for the swimming of microorganisms or the flow of plasma comprising stars. PDEs range from the concrete, such as the Black-Scholes equation which describes the price evolution of financial derivatives, to the abstract, such as the Euler-Lagrange equations which state that every physical path is a minimizer of some action. There is a unifying aspect too: the same equation describes the diffusion of a pollutant in a body of water and the diffusion of heat throughout a body. The same equation [[Laplace's equation]] describes incompressible, irrotational fluid flow, electromagnetic field configurations and the stable configurations of an elastic membrane. Heat and wave evolutions are unified through the Helmholtz equation (more on this later ;)).

But it is not the applications that interest me: mathematically, these are interesting objects to study in their own right. They are devilishly difficult to solve in generality (see the Navier-Stokes post) but we can still say so much about the solutions without knowing what they are. The tools we need to tackle them vary massively, taking ideas from functional analysis and applying them to physically motivated situations.