Introduction to partial differential equations

An ordinary differential equation involves derivatives with respect to one variable, and its solution is fixed by a handful of numbers — initial position and velocity, say. A partial differential equation involves partial derivatives of a function of several variables (position and time, typically), and that extra dimension changes everything: the general solution contains arbitrary functions, not arbitrary constants, and only gets pinned down to one specific solution once boundary conditions (what happens at the edges of the domain, for all time) and initial conditions (what the whole domain looks like at one instant) are both supplied.

Three named equations are worth recognising by their qualitative behaviour, not just their symbols. The heat equation (∂u/∂t = α∇²u) is diffusive: it takes any initial temperature profile, however jagged, and smooths it out irreversibly as time runs forward — you cannot run a heat-equation solution backward and recover a sharp initial spike. The wave equation (∂²u/∂t² = c²∇²u) is the opposite: it propagates a disturbance, sharp features and all, without decay, and it is time-reversible — a movie of a wave solution run backward is still a valid wave solution. Laplace's equation (∇²u = 0) has no time dependence at all; it describes a steady state, the shape a diffusing or vibrating system settles into once nothing is changing anymore.

Recognising which of these three archetypes a given equation resembles — first order in time and diffusive, second order in time and wave-like, or time-independent — is usually more useful up front than any solution technique, because it tells you what kind of behaviour to expect before you have solved anything.