Introduction to PDEs

A partial differential equation (PDE) involves an unknown function of two or more variables and its partial derivatives (14.03). The unknown is usually $u(x, t)$, a quantity at position $x$ and time $t$, or $u(x, y)$ on a region of the plane. Subscripts mean partial derivatives: $u_t = \frac{\partial u}{\partial t}$, $u_{xx} = \frac{\partial^2 u}{\partial x^2}$.

The four classic equations

Name	Equation	Models
transport	$u_t + cu_x = 0$	a pattern carried along at speed $c$
heat (diffusion)	$u_t = ku_{xx}$	temperature in a rod, a spreading dye
wave	$u_{tt} = c^2u_{xx}$	a vibrating string, sound, light
Laplace	$u_{xx} + u_{yy} = 0$	steady temperatures, electrostatics, fluid flow

They extend to more dimensions with the Laplacian $\nabla^2 u = \nabla\cdot\nabla u = u_{xx} + u_{yy} + u_{zz}$, the divergence of the gradient (14.09): $u_t = k\nabla^2 u$, $u_{tt} = c^2\nabla^2 u$, $\nabla^2 u = 0$.

Checking a solution

Substitute, as with ODEs. Does $u = e^{-t}\sin x$ solve the heat equation with $k = 1$? $u_t = -e^{-t}\sin x$ and $u_{xx} = -e^{-t}\sin x$, so $u_t = u_{xx}$. Yes: a sine-shaped temperature that fades.

Arbitrary functions, not constants

ODE solutions have arbitrary constants; PDE solutions have arbitrary functions. If $u(x, y)$ satisfies $u_x = 0$, then $u$ doesn't depend on $x$, so $u = f(y)$ for any function $f$. Similarly $u_{xy} = 0$ gives $u = f(x) + g(y)$.

The transport equation $u_t + cu_x = 0$ is solved by

$$u(x, t) = f(x - ct)$$

for any $f$: check $u_t = -cf'(x - ct)$ and $u_x = f'(x - ct)$. The graph of $f$ slides right at speed $c$ without changing shape. With $u(x, 0) = f(x)$ given, the solution is determined.

Conditions pick out one solution

Because the family of solutions is so large, a physical problem comes with extra information:

• Initial conditions: the state at $t = 0$, like the starting temperature $u(x, 0) = f(x)$. The wave equation, being second order in time, also needs the starting velocity $u_t(x, 0)$.
• Boundary conditions: what happens at the edges, like the ends of a rod held at $0°$: $u(0, t) = u(L, t) = 0$.

Linearity

All four classic equations are linear and homogeneous: if $u_1$ and $u_2$ are solutions, so is $c_1u_1 + c_2u_2$. Superposition is the key to almost every method in this chapter: build complicated solutions by adding simple ones.