What Is a PDE?

A partial differential equation is simply a rule for how something changes when it depends on more than one thing at once, like the temperature $u(x, t)$ of a rod, which depends on the spot $x$ along it (where) and the time $t$ (when). The name comes from partial derivatives: you measure change in one direction at a time while holding the other variable still, the way an ODE measured change in time alone. PDEs are how we describe heat spreading, strings vibrating, sound, fluids and electric fields. An equation like $u_t = k\,u_{xx}$ looks intimidating, but it's just two pieces set equal: $u_t$ (how fast the temperature at one spot is changing) and $k\,u_{xx}$ (how much that spot differs from its neighbors, scaled by how quickly heat spreads). Every equation in this course is a few simple pieces like these, so bear with the notation.

Notation. Subscripts mean partial derivatives: $u_x = \frac{\partial u}{\partial x}$, $u_{xt} = \frac{\partial^2 u}{\partial t\,\partial x}$.

Vocabulary. These words mean what they meant for ODEs:

• Order: the highest derivative that appears.
• Linear: $u$ and its derivatives appear only to the first power, never multiplied together. $u_t = u_{xx}$ is linear; $u_t + u\,u_x = 0$ is not.
• Homogeneous: every term contains $u$; there's no forcing term on its own.

Superposition. If $u_1$ and $u_2$ solve a linear homogeneous PDE, so does $c_1u_1 + c_2u_2$. You've seen this before: it's the same reason $c_1y_1 + c_2y_2$ worked for linear ODEs. The derivative of a sum is the sum of the derivatives, so each piece makes the equation balance to $0$ on its own, and so does the total. Almost everything in this course builds solutions by adding up simple ones.

Conditions. A PDE alone has infinitely many solutions, far more than an ODE, since the freedom is in whole functions rather than a couple of constants. To pin down one answer we also need initial conditions (the state at $t = 0$) and boundary conditions: the value of $u$ (Dirichlet), the value of $u_x$ (Neumann), or a mix (Robin). A problem is well-posed when a solution exists, is unique, and depends continuously on the data. That last part matters in practice: measurements are never exact, and a small error in the data shouldn't wreck the answer.

The three families. The discriminant $B^2 - 4AC$ in the classification above is the same one that sorts conic sections into ellipses, parabolas and hyperbolas, which is where the names come from. The heat equation is parabolic, the wave equation hyperbolic, and Laplace's equation elliptic. Each family behaves in its own way: heat smooths things out, waves travel, and Laplace solutions are as smooth and balanced as possible. There's an animation further down that shows heat and waves side by side, with Laplace's equation as the line the heat settles onto.

Weather and climate models, heat sinks, earthquake waves, MRI physics, option pricing and the rendering of smoke in films are all PDEs solved on computers.