PDEs

A Partial Differential Equation (PDE) involves an unknown function of multiple independent variables and its partial derivatives. While ODEs describe change along a single dimension (usually time), PDEs describe phenomena that vary across space AND time — like the temperature in a room, the vibration of a drumhead, or the flow of air around an airplane wing. The "Big Three" PDEs that appear across all of physics are the heat equation (diffusion), the wave equation (vibration), and the Laplace equation (steady-state equilibrium).

One of the most powerful techniques for solving PDEs is separation of variables. You assume the solution can be written as a product of functions, each depending on only one variable. For the wave equation, assume the solution u of x and t equals the product X of x times T of t. Substituting this into the PDE splits it into two separate ODEs that can be solved independently. The general solution is then a Fourier series — an infinite sum of these separated solutions. This method works for many boundary value problems on simple domains like rectangles, circles, and spheres.

Most real-world PDEs cannot be solved analytically and require numerical methods. The Finite Difference Method (FDM) approximates derivatives by difference quotients on a discrete grid. The Finite Element Method (FEM) divides the domain into small elements (often triangles) and solves a weak form of the equation on each element. The Finite Volume Method (FVM) conserves fluxes across cell boundaries and is preferred for fluid dynamics. These methods power weather prediction, aircraft design, semiconductor simulation, and seismic imaging.

See how waves travel and spread over time. Adjust the wave speed to observe how it changes the propagation pattern.