Partial differential equations

PDE type predicts information transport

A PDE encodes how a field evolves across space and time. Transport carries profiles, diffusion smooths them, and waves transmit disturbances without immediate smoothing; the highest derivatives reveal the qualitative behavior.

Compare canonical equations

Advection: u_t + c u_x = 0 transports a profile at speed c. Diffusion: u_t = αu_xx smooths curvature, with α units m²/s. Wave motion: u_tt = c²u_xx requires two time conditions and preserves oscillatory structure. The units check forces c to m/s and α to m²/s.

A temperature disturbance over L = 0.1 m with α = 1×10⁻⁵ m²/s diffuses on t ~ L²/α = 1000 s. A wave at c = 2 m/s crosses the same length in 0.05 s.

Real equations can mix behaviors

Navier–Stokes has advection and diffusion; damped waves have propagation and loss. Boundary conditions can dominate the observed solution even when the local PDE is unchanged.

Work it: For u_t + 3u_x = 0, write the profile at t = 2 s if u(x,0) = e^{−x²}. For u_t = 0.5u_xx, state qualitatively what curvature at a peak does.