The three canonical PDEs

Heat equation (parabolic): $\partial_t u = \kappa \nabla^2 u$ — diffusion, probability densities

Wave equation (hyperbolic): $\partial_{tt} u = c^2 \nabla^2 u$ — acoustics, seismics, electromagnetism

Laplace / Poisson equation (elliptic): $\nabla^2 u = 0$ (or $= f$) — steady states, gravitation, electrostatics

Solution via Fourier transform

For the heat equation on $\mathbb{R}$ with initial condition $u(x,0) = u_0(x)$:

\[u(x,t) = \frac{1}{\sqrt{4\pi\kappa t}} \int_{-\infty}^\infty u_0(y)\, e^{-(x-y)^2/4\kappa t}\,dy\]

The fundamental solution (Green’s function) is a Gaussian that spreads over time.

Boundary conditions

• Dirichlet: $u = g$ on boundary — prescribed values
• Neumann: $\nabla u \cdot \hat{n} = h$ — prescribed flux
• Robin: $\alpha u + \beta \nabla u \cdot \hat{n} = g$ — mixed