The Poisson equation $\nabla^2 u = f$ generalizes Laplace's equation and models electrostatics: $\nabla^2 \phi = -\frac{\rho}{\epsilon_0}$ where $\phi$ is the electric potential and $\rho$ is the charge density. The solution at any point depends on the entire domain—a characteristic elliptic property.

The diffusion equation $\frac{\partial u}{\partial t} = D\nabla^2 u$ models concentration gradients: $\frac{\partial c}{\partial t} = D\left(\frac{\partial^2 c}{\partial x^2} + \frac{\partial^2 c}{\partial y^2} + \frac{\partial^2 c}{\partial z^2}\right)$ where $c$ is concentration and $D$ is the diffusion coefficient. This equation exhibits smoothing behavior: rough initial data immediately becomes smooth.

The transport equation $\frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x} = 0$ is first-order hyperbolic with general solution $u(x,t) = f(x - ct)$, representing a traveling wave.

The wave equation $\frac{\partial^2 u}{\partial t^2} = c^2\nabla^2 u$ models vibrating strings and electromagnetic waves. D'Alembert's solution in one dimension is: $u(x,t) = \frac{1}{2}[f(x-ct) + f(x+ct)] + \frac{1}{2c}\int_{x-ct}^{x+ct} g(s)\,ds$