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A partial differential equation relates an unknown function of several independent variables to its partial derivatives. For $u(x,t)$, the notation $u_t=\partial u/\partial t$ means changing time while holding position fixed; $u_{xx}={\partial }^{2}u/\partial x^2$ measures how its spatial slope changes. Unlike an ordinary differential equation, a partial differential equation can describe an entire evolving profile rather than a single trajectory.

The heat equation $u_t=D{u}_{xx}$ smooths spatial variations, the transport equation $u_t+c{u}_x=0$ moves them, and Laplace's equation $\Delta u=0$ describes a spatial equilibrium. Here $D>0$ is a diffusivity, $c$ is a velocity, and $\Delta$ is the Laplacian. The periodic heat and transport comparison makes the difference visible using identical initial conditions.

A partial differential equation usually supplies a local rule rather than a complete answer. For example, every $u(x,t)=A+Bx$ satisfies the heat equation with zero PDE source term; data select which profile is relevant. A complete problem specifies its domain of a function, coefficients, initial conditions, boundary conditions, and meaning of solution of a partial differential equation. A well-posed problem has existence, uniqueness, and continuous dependence on initial data in stated spaces; the equation alone guarantees none of these.^[ref]

A transport equation carries a quantity along the flow of a velocity field $b$. For a smooth solution of $u_t+b\cdot \nabla u=0$, the value of $u$ is constant along each characteristic curve. With constant speed $c$, the chain rule gives $\frac{d}{dt}u(x_0+ct,t)=u_t+c{u}_x=0$, hence $u(x,t)=g(x-ct)$. This is exact translation: it preserves the shape rather than smoothing it, as the periodic heat and transport comparison shows.