What Is a PDE? Order and Linearity

A partial differential equation (PDE) relates an unknown function of several variables to its partial derivatives. We write $u_t = \partial u/\partial t$, $u_{xx} = \partial^2 u/\partial x^2$, and so on. Two features classify a PDE:

• Order: the highest derivative appearing. The heat equation $u_t = k\,u_{xx}$ is second order (because of $u_{xx}$).
• Linearity: the PDE is linear if $u$ and its derivatives appear only to the first power and are never multiplied together. $u_t + u\,u_x = 0$ is nonlinear (the term $u\,u_x$ multiplies the unknown by its own derivative).

Worked example. Classify $u_t = k\,u_{xx}$ (the heat equation). Highest derivative is $u_{xx}$ → order $2$. Each term is linear in $u$ and its derivatives → linear. Three canonical second-order linear PDEs anchor the whole subject: the heat equation $u_t = k u_{xx}$, the wave equation $u_{tt} = c^2 u_{xx}$, and the Laplace equation $u_{xx} + u_{yy} = 0$.