Partial differential equation (nonfiction)

A visualisation of a solution to the two-dimensional heat equation with temperature represented by the third dimension.

In mathematics, a partial differential equation (PDE) is a differential equation that contains beforehand unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.

PDEs can be used to describe a wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs.

Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems.

PDEs find their generalization in stochastic partial differential equations.