---
tag: GC5BDZ6QBC
kind: concept
name: Partial differential equation
msc: [35A99]
specializes: [003R0V5M7V]
sources:
  - url: https://en.wikipedia.org/wiki/Partial_differential_equation
same_as:
  wikidata: Q271977
  wikipedia: Partial differential equation
---

## Summary

A partial differential equation, or PDE, is an equation that relates an unknown function of several variables to its [[partial derivatives|09EGC6F8PT]].
It is the several-variable counterpart of an [[ordinary differential equation|0NT64JV96A]].
PDEs describe heat flow, waves, fluids, electromagnetism, quantum mechanics and many other laws of physics, and they are central objects of geometry and probability.
See the [[definition|PV8DQ41Q55]].

## Explanations

### L1 human

Many quantities depend on several variables at once, such as the temperature at a point of a room at a given time.
A law of nature often says how the rates of change of such a quantity in different directions are linked.
Written as a formula, this law is a partial differential equation.
For example, the temperature $u(t, x)$ in a thin rod obeys $\partial_t u = \partial_x^2 u$, the [[heat equation|VP2HN86VAG]].
Solving the PDE means finding all functions $u$ for which the formula holds.