---
tag: PV8DQ41Q55
kind: definition
name: Partial differential equation
msc: [35A99]
defines: GC5BDZ6QBC
uses: [02BH3MCBQB, 09EGC6F8PT, 0PJXBR6QA3, 003R0V5M7V]
sources:
  - url: https://en.wikipedia.org/wiki/Partial_differential_equation
---

## Definition

Let $\Omega$ be an [[open set|02BH3MCBQB]] in $\R^n$ with $n \ge 2$, and let $k \ge 1$ be an integer.
For a function $u \colon \Omega \to \R$ of class $C^k$, write $D^j u(x)$ for the collection of all [[partial derivatives|09EGC6F8PT]] $\partial^\alpha u(x)$ of order $\abs{\alpha} = j$, in [[multi-index notation|0PJXBR6QA3]].
Let $F$ be a given function of these derivatives and of $x$.
A *partial differential equation* of order at most $k$ in $\Omega$ is an equation of the following form.
$$F\left(D^k u(x), D^{k-1} u(x), \dots, D u(x), u(x), x\right) = 0 \quad \text{for } x \in \Omega.$$
The function $u$ is the unknown.
A *classical solution* is a function $u \in C^k(\Omega)$ that satisfies the equation at every point of $\Omega$.
A *system* of partial differential equations is defined in the same way with $u$ and $F$ taking values in $\R^N$ and $\R^M$.

## Remarks

Example: with $n = 2$ and variables $(x, y)$, the equation $\partial_x^2 u + \partial_y^2 u = 0$ is a partial differential equation.
The function $u(x, y) = x^2 - y^2$ is a classical solution on $\R^2$, since $\partial_x^2 u = 2$ and $\partial_y^2 u = -2$.
Non-example: $u'(x) = u(x)$ for $u \colon \R \to \R$ involves derivatives with respect to one variable only, so it is an [[ordinary differential equation|0NT64JV96A]].
Complex-valued unknowns are treated in the same way.
Many PDEs also have solutions in a weaker sense, such as [[weak solutions|SJZX81A783]], which need not be of class $C^k$.

## Explanations

### L1 human

A partial differential equation is a condition on a function of several variables and its partial derivatives, which must hold at every point.
The unknown is a whole function, not a number.