Editing Partial differential equation (section)

== Definition ==
A partial differential equation is an equation that involves an unknown function of <math>n\geq 2</math> variables and (some of) its partial derivatives.{{sfn|Evans|1998|pp=1-2}} That is, for the unknown function
<math display="block">u : U \rightarrow \mathbb{R},</math>
of variables <math>x = (x_1,\dots,x_n)</math> belonging to the open subset <math>U</math> of <math>\mathbb{R}^n</math>, the <math>k^{th}</math>-order partial differential equation is defined as 
<math display="block">F[D^{k} u, D^{k-1}u,\dots, D u, u, x]=0,</math>
where
<math display="block"> F: \mathbb{R}^{n^{k}}\times \mathbb{R}^{n^{k-1}}\dots \times \mathbb{R}^{n} \times \mathbb{R} \times U \rightarrow \mathbb{R},</math>
and <math>D</math> is the partial derivative operator.

=== Notation ===
{{main|Notation for differentiation#Partial derivatives}}
When writing PDEs, it is common to denote partial derivatives using subscripts. For example:
<math display="block">u_x = \frac{\partial u}{\partial x},\quad u_{xx} = \frac{\partial^2 u}{\partial x^2},\quad u_{xy} = \frac{\partial^2 u}{\partial y\, \partial x} = \frac{\partial}{\partial y } \left(\frac{\partial u}{\partial x}\right). </math>
In the general situation that {{mvar|u}} is a function of {{mvar|n}} variables, then {{math|''u''<sub>''i''</sub>}} denotes the first partial derivative relative to the {{mvar|i}}-th input, {{math|''u''<sub>''ij''</sub>}} denotes the second partial derivative relative to the {{mvar|i}}-th and {{mvar|j}}-th inputs, and so on.

The Greek letter {{math|Δ}} denotes the [[Laplace operator]]; if {{mvar|u}} is a function of {{mvar|n}} variables, then
<math display="block">\Delta u = u_{11} + u_{22} + \cdots + u_{nn}.</math>
In the physics literature, the Laplace operator is often denoted by {{math|∇<sup>2</sup>}}; in the mathematics literature, {{math|∇<sup>2</sup>''u''}} may also denote the [[Hessian matrix]] of {{mvar|u}}.