# Definitions > [!NOTE] Definition > A functional equation of the form $F(x_{1},\dots,x_{n}, u, \partial_{x_{1}}u,\dots \partial_{x_{1}x_{2}}u,\dots)=0$ > > where $F:\mathbb{R}^{n+k}\to \mathbb{R}^m$; $k$ is the number of partial derivative of $u$ (of any order: including zero - the function itself) that appear in the PDE; $m\in \mathbb{N}$ is the number of equations. **Terminology**: - The **associated differential operator**, $\mathcal{L}(u)$, is the expression in $F$ containing $u$ and its partial derivatives. > [!Example]- Examples > - [[Heat Equation]]: Given the PDE $\partial_{t}u=\partial_{xx}u+e^{-x^{2}/2}$we have $\mathcal{L}(u)=\partial_{t}(u)-\partial_{xx}u$ and $r(x)=e^{-x^{2}/2}.$ > - $u=u(x,t)$ corresponds to temperature at positiion $x$ and time $t.$ > $\partial_{t}u(x,t)=D\partial_{\times}u(x,t)$where $D>0.$ > - VisualPDEs: https://visualpde.com/explore. # Properties(s) # Application(s) **More examples**: [[Heat Equation]]. # Bibliography