Table of partial differential equations

Name	Equation	Classification(s)
Poisson	\(\nabla\cdot(\mathsf{D}\cdot\nabla u) = f\)	elliptic, second-order in space
diffusion	\(\frac{\partial u}{\partial t} = \nabla\cdot(\mathsf{D}\cdot\nabla u)\)	parabolic, second-order in space, first-order in time
advection	\(\frac{\partial u}{\partial t}+\textbf{a}\cdot\nabla u = 0\)	hyperbolic, first-order in space, first-order in time
advection-diffusion	\(\frac{\partial u}{\partial t}+\textbf{a}\cdot\nabla u = \nabla\cdot(\mathsf{D}\cdot\nabla u)\)	parabolic, second-order in space, first-order in time
advection-diffusion-reaction (nonlinear)	\(\frac{\partial u}{\partial t}+\textbf{a}\cdot\nabla u = \nabla\cdot(\mathsf{D}\cdot\nabla u) + \Sigma(u)\)	parabolic, second-order in space, first-order in time
advection-diffusion-reaction (linear)	\(\frac{\partial u}{\partial t}+\textbf{a}\cdot\nabla u = \nabla\cdot(\mathsf{D}\cdot\nabla u) + Ru + J\)	parabolic, second-order in space, first-order in time
steady advection-diffusion-reaction	\(\textbf{a}\cdot\nabla u = \nabla\cdot(\mathsf{D}\cdot\nabla u) + Ru + J\)	parabolic, second-order in space
steady diffusion-reaction	\(0 = \nabla\cdot(\mathsf{D}\cdot\nabla u) + Ru + J\)	parabolic, second-order in space
steady advection-reaction	\(\textbf{a}\cdot\nabla u = Ru + J\)	hyperbolic, first-order in space
Helmholtz	\(\nabla\cdot(\mathsf{D}\cdot\nabla u) + k^2 u = f\)	elliptic, second-order in space
wave	\(\frac{\partial^2u}{\partial t^2} = \nabla\cdot(\mathsf{D}\cdot\nabla u)\)	hyperbolic, second-order in space, second-order in time
Darcy	\(\begin{matrix}\nabla\cdot\textbf{u} = 0\\ \textbf{u} = -\frac{\mathsf{K}}{\mu}\cdot(\nabla p - \textbf{f}\,)\end{matrix}\)	mixed, first-order in space
Darcy streamfunction	\(\nabla\cdot\left(\frac{\mu\mathsf{K}^{\mathsf{T}}\cdot\nabla\psi}{\text{det}(\mathsf{K})}\right)=-\frac{\partial(f_y)}{\partial x} + \frac{\partial(f_x)}{\partial y}\)	elliptic, second-order in space
Darcy pressure	\(\nabla\cdot\left(\frac{\mathsf{K}}{\mu}\cdot\nabla p\right)=\nabla\cdot\left(\frac{\mathsf{K}}{\mu}\cdot\textbf{f}\right)\)	elliptic, second-order in space
Stokes	\(\begin{matrix}\nabla\cdot\textbf{u} = 0\\ \textbf{0}=-\nabla p + \nabla\cdot\tau + \textbf{f}\end{matrix}\)	mixed, second-order in space
Navier-Stokes	\(\begin{matrix}\nabla\cdot\textbf{u} = 0\\ \rho \left(\frac{\partial\textbf{u}}{\partial t}+\textbf{u}\cdot\nabla\textbf{u}\right)=-\nabla p + \nabla\cdot\tau + \textbf{f}\end{matrix}\)	mixed, second-order in space, first-order in time