Editing Partial differential equation (section)

=== Linear and nonlinear equations ===

A PDE is called '''linear''' if it is linear in the unknown and its derivatives. For example, for a function {{mvar|u}} of {{mvar|x}} and {{mvar|y}}, a second order linear PDE is of the form
<math display="block"> a_1(x,y)u_{xx} + a_2(x,y)u_{xy} + a_3(x,y)u_{yx} + a_4(x,y)u_{yy} + a_5(x,y)u_x + a_6(x,y)u_y + a_7(x,y)u = f(x,y)
</math>
where {{math|''a<sub>i</sub>''}} and {{mvar|''f''}} are functions of the independent variables {{mvar|x}} and {{mvar|y}} only. (Often the mixed-partial derivatives {{math|''u<sub>xy</sub>''}} and {{math|''u<sub>yx</sub>''}} will be equated, but this is not required for the discussion of linearity.)
If the {{math|''a<sub>i</sub>''}} are constants (independent of {{mvar|x}} and {{mvar|y}}) then the PDE is called '''linear with constant coefficients'''. If {{mvar|''f''}} is zero everywhere then the linear PDE is '''homogeneous''', otherwise it is '''inhomogeneous'''. (This is separate from [[asymptotic homogenization]], which studies the effects of high-frequency oscillations in the coefficients upon solutions to PDEs.)

Nearest to linear PDEs are '''semi-linear''' PDEs, where only the highest order derivatives appear as linear terms, with coefficients that are functions of the independent variables. The lower order derivatives and the unknown function may appear arbitrarily. For example, a general second order semi-linear PDE in two variables is
<math display="block"> a_1(x,y)u_{xx} + a_2(x,y)u_{xy} + a_3(x,y)u_{yx} + a_4(x,y)u_{yy} + f(u_x, u_y, u, x, y) = 0 </math>

In a '''quasilinear''' PDE the highest order derivatives likewise appear only as linear terms, but with coefficients possibly functions of the unknown and lower-order derivatives:
<math display="block"> a_1(u_x, u_y, u, x, y)u_{xx} + a_2(u_x, u_y, u, x, y)u_{xy} + a_3(u_x, u_y, u, x, y)u_{yx} + a_4(u_x, u_y, u, x, y)u_{yy} + f(u_x, u_y, u, x, y) = 0 </math>
Many of the fundamental PDEs in physics are quasilinear, such as the [[Einstein equations]] of [[general relativity]] and the [[Navier–Stokes equations]] describing fluid motion.

A PDE without any linearity properties is called '''fully [[Nonlinear partial differential equation|nonlinear]]''', and possesses nonlinearities on one or more of the highest-order derivatives. An example is the [[Monge–Ampère equation]], which arises in [[differential geometry]].<ref name="PrincetonCompanion">{{Citation|last = Klainerman|first = Sergiu|year = 2008|title =Partial Differential Equations|editor-last1 = Gowers|editor-first1 = Timothy|editor-last2 = Barrow-Green|editor-first2 = June|editor-last3 = Leader|editor-first3 = Imre|encyclopedia = The Princeton Companion to Mathematics|pages = 455–483|publisher = Princeton University Press}}</ref>