Editing Partial differential equation (section)

===Methods for non-linear equations===
{{see also|nonlinear partial differential equation}}

There are no generally applicable analytical methods to solve nonlinear PDEs. Still, existence and uniqueness results (such as the [[Cauchy–Kowalevski theorem]]) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of [[mathematical analysis|analysis]]).

Nevertheless, some techniques can be used for several types of equations. The [[h-principle|{{mvar|h}}-principle]] is the most powerful method to solve [[Underdetermined system|underdetermined]] equations. The [[Riquier–Janet theory]] is an effective method for obtaining information about many analytic [[Overdetermined system|overdetermined]] systems.

The [[method of characteristics]] can be used in some very special cases to solve nonlinear partial differential equations.<ref>{{cite book |first=J. David |last=Logan |title=An Introduction to Nonlinear Partial Differential Equations |location=New York |publisher=John Wiley & Sons |year=1994 |isbn=0-471-59916-6 |chapter=First Order Equations and Characteristics |pages=51–79 }}</ref>

In some cases, a PDE can be solved via [[perturbation analysis]] in which the solution is considered to be a correction to an equation with a known solution. Alternatives are [[numerical analysis]] techniques from simple [[finite difference]] schemes to the more mature [[multigrid]] and [[finite element method]]s. Many interesting problems in science and engineering are solved in this way using [[computer]]s, sometimes high performance [[supercomputer]]s.