Sandbox and Category:Quasilinear equations: Difference between pages

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=Sample 1=
A quasilinear equation is one that is linear in all but the terms involving the highest order derivatives (whether they are of fractional order or not). For instance, the following equations are all quasilinear (and not semilinear)


{| cellspacing="3"
\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]
|width="50%" style="border: 1px solid #c6c9ff; color: #000;" |
Hello world!
|width="50%" style="border: 1px solid #ffc9c9; color: #000;"|
Something else.
|}
{{infobox|
This is an example of a box.
}}


==References==
<center> [[Mean curvature flow]] </center>
<!-- A sample reference section -->


*{{Citation | last1=Gilbarg | first1=David | last2=Trudinger | first2=Neil S. | author2-link=Neil Trudinger | title=Elliptic partial differential equations of second order | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-3-540-41160-4 | year=2001}}
\[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * |x|^{-n+\alpha} \]  


=Sample 2: Inline references=
<center> [[Nonlocal porous medium equation]] </center>
<!-- If we wanted to be more fancy - this will not render properly until Maorung installs the Cite extension -->


This is a citation.<ref name="GT"/>


==References==
Equations which are not quasilinear are called [[Fully nonlinear equations]], which include for instance [[Monge Ampére]] and [[Fully nonlinear integro-differential equations]]. Note that all [[Semilinear equations]] are automatically quasilinear.
{{reflist|refs=
<ref name="GT">{{Citation | last1=Gilbarg | first1=David | last2=Trudinger | first2=Neil S. | author2-link=Neil Trudinger | title=Elliptic partial differential equations of second order | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-3-540-41160-4 | year=2001}}</ref>
}}

Revision as of 17:22, 3 June 2011

A quasilinear equation is one that is linear in all but the terms involving the highest order derivatives (whether they are of fractional order or not). For instance, the following equations are all quasilinear (and not semilinear)

\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]

Mean curvature flow

\[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * |x|^{-n+\alpha} \]

Nonlocal porous medium equation


Equations which are not quasilinear are called Fully nonlinear equations, which include for instance Monge Ampére and Fully nonlinear integro-differential equations. Note that all Semilinear equations are automatically quasilinear.

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