Category:Quasilinear equations: Difference between revisions
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For instance, the following equations are all quasilinear (and | For instance, the following equations are all quasilinear (and the first two are NOT semilinear) | ||
\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \] | \[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \] | ||
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<center> [[Nonlocal porous medium equation]] </center> | <center> [[Nonlocal porous medium equation]] </center> | ||
\[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\] | |||
<center> Hamilton-Jacobi with fractional diffusion </center> | |||
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. | 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. | ||
Note: In this category are listed all equations which are quasilinear and NOT semilinear. Strictly speaking, all semilinear equations ought to be listed here aswell, however, as the specific methods and questions are so different in both categories (i.e. quasilinear techniques may give results for semilinear equations which are weaker when compared to the more powerful methods tailor-made for semilinear ones) we prefer to list semilinear ones in their own category only. | Note: In this category are listed all equations which are quasilinear and NOT semilinear. Strictly speaking, all semilinear equations ought to be listed here aswell, however, as the specific methods and questions are so different in both categories (i.e. quasilinear techniques may give results for semilinear equations which are weaker when compared to the more powerful methods tailor-made for semilinear ones) we prefer to list semilinear ones in their own category only. | ||
Revision as of 16:30, 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). See also the article for Quasilinear equations
For instance, the following equations are all quasilinear (and the first two are NOT semilinear)
\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]
\[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * |x|^{-n+\alpha} \]
\[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]
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.
Note: In this category are listed all equations which are quasilinear and NOT semilinear. Strictly speaking, all semilinear equations ought to be listed here aswell, however, as the specific methods and questions are so different in both categories (i.e. quasilinear techniques may give results for semilinear equations which are weaker when compared to the more powerful methods tailor-made for semilinear ones) we prefer to list semilinear ones in their own category only.
Pages in category "Quasilinear equations"
The following 2 pages are in this category, out of 2 total.