Category:Quasilinear equations: Difference between revisions
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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) | 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 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> | ||
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), thus they are listed only in their own category. | 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), thus they are listed only in their own category. | ||
Revision as of 17:28, 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 not semilinear)
\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]
\[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * |x|^{-n+\alpha} \]
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), thus they are listed only in their own category.
Pages in category "Quasilinear equations"
The following 2 pages are in this category, out of 2 total.