Category:Quasilinear equations and Quasilinear equations: Difference between pages
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Quasilinear equations are those which are linear in all terms except for the highest order derivatives (whether they are of fractional order or not). | |||
For instance, the following equations are all quasilinear (and the first two are NOT semilinear) | |||
\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \] | |||
<center> [[Mean curvature flow]] </center> | |||
\[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * |x|^{-n+\alpha} \] | |||
<center> [[Nonlocal porous medium equation]] </center> | |||
\[ u_t + (-\Delta)^s u + H(x,t,u,\nabla u)= 0.\] | |||
<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. | |||
Revision as of 16:45, 3 June 2011
Quasilinear equations are those which are linear in all terms except for the highest order derivatives (whether they are of fractional order or not).
For instance, the following equations are all quasilinear (and the first two are NOT semilinear)
\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]
\[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * |x|^{-n+\alpha} \]
\[ u_t + (-\Delta)^s u + H(x,t,u,\nabla u)= 0.\]
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.
Pages in category "Quasilinear equations"
The following 2 pages are in this category, out of 2 total.