Quasilinear equations: Difference between revisions
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imported>Nestor (Created page with "For instance, the following equations are all quasilinear (and the first two are NOT semilinear) \[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \] ...") |
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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) | For instance, the following equations are all quasilinear (and the first two are NOT semilinear) | ||
Revision as of 17:45, 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)
\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]
\[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * |x|^{-n+\alpha} \]
\[ u_t + H(x,t,u,\nabla u) + (-\Delta)^s u = 0.\]
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.