Difference between revisions of "Semilinear"

From DispersiveWiki
Jump to navigationJump to search
 
m (Clarified antecedent of "these equations")
 
Line 8: Line 8:
 
<center><math>L u = F(u, Du, \ldots, D^k u)</math></center>
 
<center><math>L u = F(u, Du, \ldots, D^k u)</math></center>
 
where ''L'' is a linear operator, ''F'' is a nonlinear function of the first few derivatives <math>u, Du, \ldots, D^k u</math>, with
 
where ''L'' is a linear operator, ''F'' is a nonlinear function of the first few derivatives <math>u, Du, \ldots, D^k u</math>, with
''k'' strictly less than the order of ''L''.  These equations are more nonlinear than semilinear equations, but are less nonlinear than [[quasilinear]] or [[fully nonlinear]] equations.
+
''k'' strictly less than the order of ''L''.  Semilinear-with-derivatives equations are more nonlinear than semilinear equations, but are less nonlinear than [[quasilinear]] or [[fully nonlinear]] equations.
  
 
[[Category:concept]]
 
[[Category:concept]]

Latest revision as of 15:41, 12 December 2007


A semilinear equation is a PDE of the form

where L is a linear operator and F is a nonlinear operator which does not involve any derivatives of u.

A semilinear-with-derivatives equation is a PDE of the form

where L is a linear operator, F is a nonlinear function of the first few derivatives , with k strictly less than the order of L. Semilinear-with-derivatives equations are more nonlinear than semilinear equations, but are less nonlinear than quasilinear or fully nonlinear equations.