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Difference between revisions of "Quasi-linear equation"

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A partial differential equation (cf. [[Differential equation, partial|Differential equation, partial]]) that is linear with respect to the leading derivatives of the unknown function. For example, the equation
 
A partial differential equation (cf. [[Differential equation, partial|Differential equation, partial]]) that is linear with respect to the leading derivatives of the unknown function. For example, the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076570/q0765701.png" /></td> </tr></table>
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$$\left(\frac{\partial u}{\partial x}\right)^2\frac{\partial^2u}{\partial x^2}+\frac{\partial u}{\partial y}\frac{\partial^2u}{\partial y^2}+u^2=0$$
  
is a second-order quasi-linear equation with respect to the unknown function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076570/q0765702.png" />.
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is a second-order quasi-linear equation with respect to the unknown function $u$.
  
 
====Comments====
 
====Comments====

Latest revision as of 14:48, 10 August 2014

A partial differential equation (cf. Differential equation, partial) that is linear with respect to the leading derivatives of the unknown function. For example, the equation

$$\left(\frac{\partial u}{\partial x}\right)^2\frac{\partial^2u}{\partial x^2}+\frac{\partial u}{\partial y}\frac{\partial^2u}{\partial y^2}+u^2=0$$

is a second-order quasi-linear equation with respect to the unknown function $u$.

Comments

References

[a1] H.M. Luberstein, "Theory of partial differential equations" , Acad. Press (1972) pp. 10; 12; 27
[a2] G.F. Carrier, C.E. Pearson, "Partial differential equations" , Acad. Press (1976) pp. Sect. 6.3; 89; 252
How to Cite This Entry:
Quasi-linear equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-linear_equation&oldid=18823