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Difference between revisions of "Non-linear boundary value problem"

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The determination in a certain domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067070/n0670701.png" /> of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067070/n0670702.png" /> of a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067070/n0670703.png" /> of a differential equation
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from its values on (a part of) the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067070/n0670705.png" /> of this domain:
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The determination in a certain domain  $  D $
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of variables  $  x = ( x _ {1} \dots x _ {n} ) $
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of a solution  $  u( x) $
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of a differential equation
  
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$$
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( Lu ) ( x)  = f ( x) ,\  x \in D ,
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$$
  
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where at least one of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067070/n0670707.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067070/n0670708.png" /> is non-linear.
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from its values on (a part of) the boundary  $  S $
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of this domain:
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$$
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( Bu ) ( y)  =  \phi ( y) ,\  y \in S ,
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$$
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where at least one of the operators $  L $
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or $  B $
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is non-linear.
  
 
See also [[Boundary value problem, ordinary differential equations|Boundary value problem, ordinary differential equations]]; [[Boundary value problem, partial differential equations|Boundary value problem, partial differential equations]].
 
See also [[Boundary value problem, ordinary differential equations|Boundary value problem, ordinary differential equations]]; [[Boundary value problem, partial differential equations|Boundary value problem, partial differential equations]].

Latest revision as of 08:03, 6 June 2020


The determination in a certain domain $ D $ of variables $ x = ( x _ {1} \dots x _ {n} ) $ of a solution $ u( x) $ of a differential equation

$$ ( Lu ) ( x) = f ( x) ,\ x \in D , $$

from its values on (a part of) the boundary $ S $ of this domain:

$$ ( Bu ) ( y) = \phi ( y) ,\ y \in S , $$

where at least one of the operators $ L $ or $ B $ is non-linear.

See also Boundary value problem, ordinary differential equations; Boundary value problem, partial differential equations.

How to Cite This Entry:
Non-linear boundary value problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-linear_boundary_value_problem&oldid=15674
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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