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''of a first-order partial differential equation''
 
''of a first-order partial differential equation''
  
 
A family
 
A family
  
<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/c/c021/c021730/c0217301.png" /></td> </tr></table>
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$$
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= x ( t),\ \
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= y ( t),\ \
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u _ {x}  = p ( t)
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$$
  
of continuously-differentiable functions in an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021730/c0217302.png" />, satisfying the equations
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of continuously-differentiable functions in an interval $  \alpha < t < \beta $,  
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satisfying the equations
  
<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/c/c021/c021730/c0217303.png" /></td> </tr></table>
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$$
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x  ^  \prime  ( t)  = F _ {p} ,\ \
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y  ^  \prime  ( t)  = pF _ {p} ,\ \
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p  ^  \prime  ( t)  = - F _ {x} - p F _ {y} ,
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$$
  
 
where the multiplication of the vectors is the scalar product, and where
 
where the multiplication of the vectors is the scalar product, and where
  
<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/c/c021/c021730/c0217304.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$ \tag{* }
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F ( x, u , u _ {x} )  = 0
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$$
  
is a non-linear first-order partial differential equation in the unknown function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021730/c0217305.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021730/c0217306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021730/c0217307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021730/c0217308.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021730/c0217309.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021730/c02173010.png" />.
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is a non-linear first-order partial differential equation in the unknown function $  u: \Omega \subseteq \mathbf R  ^ {n} \rightarrow \mathbf R $.  
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Here $  u _ {x} = \mathop{\rm grad}  u $,
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$  F ( x, y, p): \Omega \times \mathbf R \times \mathbf R  ^ {n} \rightarrow \mathbf R $,  
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$  x, p \in \mathbf R  ^ {n} $,  
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$  y \in \mathbf R $,  
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$  n \in \mathbf N $.
  
 
The importance of a characteristic strip consists in the fact that it is used in the study of, and in the search for, solutions of equation (*).
 
The importance of a characteristic strip consists in the fact that it is used in the study of, and in the search for, solutions of equation (*).
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''2. Partielle Differentialgleichungen erster Ordnung für die gesuchte Funktion''' , Akad. Verlagsgesell.  (1944)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Hartman,  "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''2. Partielle Differentialgleichungen erster Ordnung für die gesuchte Funktion''' , Akad. Verlagsgesell.  (1944)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Hartman,  "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 16:43, 4 June 2020


of a first-order partial differential equation

A family

$$ x = x ( t),\ \ u = y ( t),\ \ u _ {x} = p ( t) $$

of continuously-differentiable functions in an interval $ \alpha < t < \beta $, satisfying the equations

$$ x ^ \prime ( t) = F _ {p} ,\ \ y ^ \prime ( t) = pF _ {p} ,\ \ p ^ \prime ( t) = - F _ {x} - p F _ {y} , $$

where the multiplication of the vectors is the scalar product, and where

$$ \tag{* } F ( x, u , u _ {x} ) = 0 $$

is a non-linear first-order partial differential equation in the unknown function $ u: \Omega \subseteq \mathbf R ^ {n} \rightarrow \mathbf R $. Here $ u _ {x} = \mathop{\rm grad} u $, $ F ( x, y, p): \Omega \times \mathbf R \times \mathbf R ^ {n} \rightarrow \mathbf R $, $ x, p \in \mathbf R ^ {n} $, $ y \in \mathbf R $, $ n \in \mathbf N $.

The importance of a characteristic strip consists in the fact that it is used in the study of, and in the search for, solutions of equation (*).

See also Characteristic.

References

[1] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 2. Partielle Differentialgleichungen erster Ordnung für die gesuchte Funktion , Akad. Verlagsgesell. (1944)
[2] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)

Comments

A characteristic strip is sometimes called a bicharacteristic.

In the modern theory, the characteristic strips of a partial differential equation carry the wave front sets of solutions of a partial differential equation.

References

[a1] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1962) (Translated from German)
[a2] L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983)
How to Cite This Entry:
Characteristic strip. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Characteristic_strip&oldid=46323
This article was adapted from an original article by Yu.V. Komlenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article