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Difference between revisions of "Linear partial differential equation"

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$  i _ {1} \dots i _ {n} $
 
$  i _ {1} \dots i _ {n} $
are non-negative integer indices,  $  \sum _ {j=} ^ {n} i _ {j} = k $,  
+
are non-negative integer indices,  $  \sum_{j=1}^ {n} i _ {j} = k $,  
 
$  k = 0 \dots m $,  
 
$  k = 0 \dots m $,  
 
$  m \geq  1 $,  
 
$  m \geq  1 $,  
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\frac{\partial  F }{\partial  p _ {i _ {1}  \dots i _ {n} } }
 
\frac{\partial  F }{\partial  p _ {i _ {1}  \dots i _ {n} } }
 
  ,\ \  
 
  ,\ \  
\sum _ { j= } 1 ^ { n }  i _ {j} = m ,
+
\sum_{j=1}^ { n }  i _ {j} = m ,
 
$$
 
$$
  

Latest revision as of 19:50, 18 January 2024


An equation of the form

$$ F ( x \dots p _ {i _ {1} \dots i _ {n} } , . . . ) = 0 , $$

where $ F $ is a linear function of real variables,

$$ p _ {i _ {1} \dots i _ {n} } \equiv \ \frac{\partial ^ {k} }{\partial x _ {1} ^ {i _ {1} } \dots d x _ {n} ^ {i _ {n} } } , $$

$ i _ {1} \dots i _ {n} $ are non-negative integer indices, $ \sum_{j=1}^ {n} i _ {j} = k $, $ k = 0 \dots m $, $ m \geq 1 $, and at least one of the derivatives

$$ \frac{\partial F }{\partial p _ {i _ {1} \dots i _ {n} } } ,\ \ \sum_{j=1}^ { n } i _ {j} = m , $$

is non-zero.

For more details, see Differential equation, partial.

How to Cite This Entry:
Linear partial differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_partial_differential_equation&oldid=55201
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article