Difference between revisions of "Linear partial differential equation"
From Encyclopedia of Mathematics
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$ i _ {1} \dots i _ {n} $ | $ i _ {1} \dots i _ {n} $ | ||
| − | are non-negative integer indices, $ \ | + | 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 , |
$$ | $$ | ||
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
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