# Linear partial differential equation

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=47662
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article