Cauchy problem
التعريفات والمعاني
== English ==
=== Etymology ===
After French mathematician Augustin Louis Cauchy.
=== Noun ===
Cauchy problem (plural Cauchy problems)
(mathematics, mathematical analysis) For a given m-order partial differential equation, the problem of finding a solution function
u
{\displaystyle u}
on
R
n
{\displaystyle \mathbb {R} ^{n}}
that satisfies the boundary conditions that, for a smooth manifold
S
⊂
R
n
{\displaystyle S\subset \mathbb {R} ^{n}}
,
u
(
x
)
=
f
0
(
x
)
{\displaystyle \textstyle u(x)=f_{0}(x)}
and
∂
k
u
(
x
)
∂
n
k
=
f
k
(
x
)
{\displaystyle \textstyle {\frac {\partial ^{k}u(x)}{\partial n^{k}}}=f_{k}(x)}
,
∀
x
∈
S
{\displaystyle \forall x\in S}
,
k
=
1
…
m
−
1
{\displaystyle k=1\dots m-1}
, given specified functions
f
k
{\displaystyle f_{k}}
defined on, and vector
n
{\displaystyle n}
normal to, the manifold.
==== Usage notes ====
The hypersurface S is called the Cauchy surface. The functions fk defined on S are collectively known as the Cauchy data of the problem.
==== Translations ====
=== Further reading ===
Cauchy boundary condition on Wikipedia.Wikipedia
Cauchy-Kowalevski theorem on Wikipedia.Wikipedia