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}}
, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wiktionary.org/v1/":): {\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.
The Cauchy–Kowalevski theorem states that the Cauchy problem for any partial differential equation whose coefficients are analytic in the unknown function and its derivatives, has a locally unique analytic solution. Source: Internet
However, such theories in general do not have a well-defined Cauchy problem (for reasons related to the issues of causality discussed above), and are probably inconsistent quantum mechanically. Source: Internet