cauchy initial value problem
Let be a subset of
,
a point of
, and
be a function.
We say that a function is a solution to the Cauchy (or initial value) problem
![]() |
(1) |
if
is a differentiable function
defined on a interval
;
- one has
for all
and
;
- one has
and
for all
.
We say that a solution
is a maximal solution if it cannot be extended to a bigger interval. More precisely given any other solution
defined on an interval
and such that
for all
, one has
(and hence
and
are the same function).
We say that a solution
is a global solution if
.
We say that a solution
is unique if given any other solution
one has
for all
(i.e.
is the unique solution defined on the interval
).
Notation
Usually the differential equation in (1) is simply written as



Examples
- The function
defined on
is the unique maximal solution to the Cauchy problem:
In this case,
,
,
.
- The function
is a global (and hence maximal), unique solution to the Cauchy problem:
- Consider the Cauchy problem
The function
defined on
is a global solution. However the function
defined on
is also a solution and so are the functions
for every. So there are no unique solutions. Moreover
is not a maximal solution.
posted on 2007-10-24 23:45 cloudseawang 阅读(298) 评论(0) 编辑 收藏 举报