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
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 ).
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.
|Title||Cauchy initial value problem|
|Date of creation||2013-03-22 14:57:18|
|Last modified on||2013-03-22 14:57:18|
|Last modified by||paolini (1187)|
|Synonym||initial value problem|
|Defines||solution to the Cauchy problem|
|Defines||solution to the initial value problem|