cauchy initial value problem

Let $ D$ be a subset of $ \mathbb{R}^n\times \mathbb{R}$, $ (x_0,t_0)$ a point of $ D$, and $ f\colon D\to \mathbb{R}$ be a function.

We say that a function $ x(t)$ is a solution to the Cauchy (or initial value) problem

\begin{displaymath}\begin{cases}x'(t)=f(x(t),t)\\ x(t_0)=x_0 \end{cases}\end{displaymath} (1)

if
  1. $ x$ is a differentiable function $ x\colon I\to \mathbb{R}^n$ defined on a interval $ I\subset \mathbb{R}$;
  2. one has $ (x(t),t)\in D$ for all $ t\in I$ and $ t_0\in I$;
  3. one has $ x(t_0)=x_0$ and $ x'(t)=f(x(t),t)$ for all $ t\in I$.

We say that a solution $ x\colon I\to\mathbb{R}^n$ is a maximal solution if it cannot be extended to a bigger interval. More precisely given any other solution $ y\colon J\to \mathbb{R}^n$ defined on an interval $ J\supset I$ and such that $ y(t)=x(t)$ for all $ t\in I$, one has $ I=J$ (and hence $ x$ and $ y$ are the same function).

We say that a solution $ x\colon I\to\mathbb{R}^n$ is a global solution if $ I=\mathbb{R}$.

We say that a solution $ x\colon I\to\mathbb{R}^n$ is unique if given any other solution $ y\colon I\to\mathbb{R}^n$ one has $ x(t)=y(t)$ for all $ t\in I$ (i.e. $ x$ is the unique solution defined on the interval $ I$).

Notation

Usually the differential equation in (1) is simply written as $ x'=f(x,t)$. Also, depending on the topics, the name chosen for the function and for the variable, can change. Other common choices are $ y'=f(y,t)$ or $ y'=f(y,x)$. It is also common to write $ \dot x=f(x,t)$ when the independent variable represents a time value.

Examples

  1. The function $ x(t)=\log t$ defined on $ I=(0,+\infty)$ is the unique maximal solution to the Cauchy problem:
    \begin{displaymath} \begin{cases} x'(t) = 1/t\ x(1)=0. \end{cases}\end{displaymath}
    In this case $ f(x,t)=1/t$, $ D=\{(x,t)\colon t\neq 0\}$, $ t_0=1$, $ x_0=0$.
  2. The function $ x(t)=e^x$ is a global (and hence maximal), unique solution to the Cauchy problem:
    \begin{displaymath} \begin{cases} x'(t) = x(t)\ x(0)=1. \end{cases}\end{displaymath}
  3. Consider the Cauchy problem
    \begin{displaymath} \begin{cases} x'(t) = \frac 3 2 \sqrt[3] x\ x(0)=0. \end{cases}\end{displaymath}
    The function $ x(t)=0$ defined on $ I=\mathbb{R}$ is a global solution. However the function $ y(t)=\sqrt{t^3}$ defined on $ I=[0,+\infty)$ is also a solution and so are the functions
    $\displaystyle z(t)=\begin{cases}\sqrt{(t-c)^3}&\text{if $t\ge c$} \\ 0 &\text{if $t < c$}.\end{cases}$
    for every $ c\ge 0$. So there are no unique solutions. Moreover $ y$ 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

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