Initial conditions

Conditions imposed in formulating the Cauchy problem for differential equations. For an ordinary differential equation in the form

$$\begin{array}{}\text{(1)}& u^{(}m)=F(t,u,u^{\prime }\dots u^{(m-1)}),\end{array}$$

the initial conditions prescribe the values of the derivatives (Cauchy data):

$$\begin{array}{}\text{(2)}& u(t_0)=u_0\dots u^{(m-1)}(t_0)=u_0^{(m-1)},\end{array}$$

where $(t_0,u_0\dots u_0^{(m-1)})$ is an arbitrary fixed point of the domain of definition of the function $F$; this point is known as the initial point of the required solution. The Cauchy problem (1), (2) is often called an initial value problem.

For a partial differential equation, written in normal form with respect to a distinguished variable $t$,

$$Lu=\frac{{\partial }^{m}u}{\partial t^m}-F(x,t,\frac{{\partial }^{\alpha +k}u}{\partial x^{\alpha }\partial t^k})=0,$$

$$|\alpha |+k\le N,0\le k<m,x=(x_1\dots x_n),$$

the initial conditions consist in prescribing the values of the derivatives (Cauchy data)

$${\frac{{\partial }^{k}u}{\partial t^k}|}_{t=0}={\varphi }_k(x),k=0\dots m-1,$$

of the required solution $u(x,t)$ on the hyperplane $t=0$( the support of the initial conditions).

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