# Initial conditions

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

$$\tag{1 } u ^ {(} m) = F ( t, u , u ^ \prime \dots u ^ {( m - 1) } ) ,$$

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

$$\tag{2 } u ( t _ {0} ) = u _ {0} \dots u ^ {( m - 1) } ( t _ {0} ) = u _ {0} ^ {( m - 1) } ,$$

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 $t$,

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

$$| \alpha | + k \leq N,\ 0 \leq k < m,\ x = ( x _ {1} \dots x _ {n} ),$$

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

$$\left . \frac{\partial ^ {k} u }{\partial t ^ {k} } \right | _ {t = 0 } = \ \phi _ {k} ( x),\ \ k = 0 \dots m - 1,$$

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