Initial Conditions

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).

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