# Initial conditions

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

(1) |

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

(2) |

where is an arbitrary fixed point of the domain of definition of the function ; 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 ,

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

of the required solution on the hyperplane (the support of the initial conditions).

#### Comments

#### References

[a1] | E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956) |

[a2] | S. Mizohata, "The theory of partial differential equations" , Cambridge Univ. Press (1973) (Translated from Japanese) |

**How to Cite This Entry:**

Initial conditions.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Initial_conditions&oldid=12913