Initial conditions

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


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


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).



[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. A.P. Soldatov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098