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)|
Initial conditions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Initial_conditions&oldid=12913