By the mid eighty's the initial value problem of integrable evolution equations in one and two space variables was solved via the so called inverse scattering transform.
22.
Gr�nwall's inequality is an important tool to obtain various estimates in the theory of uniqueness of a solution to the initial value problem; see the Picard Lindel�f theorem.
23.
Walter Craig and Steven Weinstein proved the existence of a well-posed initial value problem for the ultrahyperbolic equation ( a wave equation in more than one time dimension ).
24.
Abstractly, if " A " is a linear operator acting on functions of " x ", then a convolution semigroup arises by solving the initial value problem
25.
In the theory of differential equations, Lipschitz continuity is the central condition of the Picard Lindel�f theorem which guarantees the existence and uniqueness of the solution to an initial value problem.
26.
In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem.
27.
An older proof of the Picard Lindel�f theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem.
28.
His book gives a necessary and sufficient condition for solutions of ordinary initial value problems to be unique and to depend on a class C 1 manner on the initial conditions for solutions.
29.
In a joint paper with Walter Craig, they gave the first well-posed initial value problem for the wave equation in more than one time dimension ( the ultrahyperbolic equation ).
30.
For the initial value problem of the heat equation, using the Fourier transform pair, it is straighforward to obtain both the relevant contour and the function U ( \ lambda ):
