Cauchy problem

for a PDE on $R^{n + 1}$ . Taken from Wikipedia.

Given a partial differential equation defined on $R^{n + 1}$ and a smooth manifold $S \subset R^{n + 1}$ of dimension $n$ (called the Cauchy surface), the Cauchy problem is the task of finding the unknown functions $u_{1}, \dots, u_{N}$ of the differential equation with respect to the independent variables $t, x_{1}, \dots, x_{n}$ , such that:

\frac{\partial^{n_{i}} u_{i}}{\partial t^{n_{i}}} = F_{i} (t, x_{1}, \dots, x_{n}, u_{1}, \dots, u_{N}, \dots, \frac{\partial^{k} u_{j}}{\partial t^{k_{0}} \partial x_{1}^{k_{1}} \dots \partial x_{n}^{k_{n}}}, \dots)

for $i, j = 1, 2, \dots, N$ , subject to the following conditions:

$k_{0} + k_{1} + \dots + k_{n} = k \leq n_{j}$ ,
$k_{0} < n_{j}$ .

The solution must satisfy the given initial conditions at some value $t = t_{0}$ :

\frac{\partial^{k} u_{i}}{\partial t^{k}} = ϕ_{i}^{(k)} (x_{1}, \dots, x_{n}) for k = 0, 1, 2, \dots, n_{i} - 1,

where:

$ϕ_{i}^{(k)} (x_{1}, \dots, x_{n})$ are given functions defined on the surface $S$ (collectively known as the Cauchy data of the problem).
The derivative of order zero indicates that the function itself is specified.

Existence of solution is guaranteed by Cauchy–Kowalevski theorem.