Cauchy Problem, Initial Value Problem 📂Partial Differential Equations

Cauchy Problem, Initial Value Problem

Definition¹

Let’s say we have a partial differential equation defined on an open set $\Omega=\mathbb{R}^{n}$. When the time is $t=0$, the value of the unknown $u$ at $\Omega$, that is, the initial value, is given. The problem of finding solutions to such partial differential equations is called the Cauchy problem or the initial value problem.

Explanation

The acronym IVP is commonly used.

Example

Solving the Cauchy problem for the heat equation means finding the solution under the following condition:

$$ \left\{ \begin{align*} u_{t} -\Delta u &= 0 && \text{in } \mathbb{R}^{n} \times (0,\infty) \\ u &= g && \text{on } \mathbb{R}^{n} \times \left\{ t=0 \right\} \end{align*} \right. $$