Encyclopedia

Cauchy problem

Class of problems for PDEs

Wikipedia contributorsAdapted for Fathomly · View authors

1 min read
Source updated May 24, 2026

A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain. A Cauchy problem may involve initial or boundary values. It is named after Augustin-Louis Cauchy.

01Formal statement

For a partial differential equation defined on $\mathbb {R} ^{n+1}$ and a smooth manifold $S\subset \mathbb {R} ^{n+1}$ of dimension $n$ ( $S$ is called the Cauchy surface), the Cauchy problem consists of finding the unknown functions $u_{1},\dots ,u_{N}$ of the differential equation with respect to the independent variables $t,x_{1},\dots ,x_{n}$ that satisfies ${\begin{aligned}&{\frac {\partial ^{n_{i}}u_{i}}{\partial t^{n_{i}}}}=F_{i}\left(t,x_{1},\dots ,x_{n},u_{1},\dots ,u_{N},\dots ,{\frac {\partial ^{k}u_{j}}{\partial t^{k_{0}}\partial x_{1}^{k_{1}}\dots \partial x_{n}^{k_{n}}}},\dots \right)\\&{\text{for }}i,j=1,2,\dots ,N;\,k_{0}+k_{1}+\dots +k_{n}=k\leq n_{j};\,k_{0}<n_{j}\end{aligned}}$ subject to the condition, for some value $t=t_{0}$ ,

${\frac {\partial ^{k}u_{i}}{\partial t^{k}}}=\phi _{i}^{(k)}(x_{1},\dots ,x_{n})\quad {\text{for }}k=0,1,2,\dots ,n_{i}-1$

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

02Cauchy-Kowalevski theorem

The Cauchy-Kovalevskaya theorem, named in honor of Cauchy and Sofya Kovalevskaya, states: If all the functions $F_{i}$ are analytic in some neighborhood of the point $(t^{0},x_{1}^{0},x_{2}^{0},\dots ,\phi _{j,k_{0},k_{1},\dots ,k_{n}}^{0},\dots )$ , and if all the functions $\phi _{j}^{(k)}$ are analytic in some neighborhood of the point $(x_{1}^{0},x_{2}^{0},\dots ,x_{n}^{0})$ , then the Cauchy problem has a unique analytic solution in some neighborhood of the point $(t^{0},x_{1}^{0},x_{2}^{0},\dots ,x_{n}^{0})$ .

Watch videos about Cauchy problemExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Cauchy problem”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Cauchy problem

01Formal statement

02Cauchy-Kowalevski theorem

Sources and credits

Related topics

Cauchy boundary condition

Cauchy horizon