Cauchy problem

English

Etymology

After French mathematician Augustin Louis Cauchy.

Noun

Cauchy problem (plural Cauchy problems)

1. (mathematics, mathematical analysis) For a given m-order partial differential equation, the problem of finding a solution function $u$ on $\mathbb {R} ^{n}$ that satisfies the boundary conditions that, for a smooth manifold $S\subset \mathbb {R} ^{n}$, $\textstyle u(x)=f_{0}(x)$ and $\textstyle {\frac {\partial ^{k}u(x)}{\partial n^{k}}}=f_{k}(x)$, $\forall x\in S$, $k=1\dots m-1$, given specified functions $f_{k}$ defined on, and vector $n$ normal to, the manifold.
   • 2006, Victor Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Springer, page 41:

     In this chapter we formulate and in many cases prove results on uniqueness and stability of solutions of the Cauchy problem for general partial differential equations.

   • 2006, S. Albeverio, Ya. Belopolskaya, “Probabilistic Interpretation of the VV-Method for PDE Systems”, in Olga S. Rozanova, editor, Analytical Approaches to Multidimensional Balance Laws, Nova Science Publishers, page 2:

     To emphasize the similarity between the characteristic method and the probabilistic approach we recall that the method of characteristics allows [one] to reduce the Cauchy problem for a first order PDE to a Cauchy problem for an ODE while the probabilistic approach allows [one] to reduce the Cauchy problem for a second order PDE to a Cauchy problem for an SDE (stochastic differential equation).

   • 2014, Tatsuo Nishitani, Hyperbolic Systems with Analytic Coefficients: Well-posedness of the Cauchy Problem, Springer, page v:

     In this monograph we discuss the $C^{\infty }$ well-posedness of the Cauchy problem for hyperbolic systems.

Usage notes

The hypersurface S is called the Cauchy surface. The functions f_k defined on S are collectively known as the Cauchy data of the problem.

Translations

class of problem involving simultaneous PDEs

• Italian: problema di Cauchy m

Further reading

• Cauchy boundary condition on Wikipedia.Wikipedia
• Cauchy-Kowalevski theorem on Wikipedia.Wikipedia