Cauchy problem

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Cauchy problem

Wikipedia

Etymology

After French mathematician Augustin Louis Cauchy.

Noun

Cauchy problem (plural Cauchy problems)

(mathematics, mathematical analysis) For a given m-order partial differential equation, the problem of finding a solution function $u$ $u$ on $\mathbb {R} ^{n}$ $\mathbb {R} ^{n}$ that satisfies the boundary conditions that, for a smooth manifold $S\subset \mathbb {R} ^{n}$ $S\subset \mathbb {R} ^{n}$ , $\textstyle u(x)=f_{0}(x)$ $\textstyle u(x)=f_{0}(x)$ and $\textstyle {\frac {\partial ^{k}u(x)}{\partial n^{k}}}=f_{k}(x)$ $\textstyle {\frac {\partial ^{k}u(x)}{\partial n^{k}}}=f_{k}(x)$ , $\forall x\in S$ $\forall x\in S$ , $k=1\dots m-1$ $k=1\dots m-1$ , given specified functions $f_{k}$ $f_{k}$ defined on, and vector $n$ $n$ 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.