Cauchy problem

English

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 ${\\displaystyle u}$ $\text{[math]}$ on ${\\displaystyle \\mathbb {R} ^{n}}$ $\text{[math]}$ that satisfies the boundary conditions that, for a smooth manifold ${\\displaystyle S\\subset \\mathbb {R} ^{n}}$ $\text{[math]}$ , ${\\displaystyle \\textstyle u(x)=f_{0}(x)}$ $\text{[math]}$ and ${\\displaystyle \\textstyle {\\frac {\\partial ^{k}u(x)}{\\partial n^{k}}}=f_{k}(x)}$ $\text{[math]}$ , ${\\displaystyle \\forall x\\in S}$ $\text{[math]}$ , ${\\displaystyle k=1\\dots m-1}$ $\text{[math]}$ , given specified functions ${\\displaystyle f_{k}}$ $\text{[math]}$ defined on, and vector ${\\displaystyle n}$ $\text{[math]}$ normal to, the manifold.
- 2006, Victor Isakov, Inverse Problems for Partial Differential Equations, Springer, 2nd Edition, 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, 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 ${\\displaystyle C^{\\infty }}$ 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