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Cauchy problem

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From Wikipedia, the free encyclopedia
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  1. Cauchy problem

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.1 A Cauchy problem may involve initial or boundary values. It is named after Augustin-Louis Cauchy.

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Contents
  • (Top)
  • Formal statement
  • Cauchy–Kowalevski theorem
  • See also
  • References
  • Further reading
  • External links
  • Recent changes
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Contents8 sections
  • (Top)
  • Formal statement
  • Cauchy–Kowalevski theorem
  • See also
  • References
  • Further reading
  • External links
  • Recent changes
  • Rabbit hole

Formal statement

For a partial differential equation defined on and a smooth manifold of dimension ( is called the Cauchy surface), the Cauchy problem consists of finding the unknown functions of the differential equation with respect to the independent variables that satisfies2

subject to the condition, for some value ,

where are given functions defined on the surface (collectively known as the Cauchy data of the problem). The derivative of order zero means that the function itself is specified.

Cauchy–Kowalevski theorem

The Cauchy–Kovalevskaya theorem, named in honor of Cauchy and Sofya Kovalevskaya, states: If all the functions are analytic in some neighborhood of the point , and if all the functions are analytic in some neighborhood of the point , then the Cauchy problem has a unique analytic solution in some neighborhood of the point .

See also

  • Cauchy boundary condition
  • Cauchy horizon

References

  1. 1Hadamard, Jacques (1923). Lectures on Cauchy's Problem in Linear Partial Differential Equations. New Haven: Yale University Press. pp. 4–5. OCLC 1880147.
  2. 2Petrovsky, I. G. (1991) [1954]. Lectures on Partial Differential Equations. Translated by Shenitzer, A. (Dover ed.). New York: Interscience. p. 14. ISBN 0-486-66902-5.

Further reading

  • Hille, Einar (1956)[1954]. Some Aspect of Cauchy's Problem Proceedings of 1954 ICM vol III section II (analysis half-hour invited address) p. 1 0 9 ~ 1 6.
  • Sigeru Mizohata(溝畑 茂 1965). Lectures on Cauchy Problem. Tata Institute of Fundamental Research.
  • Sigeru Mizohata (1985).On the Cauchy Problem. Notes and Reports in Mathematics in Science and Engineering. 3. Academic Press, Inc.. ISBN 9781483269061
  • Arendt, Wolfgang; Batty, Charles; Hieber, Matthias; Neubrander, Frank (2001), Vector-valued Laplace Transforms and Cauchy Problems, Birkhauser.

External links

  • Cauchy problem at MathWorld.
Study aids for Cauchy problem

Terminology

Cauchy data
where are given functions defined on the surface (collectively known as the Cauchy data of the problem).

Terms the article defines in bold, quoted in context.

Categories:
  • Partial differential equations
  • Mathematical problems
  • Boundary value problems

174 words · 2 references · revision 1355951422 · CC BY-SA 4.0

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Key concepts

  • Partial differential equation1
  • Hypersurface1
  • Initial condition1
  • Boundary value problem1
  • Augustin-Louis Cauchy1
  • Smooth manifold1
  • Cauchy surface1
  • Cauchy–Kovalevskaya theorem1
  • Sofya Kovalevskaya1
  • Analytic function1

The articles this one links to most, weighted towards the introduction.

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Created
11 Apr 2005
History
88 edits
63 contributors
No edits in the last 30 days
Last edited
24 May 2026

88 revisions in total · 63 contributors

  1. 24 May 2026, 21:56 UTCMaxwellsdaemon2Add page number to reference+9diff
  2. 20 Feb 2026, 15:35 UTCNerd271Miscellaneous improvements.+72diff
  3. 20 Feb 2026, 15:30 UTCNerd271Cleaning up.−35diff
  4. 19 Nov 2025, 13:14 UTCBunnysBot§ Further reading Fix CW Errors with GenFixes (T1)−11diff
  5. 17 Aug 2025, 20:31 UTCTurtlensm§ Further reading+1diff
  6. 23 Apr 2025, 22:54 UTCRich Farmbrough§ References+17diff
  7. 9 Oct 2024, 20:46 UTCRetroflexivityModified formatting in added references.+14diff
  8. 9 Oct 2024, 20:39 UTCRetroflexivityUndid revision 1250336972 by Retroflexivity (talk)+1diff

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Created
11 Apr 2005
History
88 edits
63 contributors
No edits in the last 30 days
Last edited
24 May 2026

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