Cauchy problem

 ( Partial Differential Equations ) 

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 can be an initial value problem or a boundary value problem (for this case see also Cauchy boundary condition). It is named after Augustin-Louis Cauchy.

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Formal statement For a partial differential equation defined on R ⁿ⁺¹ and a smooth manifold S ⊂ R ⁿ⁺¹ of dimension n ( S is called the Cauchy surface), the Cauchy problem consists of finding the unknown functions $u\_1,\backslash dots,u\_N$ of the differential equation with respect to the independent variables $t,x\_1,\backslash dots,x\_n$ that satisfies

I. G. Petrovsky (1991). 9780486669021, Interscience. ISBN 9780486669021

$$\backslash frac\{\backslash partial^k\; u\_i\}\{\backslash partial\; t^k\}=\backslash phi\_i^\{(k)\}(x\_1,\backslash dots,x\_n)\; \backslash quad\; \backslash text\{for\; \}\; k=0,1,2,\backslash dots,n\_i-1$$

where $\backslash phi\_i^\{(k)\}(x\_1,\backslash dots,x\_n)$ 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.

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Cauchy–Kowalevski theorem The Cauchy–Kowalevski theorem states that If all the functions $F\_i$ are analytic in some neighborhood of the point $(t^0,x\_1^0,x\_2^0,\backslash dots,\backslash phi\_\{j,k\_0,k\_1,\backslash dots,k\_n\}^0,\backslash dots)$, and if all the functions $\backslash phi\_j^\{(k)\}$ are analytic in some neighborhood of the point $(x\_1^0,x\_2^0,\backslash dots,x\_n^0)$, then the Cauchy problem has a unique analytic solution in some neighborhood of the point $(t^0,x\_1^0,x\_2^0,\backslash dots,x\_n^0)$.

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See also

• Cauchy boundary condition
• Cauchy horizon

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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.
  

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External links

• Cauchy problem at MathWorld.

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