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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 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.

Formal statement

For a partial differential equation defined on Rⁿ⁺¹ and a smooth manifold S ⊂ Rⁿ⁺¹ of dimension n (S is called the Cauchy surface), the Cauchy problem consists of finding the unknown functions $u_{1},\dots ,u_{N}$ of the differential equation with respect to the independent variables $t,x_{1},\dots ,x_{n}$ that satisfies^[2] ${\begin{aligned}&{\frac {\partial ^{n_{i}}u_{i}}{\partial t^{n_{i}}}}=F_{i}\left(t,x_{1},\dots ,x_{n},u_{1},\dots ,u_{N},\dots ,{\frac {\partial ^{k}u_{j}}{\partial t^{k_{0}}\partial x_{1}^{k_{1}}\dots \partial x_{n}^{k_{n}}}},\dots \right)\\&{\text{for }}i,j=1,2,\dots ,N;\,k_{0}+k_{1}+\dots +k_{n}=k\leq n_{j};\,k_{0}<n_{j}\end{aligned}}$ subject to the condition, for some value $t=t_{0}$ ,

${\frac {\partial ^{k}u_{i}}{\partial t^{k}}}=\phi _{i}^{(k)}(x_{1},\dots ,x_{n})\quad {\text{for }}k=0,1,2,\dots ,n_{i}-1$

where $\phi _{i}^{(k)}(x_{1},\dots ,x_{n})$ 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.

Cauchy–Kowalevski theorem

The Cauchy–Kowalevski theorem states that If all the functions $F_{i}$ are analytic in some neighborhood of the point $(t^{0},x_{1}^{0},x_{2}^{0},\dots ,\phi _{j,k_{0},k_{1},\dots ,k_{n}}^{0},\dots )$ , and if all the functions $\phi _{j}^{(k)}$ are analytic in some neighborhood of the point $(x_{1}^{0},x_{2}^{0},\dots ,x_{n}^{0})$ , then the Cauchy problem has a unique analytic solution in some neighborhood of the point $(t^{0},x_{1}^{0},x_{2}^{0},\dots ,x_{n}^{0})$ .

References

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

3. Hille,Einar (1956)[1954]. Some Aspect of Cauchy's Problem Proceedings of '5 4 ICM vol III section II (analysis half-hour invited address) p.1 0 9 ~ 1 6.
4. Sigeru Mizohata(溝畑茂 1965). Lectures on Cauchy Problem. Tata Institute of Fundamental Research.
5. Sigeru Mizohata (1985).On the Cauchy Problem. Notes and Reports in Mathematics in Science and Engineering. 3. Academic Press, Inc.. ISBN 9781483269061
6. Arendt, Wolfgang; Batty, Charles; Hieber, Matthias; Neubrander, Frank (2001), Vector-valued Laplace Transforms and Cauchy Problems, Birkhauser.

External links

Cauchy problem at MathWorld.

[1] Hadamard, Jacques (1923). Lectures on Cauchy's Problem in Linear Partial Differential Equations. New Haven: Yale University Press. pp. 4–5. OCLC 1880147.

[2] Petrovsky, I. G. (1991) [1954]. Lectures on Partial Differential Equations. Translated by Shenitzer, A. (Dover ed.). New York: Interscience. ISBN 0-486-66902-5.

[1]

[2]

@@ Line 1: / Line 1: @@
+{{Short description|Class of problems for PDEs}}
-A '''Cauchy problem''' in mathematics asks for the solution of a [[partial differential equation]] that satisfies certain conditions which are given on a [[hypersurface]] in the domain.
+{{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.<ref>{{cite book |first=Jacques |last=Hadamard |authorlink=Jacques Hadamard |year=1923 |title=Lectures on Cauchy's Problem in Linear Partial Differential Equations |location=New Haven |publisher=Yale University Press |pages=4–5 |oclc=1880147 }}</ref> 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]].
-==Introduction==
-A Cauchy problem can be an [[initial value problem]] or a [[boundary value problem]] (for this case see also [[Cauchy boundary condition]]), but it can be none of them. They are named after [[Augustin Louis Cauchy]].
 ==Formal statement==
-Suppose that the partial differential equation is defined on '''R'''<sup>''n''</sup> and consider a [[smooth manifold]] ''S'' ⊂ '''R'''<sup>''n''</sup> of dimension ''n'' &minus; 1 (''S'' is called the [[Cauchy surface]]). Then the Cauchy problem consists of finding the solution ''u'' of the differential equation that satisfies
+For a partial differential equation defined on '''R'''<sup>''n+1''</sup> and a [[smooth manifold]] ''S'' ⊂ '''R'''<sup>''n+1''</sup> of dimension ''n'' (''S'' is called the [[Cauchy surface]]), the Cauchy problem consists of finding the unknown functions <math>u_1,\dots,u_N</math> of the differential equation with respect to the independent variables <math>t,x_1,\dots,x_n</math> that satisfies<ref>{{cite book |last=Petrovsky |first=I. G. |authorlink=Ivan Petrovsky |orig-year=1954 |title=Lectures on Partial Differential Equations |publisher=Interscience |location=New York |translator-first=A. |translator-last=Shenitzer |edition=Dover |year=1991 |isbn=0-486-66902-5 }}</ref>
-:<math> \begin{align}
+<math display="block">\begin{align}&\frac{\partial^{n_i}u_i}{\partial t^{n_i}} = F_i\left(t,x_1,\dots,x_n,u_1,\dots,u_N,\dots,\frac{\partial^k u_j}{\partial t^{k_0}\partial x_1^{k_1}\dots\partial x_n^{k_n}},\dots\right) \\
-u(x) &= f_0(x) \qquad && \text{for all } x\in S; \\
+&\text{for } i,j = 1,2,\dots,N;\, k_0+k_1+\dots+k_n=k\leq n_j;\, k_0<n_j
-\frac{\part^k u(x)}{\part n^k} &= f_k(x) \qquad && \text{for } k=1,\ldots,\kappa-1 \text{ and all } x\in S,
-\end{align} </math>
+\end{align}</math>
+subject to the condition, for some value <math>t=t_0</math>,
-where <math>f_k</math> are given functions defined on the surface <math>S</math> (collectively known as the '''Cauchy data''' of the problem), ''n'' is a [[normal vector]] to ''S'', and κ denotes the order of the differential equation.
+<math display="block">\frac{\partial^k u_i}{\partial t^k}=\phi_i^{(k)}(x_1,\dots,x_n)
+\quad \text{for } k=0,1,2,\dots,n_i-1</math>
+where <math>\phi_i^{(k)}(x_1,\dots,x_n)</math> are given functions defined on the surface <math>S</math> (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–Kowalevski theorem]] states that ''If all the functions <math>F_i</math> are [[analytic function|analytic]] in some neighborhood of the point <math>(t^0,x_1^0,x_2^0,\dots,\phi_{j,k_0,k_1,\dots,k_n}^0,\dots)</math>, and if all the functions <math>\phi_j^{(k)}</math> are analytic in some neighborhood of the point <math>(x_1^0,x_2^0,\dots,x_n^0)</math>, then the Cauchy problem has a unique analytic solution in some neighborhood of the point <math>(t^0,x_1^0,x_2^0,\dots,x_n^0)</math>''.
-The [[Cauchy–Kowalevski theorem]] says that Cauchy problems have unique solutions under certain conditions, the most important of which being that the Cauchy data and the coefficients of the partial differential equation be [[real analytic]] functions.
 == See also ==
+{{Portal|Mathematics}}
 *[[Cauchy boundary condition]]
+*[[Cauchy horizon]]
 ==References==
+{{Reflist}}
-*{{citation|title=Lectures on Cauchy's Problem in Linear Partial Differential Equations|first=Jacques|last= Hadamard|year=2003|origyear= 1923|isbn=0-486-49549-3|series=Dover Phoenix editions|url=http://books.google.com/books?id=B25O-x21uqkC|mr=0051411 {{JFM|49.0725.04}}|publisher=Dover Publications|place= New York}}
+. Hille,Einar (1956)[1954]. Some Aspect of Cauchy's Problem Proceedings of '5 4 ICM vol III section II (analysis half-hour invited address) p.1 0 9 ~ 1 6. <br>
+. Sigeru Mizohata(溝畑 茂 1965). Lectures on Cauchy Problem. Tata Institute of Fundamental Research.<br>
+. Sigeru Mizohata (1985).On the Cauchy Problem. Notes and Reports in Mathematics in Science and Engineering. 3. Academic Press, Inc.. ISBN 9781483269061 <br>
+. Arendt, Wolfgang; Batty, Charles; Hieber, Matthias; Neubrander, Frank (2001), Vector-valued Laplace Transforms and Cauchy Problems, Birkhauser. <br>
 == External links ==
 * [http://mathworld.wolfram.com/CauchyProblem.html Cauchy problem] at [[MathWorld]].
+{{Authority control}}
 [[Category:Partial differential equations]]
 [[Category:Mathematical problems]]
+[[Category:Boundary value problems]]
-[[de:Anfangswertproblem#Partielle_Differentialgleichungen]]
+[[de:Anfangswertproblem#Partielle Differentialgleichungen]]