Cauchy Problem - Revision history

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2021-11-06T03:50:26Z

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Lila at 21:57, 14 October 2021

2021-10-14T21:57:53Z

Lila: Created page with "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..."

2021-10-14T21:57:19Z

Created page with "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..."

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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. A Cauchy problem can be an initial value problem or a boundary value problem. It is named after Augustin-Louis Cauchy.

==Formal statement==

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
:<math>\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) \\
&\text{for } i,j = 1,2,\dots,N;\, k_0+k_1+\dots+k_n=k\leq n_j;\, k_0<n_j
\end{align}</math>
subject to the condition, for some value <math>t=t_0</math>,

:<math>\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 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>''.

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 The Cauchy–Kowalevski theorem states that ''If all the functions <math>F_i</math> are 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>''.
-==Resources==
+== Licensing ==
-* [https://en.wikipedia.org/wiki/Cauchy_problem Cauchy problem], Wikipedia
+Content obtained and/or adapted from:

← Older revision		Revision as of 21:57, 14 October 2021
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	==Cauchy–Kowalevski theorem==		==Cauchy–Kowalevski theorem==
	The Cauchy–Kowalevski theorem states that ''If all the functions <math>F_i</math> are 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 states that ''If all the functions <math>F_i</math> are 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>''.
		+
		+	==Resources==
		+	* [https://en.wikipedia.org/wiki/Cauchy_problem Cauchy problem], Wikipedia