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. 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 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_1,\dots,u_N} of the differential equation with respect to the independent variables $t,x_{1},\dots ,x_{n}$ that satisfies

${\begin{aligned}&{\frac {\partial ^{n_{i}}u_{i}}{\partial t^{n_{i}}}}=F_{i}\left(t,x_{1},\dots ,x_{n},u_{1},\dots ,u_{N},\dots ,{\frac {\partial ^{k}u_{j}}{\partial t^{k_{0}}\partial x_{1}^{k_{1}}\dots \partial x_{n}^{k_{n}}}},\dots \right)\\&{\text{for }}i,j=1,2,\dots ,N;\,k_{0}+k_{1}+\dots +k_{n}=k\leq n_{j};\,k_{0}<n_{j}\end{aligned}}$

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_i^{(k)}(x_1,\dots,x_n)} are given functions defined on the surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_i} are analytic in some neighborhood of the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_j^{(k)}} are analytic in some neighborhood of the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (t^0,x_1^0,x_2^0,\dots,x_n^0)} .

Licensing

Content obtained and/or adapted from:

• Cauchy problem, Wikipedia under a CC BY-SA license