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 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.

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 ${\displaystyle u_{1},\dots ,u_{N}}${\displaystyle u_{1},\dots ,u_{N}} of the differential equation with respect to the independent variables ${\displaystyle t,x_{1},\dots ,x_{n}}${\displaystyle t,x_{1},\dots ,x_{n}} that satisfies^[2] $${\displaystyle {\begin{aligned}&{\frac {\partial ^{n_{i}}u_{i}}{\partial t^{n_{i}}}}=F_{i}\left(t,x_{1},\dots ,x_{n},u_{1},\dots ,u_{N},\dots ,{\frac {\partial ^{k}u_{j}}{\partial t^{k_{0}}\partial x_{1}^{k_{1}}\dots \partial x_{n}^{k_{n}}}},\dots \right)\\&{\text{for }}i,j=1,2,\dots ,N;,円k_{0}+k_{1}+\dots +k_{n}=k\leq n_{j};,円k_{0}<n_{j}\end{aligned}}}$${\displaystyle {\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;,円k_{0}+k_{1}+\dots +k_{n}=k\leq n_{j};,円k_{0}<n_{j}\end{aligned}}} subject to the condition, for some value ${\displaystyle t=t_{0}}${\displaystyle t=t_{0}},

$${\displaystyle {\frac {\partial ^{k}u_{i}}{\partial t^{k}}}=\phi _{i}^{(k)}(x_{1},\dots ,x_{n})\quad {\text{for }}k=0,1,2,\dots ,n_{i}-1}$${\displaystyle {\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 ${\displaystyle \phi _{i}^{(k)}(x_{1},\dots ,x_{n})}${\displaystyle \phi _{i}^{(k)}(x_{1},\dots ,x_{n})} are given functions defined on the surface ${\displaystyle S}${\displaystyle S} (collectively known as the Cauchy data of the problem). The derivative of order zero means that the function itself is specified.

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

• Cauchy boundary condition
• Cauchy horizon

• 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.

• Cauchy problem at MathWorld.

• Partial differential equations
• Mathematical problems
• Boundary value problems

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