Movatterモバイル変換

Jump to content

Cauchy problem

From Wikipedia, the free encyclopedia

Class of problems for PDEs

Differential equations

Scope

Fields

Classification

Types

By variable type
Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear
Dependent and independent variables Autonomous Coupled / Decoupled Exact Homogeneous / Nonhomogeneous
Features
Order Operator Notation

Relation to processes

Difference(discrete analogue)

Solution

Existence and uniqueness

General topics

Initial conditions
Boundary values
Wronskian
Phase portrait
Lyapunov / Asymptotic / Exponential stability
Rate of convergence
Series / Integral solutions
Numerical integration
Dirac delta function

Solution methods

People

List

Specific equations

Named equations

List of named differential equations

v
t
e

ACauchy problem in mathematics asks for the solution of apartial differential equation that satisfies certain conditions that are given on ahypersurface in the domain.^[1] A Cauchy problem can be aninitial value problem or aboundary value problem (for this case see alsoCauchy boundary condition). It is named afterAugustin-Louis Cauchy.

Formal statement

For a partial differential equation defined onRⁿ⁺¹ and asmooth manifoldS ⊂Rⁿ⁺¹ of dimensionn (S is called theCauchy 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 $S {\displaystyle S}$ (collectively known as theCauchy data of the problem). The derivative of order zero means that the function itself is specified.

Cauchy–Kowalevski theorem

TheCauchy–Kowalevski theorem states thatIf all the functions $F_{i}$ areanalytic 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 $(t^{0},x_{1}^{0},x_{2}^{0},\dots ,x_{n}^{0})$ .

See also

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.

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.

External links

Cauchy problem atMathWorld.

Retrieved from "https://en.wikipedia.org/w/index.php?title=Cauchy_problem&oldid=1323062476"

Hidden categories:

[8]ページ先頭

©2009-2026 Movatter.jp