Cauchy problem

Short description: Class of problems for PDEs

Differential equations

Navier–Stokes differential equations used to simulate airflow around an obstruction.

Scope

* Natural sciences
* Engineering

* Astronomy
* Physics
* Chemistry

Biology
* Geology

Applied mathematics

* Continuum mechanics
* Chaos theory
* Dynamical systems

Social sciences

* Economics
* Population dynamics

Classification

Types

* Ordinary
* Partial
* Differential-algebraic
* Integro-differential
* Fractional
* Linear
* Non-linear

By variable type

* Dependent and independent variables

* Autonomous
* Complex
* Coupled / Decoupled
* Exact
* Homogeneous / Nonhomogeneous

Features

* Order
* Operator

* Notation

Relation to processes

* Difference (discrete analogue)

* Stochastic
* Stochastic partial
* Delay

Solution

General topics

* Picard–Lindelöf theorem
* Wronskian

* Phase portrait
* Phase space

* Lyapunov / Asymptotic / Exponential stability
* Rate of convergence
* Series / Integral solutions

* Numerical integration
* Dirac delta function

Solution methods

* Inspection
* Method of characteristics

Euler
* Exponential response formula
* Finite difference (Crank–Nicolson)
* Finite element
* Infinite element
* Finite volume
* Galerkin
* Petrov–Galerkin
* Integrating factor
* Integral transforms
* Perturbation theory
* Runge–Kutta

* Separation of variables
* Undetermined coefficients
* Variation of parameters

People

* Isaac Newton
* Gottfried Leibniz
* Leonhard Euler
* Émile Picard
* Józef Maria Hoene-Wroński
* Ernst Lindelöf
* Rudolf Lipschitz
* Augustin-Louis Cauchy
* John Crank
* Phyllis Nicolson
* Carl David Tolmé Runge
* Martin Kutta

* v
* t
* e

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.

## Contents

* 1 Formal statement
* 2 Cauchy–Kowalevski theorem
* 3 See also
* 4 References
* 5 Further reading
* 6 External links

## 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 u1,…,uN of the differential equation with respect to the independent variables t,x1,…,xn that satisfies[2] ∂niui∂tni=Fi(t,x1,…,xn,u1,…,uN,…,∂kuj∂tk0∂x1k1…∂xnkn,…)for i,j=1,2,…,N;k0+k1+…+kn=k≤nj;k0<nj subject to the condition, for some value t=t0,

∂kui∂tk=ϕi(k)(x1,…,xn)for k=0,1,2,…,ni−1

where ϕi(k)(x1,…,xn) 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 Fi are analytic in some neighborhood of the point (t0,x10,x20,…,ϕj,k0,k1,…,kn0,…), and if all the functions ϕj(k) are analytic in some neighborhood of the point (x10,x20,…,xn0), then the Cauchy problem has a unique analytic solution in some neighborhood of the point (t0,x10,x20,…,xn0).

## See also

* Cauchy boundary condition
* Cauchy horizon

## References

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). Lectures on Partial Differential Equations (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 at MathWorld.

de:Anfangswertproblem#Partielle Differentialgleichungen

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