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 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 [math]\displaystyle{ u_1,\dots,u_N }[/math] of the differential equation with respect to the independent variables [math]\displaystyle{ t,x_1,\dots,x_n }[/math] that satisfies[2] [math]\displaystyle{ \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\lt n_j \end{align} }[/math] subject to the condition, for some value [math]\displaystyle{ t=t_0 }[/math],

[math]\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 }[/math]

where [math]\displaystyle{ \phi_i^{(k)}(x_1,\dots,x_n) }[/math] are given functions defined on the surface [math]\displaystyle{ 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]\displaystyle{ F_i }[/math] are analytic in some neighborhood of the point [math]\displaystyle{ (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]\displaystyle{ \phi_j^{(k)} }[/math] are analytic in some neighborhood of the point [math]\displaystyle{ (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]\displaystyle{ (t^0,x_1^0,x_2^0,\dots,x_n^0) }[/math].

## See also

* Cauchy boundary condition
* Cauchy horizon

## References

1. ↑ Jacques Hadamard (1923), Lectures on Cauchy's Problem in Linear Partial Differential Equations, Dover Phoenix editions
2. ↑ Petrovskii, I. G. (1954). Lectures on partial differential equations. Interscience Publishers, Inc, Translated by A. Shenitzer, (Dover publications, 1991)

## External links

* Cauchy problem at MathWorld.

de:Anfangswertproblem#Partielle Differentialgleichungen

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