A Cauchy problem in mathematics asks for the solution of a

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...

that satisfies certain conditions that are given on a

hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...

in the domain. A Cauchy problem can be an

initial value problem In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or oth ...

or a boundary value problem (for this case see also

Cauchy boundary condition In mathematics, a Cauchy () boundary condition augments an ordinary differential equation or a partial differential equation with conditions that the solution must satisfy on the boundary; ideally so as to ensure that a unique solution exists. A Ca ...

). It is named after Augustin-Louis Cauchy.

Formal statement

For a partial differential equation defined on R^''n+1'' and a

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

''S'' ⊂ R^''n+1'' of dimension ''n'' (''S'' is called the

Cauchy surface In the mathematical field of Lorentzian geometry, a Cauchy surface is a certain kind of submanifold of a Lorentzian manifold. In the application of Lorentzian geometry to the physics of general relativity, a Cauchy surface is usually interpreted as ...

), 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 satisfiesPetrovskii, I. G. (1954). Lectures on partial differential equations. Interscience Publishers, Inc, Translated by A. Shenitzer, (Dover publications, 1991)

\begin&\frac = F_i\left(t,x_1,\dots,x_n,u_1,\dots,u_N,\dots,\frac,\dots\right) \\
&\text i,j = 1,2,\dots,N;\, k_0+k_1+\dots+k_n=k\leq n_j;\, k_0 subject to the condition, for some value t=t_0, \frac=\phi_i^(x_1,\dots,x_n)
\quad \text k=0,1,2,\dots,n_i-1 where \phi_i^(x_1,\dots,x_n) 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 In mathematics, the Cauchy–Kovalevskaya theorem (also written as the Cauchy–Kowalevski theorem) is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. A ...

states that ''If all the functions

F_i

are analytic in some neighborhood of the point

(t^0,x_1^0,x_2^0,\dots,\phi_^0,\dots)

, and if all the functions

\phi_j^

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)

''.

References

{{reflist

External links

Cauchy problem
at

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. Partial differential equations Mathematical problems Boundary value problems de:Anfangswertproblem#Partielle Differentialgleichungen

Formal statement

Cauchy–Kowalevski theorem

See also

References

External links