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
of the differential equation with respect to the independent variables
that satisfies
[Petrovskii, I. G. (1954). Lectures on partial differential equations. Interscience Publishers, Inc, Translated by A. Shenitzer, (Dover publications, 1991)]