Cauchy problem

Another example is the Cauchy problem for the wave equation in R:

The following theorem connects abstract Cauchy problems and strongly continuous semigroups.

His first main works were devoted to the well-posedness of the Cauchy problem for weakly hyperbolic equations.

The authors of Cauchy problem in spacetimes with closed timelike curves write:

More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface.

The corresponding scalar Cauchy problem involving this function instead of the As and b has an explicit local analytic solution.

The Cauchy problem (sometimes called the initial value problem) is the attempt at finding a solution to a differential equation given initial conditions.

Consider the abstract Cauchy problem:

Cauchy problem at MathWorld.

The Cauchy problem for this equation consists in prescribing the initial displacement and velocity of a string or other medium:

The first paper where a sufficient condition for the solvability of the Cauchy problem for holomorphic functions of several complex variables is given.

As a result, the nascent delta functions that arise as fundamental solutions of the associated Cauchy problems are generally oscillatory integrals.

In 1958 Calderón published one of his most important results, on uniqueness of solution of the Cauchy problem for partial differential equations.

In a 1990 paper by Novikov and several others, Cauchy problem in spacetimes with closed timelike curves, the authors state:

In contrast, Cauchy problems impose a point-to-point convergence to a given boundary function and to all its derivatives (and this is a quite strong condition!)

Studying the Cauchy problem allows one to formulate the concept of causality in general relativity, as well as 'parametrising' solutions of the field equations.

The method has successfully been applied to inverse Cauchy problem associated with Poisson and nonhomogeneous Helmholtz equations.

It can be viewed as a Cauchy problem for minimal surfaces, allowing one to find a surface if a geodesic, asymptote or lines of curvature is known.

However, such theories in general do not have a well-defined Cauchy problem (for reasons related to the issues of causality discussed above), and are probably inconsistent quantum mechanically.

In connection with Cauchy problems, usually a linear operator A is given and the question is whether this is the generator of a strongly continuous semigroup.

In particular he discovered a necessary (later proven to be sufficient) condition for Cauchy problem to be well-posed no matter what the lower terms in the equation are.

A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions which are given on a hypersurface in the domain.

A paper describing the ideas of , giving some extensions of those ideas and a solution for a particular Cauchy problem for holomorphic functions of several variables.

They are one of several types of classes of PDE problems defined by the information given at the boundary, including Neumann problems and Cauchy problems.

The philosophy underlying Duhamel's principle is that it is possible to go from solutions of the Cauchy problem (or initial value problem) to solutions of the inhomogeneous problem.