Integral equation

An equation is in mathematics called integral equation, when it occurs the unknown function in an integral. Integral equations can be used in science and technology to describe various phenomena. A well-known example of an integral equation with some applications is the Abelian integral equation, which is also historically one of the first studied integral equations.

The branch of mathematics that deals with integral equations and the below-mentioned compact operators, the functional analysis.

Definition

Linear integral equation

A linear integral equation is an equation for an unknown function, and has the form

Which, given functions and are compact. The function is called core.

Nonlinear integral equation

A non-linear integral equation is of the form

With a suitable domain of the kernel function K and a suitable integration.

Classification of linear integral equations

Linear integral equations can be seen in

• Integral equations of first kind if,
• Integral equations of second kind if and
• Integral equations of third kind, for everyone else,

Divided.

This classification is arbitrary, but it is necessary due to the different analytical properties of the respective types of integral equations. For example, integral equations of second kind ( under weak assumptions on the core ) for almost all values ​​of uniquely solvable, and the solution depends continuously on. This does not apply to integral equations of first kind ( under the same conditions to the core ) in general. Integral equations of first kind are such as the Laplace transform almost always incorrectly posed problems. The Fourier transform is one of the few exceptions. Also integral equations of type 3 are incorrectly posed problems in general.

Is occurring at a known function of the integral equation, the equation is homogeneous, non-homogeneous otherwise.

In addition, one can classify integral equations according to their integration limits. Are all boundaries constant, then one speaks of Fredholm integral equations, one of the limits is variable, it is called the equation is a Volterra integral equation.

Another classification is based on characteristics of the core. There are weakly singular and strongly singular integral equations.

Theoretical operator access

With

Is defined for a sufficiently be integrated core is a linear operator. Essential to the theory of (non- strongly singular ) integral equations of the theory of compact operators. This theory is somewhat similar to that of linear equations in the finite. Compact operators have, in essence, pure eigenvalue spectra. Specifically, this means that the spectrum is (possibly apart from the zero ) only of eigenvalues ​​and these are piling up in at most one point of zero. All eigenspaces (possibly apart from that of the zero) are finite-dimensional.

Duality of integral and differential equations

Integral operators occur often (but not exclusively) in the solution of differential equations, for example, Sturm-Liouville problems, or for partial differential equations in the form of the Green's function.

Integro-differential equation

An integro - differential equation is an equation that is found in which both the derivative of the function to be determined as well as an integral, whose integrant this desired function occurs.

Such equations can be just as integral or differential equations, linear or non -linear. If only ordinary derivatives of the unknown function, it is called from an ordinary integro- differential equations, partial derivatives occur, then one speaks of a partial integro-differential equation.

An example of this is derived from the kinetic theory of gases, Boltzmann equation.