In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
, integral equations are equations in which an unknown
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
appears under an
integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
sign.
In mathematical notation, integral equations may thus be expressed as being of the form:
where
is an
integral operator
An integral operator is an operator that involves integration. Special instances are:
* The operator of integration itself, denoted by the integral symbol
* Integral linear operators, which are linear operators induced by bilinear forms involvi ...
acting on ''u.'' Hence, integral equations may be viewed as the analog to
differential equations where instead of the equation involving derivatives, the equation contains integrals. A direct comparison can be seen with the mathematical form of the general integral equation above with the general form of a differential equation which may be expressed as follows:
where
may be viewed as a
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
of order ''i''.
Due to this close connection between differential and integral equations, one can often convert between the two. For example, one method of solving a boundary value problem is by converting the differential equation with its boundary conditions into an integral equation and solving the integral equation.
In addition, because one can convert between the two, differential equations in physics such as
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
often have an analog integral and differential form. See also, for example,
Green's function
In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if L is a linear dif ...
and
Fredholm theory
In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given ...
.
Classification and overview
Various classification methods for integral equations exist. A few standard classifications include distinctions between linear and nonlinear; homogeneous and inhomogeneous; Fredholm and Volterra; first order, second order, and third order; and singular and regular integral equations.
These distinctions usually rest on some fundamental property such as the consideration of the linearity of the equation or the homogeneity of the equation.
These comments are made concrete through the following definitions and examples:
Linearity
: An integral equation is linear if the unknown function ''u''(''x'') and its integrals appear linearly in the equation.
Hence, an example of a linear equation would be:
As a note on naming convention: i) ''u''(''x'') is called the unknown function, ii) ''f''(''x'') is called a known function, iii) ''K''(''x'',''t'') is a function of two variables and often called the
Kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...
function, and iv) ''λ'' is an unknown factor or parameter, which plays the same role as the
eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
in
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mathemat ...
.
: An integral equation is nonlinear if the unknown function 'u''(''x'') or any of its integrals appear nonlinear in the equation.
Hence, examples of nonlinear equations would be the equation above if we replaced ''u''(''t'') with
, such as:
Certain kinds of nonlinear integral equations have specific names.
A selection of such equations are:
* Nonlinear Volterra integral equations of the second kind which have the general form:
where ' is a known function.
* Nonlinear Fredholm integral equations of the second kind which have the general form:
.
* A special type of nonlinear Fredholm integral equations of the second kind is given by the form:
, which has the two special subclasses:
** Urysohn equation:
.
** Hammerstein equation:
.
More information on the Hammerstein equation and different versions of the Hammerstein equation can be found in the Hammerstein section below.
Location of the unknown equation
: An integral equation is called an integral equation of the first kind if the unknown function appears only under the integral sign.
An example would be:
.
: An integral equation is called an integral equation of the second kind if the unknown function also appears outside the integral.
: An integral equation is called an integral equation of the third kind if it is a linear Integral equation of the following form:
where ''g''(''t'') vanishes at least once in the interval
'a'',''b''or where ''g''(''t'') vanishes at a finite number of points in (''a'',''b'').
Limits of Integration
Fredholm: An integral equation is called a
Fredholm integral equation
In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to ...
if both of the limits of integration in all integrals are fixed and constant.
An example would be that the integral is taken over a fixed subset of
.
Hence, the following two examples are Fredholm equations:
* Fredholm equation of the first type:
.
* Fredholm equation of the second type:
Note that we can express integral equations such as those above also using integral operator notation.
For example, we can define the Fredholm integral operator as:
Hence, the above Fredholm equation of the second kind may be written compactly as:
: An integral equation is called a
Volterra integral equation
In mathematics, the Volterra integral equations are a special type of integral equations. They are divided into two groups referred to as the first and the second kind.
A linear Volterra equation of the first kind is
: f(t) = \int_a^t K(t,s)\,x( ...
if at least one of the limits of integration is a variable.
Hence, the integral is taken over a domain varying with the variable of integration.
Examples of Volterra equations would be:
* Volterra integral equation of the first kind:
* Volterra integral equation of the second kind:
As with Fredholm equations, we can again adopt operator notation. Thus, we can define the linear Volterra integral operator
, as follows:
where