In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an
integral equation is an equation in which an unknown
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
appears under an
integralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
sign. There is a close connection between
differentialA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
and integral equations, and some problems may be formulated either way. See, for example,
Maxwell's equationsMaxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
.
Overview
The most basic type of integral equation is a
Fredholm equationIn 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.-Equation of the first kind :...
of the first type:
The notation follows Arfken.
Here φ; is an unknown function,
f is a known function,
and
K is another known function of two variables,
often called the kernel function.
Note that the limits of integration are constant; this is what characterizes a Fredholm equation.
If the unknown function occurs both inside and outside of the integral, it is known as a
Fredholm equation of the second type:
The parameter λ is an unknown factor,
which plays the same role as the eigenvalue in
linear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
.
If one limit of integration is variable, it is called a
Volterra equationIn 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 = \int_a^t K\,x\,ds...
. Thus
Volterra equations of the first and second types, respectively, would appear as:
In all of the above, if the known function
f is identically zero, it is called a
homogeneous integral equation. If
f is nonzero, it is called an
inhomogeneous integral equation.
φ
Classification
Integral equations are classified according to three different dichotomies, creating eight different kinds:
Limits of integration
- both fixed: Fredholm equation
- one variable: Volterra equation
Placement of unknown function
- only inside integral: first kind
- both inside and outside integral: second kind
Nature of known function
f
- identically zero: homogeneous
- not identically zero: inhomogeneous
Integral equations are important in many applications. Problems in which integral equations are encountered include radiative energy transfer and the
oscillationOscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...
of a string, membrane, or axle. Oscillation problems may also be solved as differential equations.
Both Fredholm and Volterra equations are linear integral equations, due to the linear behaviour of φ(x) under the integral. A nonlinear Volterra integral equation has the general form:
,
where F is a known function.
Integral equations as a generalization of eigenvalue equations
Certain homogeneous linear integral equations can be viewed as the continuum limit of
eigenvalue equationsThe eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix...
. Using
index notationIndex notation is used in mathematics and computer programming to specify the elements of matrices or the components of a vector. The formalism of how indices are used varies according to the discipline...
, an eigenvalue equation can be written as
,
where
is a matrix,
is one of its eigenvectors, and
is the associated eigenvalue.
Taking the continuum limit, by replacing the discrete indices
and
with continuous variables
and
, gives
,
where the sum over
has been replaced by an integral over
and the matrix
and vector
have been replaced by the 'kernel'
and the
eigenfunctionIn mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
. (The limits on the integral are fixed, analogously to the limits on the sum over
.) This gives a linear homogeneous Fredholm equation of the second type.
In general,
can be a
distributionIn mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
, rather than a function in the strict sense. If the distribution
has support only at the point
, then the integral equation reduces to a
differential eigenfunction equationIn mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
.
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