In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way. See, for example, Maxwell's equations.
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The most basic type of integral equation is a Fredholm equation 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 algebra.
If one limit of integration is variable, it is called a Volterra equation. 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
Integral equations are classified according to three different dichotomies, creating eight different kinds:
Integral equations are important in many applications. Problems in which integral equations are encountered include radiative energy transfer and the oscillation 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.
Certain homogeneous linear integral equations can be viewed as the continuum limit of eigenvalue equations. Using index notation, 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where the sum over has been replaced by an integral over and the matrix and vector have been replaced by the 'kernel' and the eigenfunction . (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 distribution, 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 equation.