The most basic type of integral equation is called 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, the equation is known as a Fredholm equation of the second type,
Originally, such equations were studied in connection with problems in radiative transfer, and more recently, they have been related to the solution of boundary integral equations for planar problems in which the boundary is only piecewise smooth.
Power series solution for integral equations
In many cases, if the Kernel of the integral equation is of the form K(xt) and the Mellin transform of K(t) exists, we can find the solution of the integral equation
in the form of a power series
are the Z-transform of the function g(s), and M(n + 1) is the Mellin transform of the Kernel.
Integral equations as a generalization of eigenvalue equations
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 M = [Mi,j] is a matrix, v is one of its eigenvectors, and λ is the associated eigenvalue.
Taking the continuum limit, i.e., replacing the discrete indices i and j with continuous variables x and y, yields
where the sum over j has been replaced by an integral over y and the matrix M and the vector v have been replaced by the kernelK(x, y) and the eigenfunctionφ(y). (The limits on the integral are fixed, analogously to the limits on the sum over j.) This gives a linear homogeneous Fredholm equation of the second type.