The integral equation was studied by Ivar Fredholm.
A Fredholm equation is an integral equation in which the term containing the kernel function (defined below) has constants as integration limits.
An inhomogeneous Fredholm equation of the first kind is written as
, and the limits of integration are ±∞, then the right hand side of the equation can be rewritten as a convolution of the functions
This case would not be typically included under the umbrella of Fredholm integral equations, a name that is usually reserved for when the integral operator defines a compact operator (convolution operators on non-compact groups are non-compact, since, in general, the spectrum of the operator of convolution with
, which is usually a non-countable set, whereas compact operators have discrete countable spectra).
An inhomogeneous Fredholm equation of the second kind is given as
A standard approach to solving this is to use iteration, amounting to the resolvent formalism; written as a series, the solution is known as the Liouville–Neumann series.
One of the principal results is that the kernel K yields a compact operator.
As an operator, it has a spectral theory that can be understood in terms of a discrete spectrum of eigenvalues that tend to 0.
Fredholm equations arise naturally in the theory of signal processing, for example as the famous spectral concentration problem popularized by David Slepian.
They also commonly arise in linear forward modeling and inverse problems.
In physics, the solution of such integral equations allows for experimental spectra to be related to various underlying distributions, for instance the mass distribution of polymers in a polymeric melt, [1] or the distribution of relaxation times in the system.
[2] In addition, Fredholm integral equations also arise in fluid mechanics problems involving hydrodynamic interactions near finite-sized elastic interfaces.
[3][4] A specific application of Fredholm equation is the generation of photo-realistic images in computer graphics, in which the Fredholm equation is used to model light transport from the virtual light sources to the image plane.