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 solve such equations, the Adomian decomposition method, is due to George Adomian.
Equation of the first kind
A Fredholm equation is an integral equation in which the term containing the kernel function (defined below) has constants as integration limits. A closely related form is the Volterra integral equation which has variable integral limits.
An inhomogeneous Fredholm equation of the first kind is written as
and the problem is, given the continuous kernel function K(t,s) and the function g(t), to find the function f(s).
If the kernel is a function only of the difference of its arguments, namely K(t,s) = K(t−s), and the limits of integration are ±∞, then the right hand side of the equation can be rewritten as a convolution of the functions K and f and therefore the solution is given by
where Ft and Fω−1 are the direct and inverse Fourier transforms, respectively.
Equation of the second kind
An inhomogeneous Fredholm equation of the second kind is given as
Given the kernel K(t,s), and the function f(t), the problem is typically to find the function φ(t).
The general theory underlying the Fredholm equations is known as Fredholm theory. One of the principal results is that the kernel K yields a compact operator. Compactness may be shown by invoking equicontinuity. 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. The operators involved are the same as linear filters. 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, or the distribution of relaxation times in the system.
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