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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.

Equation of the first kind

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:

$$g(t)=\int_a^b K(t,s)f(s)\,\mathrm{d}s$$

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 $$\pm \infty$$ , then the right hand side of the equation can be rewritten as a convolution of the functions K and f and therefore the solution will be given by

$$f(t) = \mathcal{F}_\omega^{-1}\left[ {\mathcal{F}_t[g(t)](\omega)\over \mathcal{F}_t[K(t)](\omega)} \right]=\int_{-\infty}^\infty {\mathcal{F}_t[g(t)](\omega)\over \mathcal{F}_t[K(t)](\omega)}e^{2\pi i \omega t} \mathrm{d}\omega$$

where $$\mathcal{F}_t and \mathcal{F}_\omega^{-1}$$ are the direct and inverse Fourier transforms respectively.

Equation of the second kind

An inhomogeneous Fredholm equation of the second kind is given as

$$\phi(t)= f(t) + \lambda \int_a^bK(t,s)\phi(s)\,\mathrm{d}s.$$

Given the kernel K(t,s), and the function f(t), the problem is typically to find the function $$\phi(t)$$ . A standard approach to solving this is to use the resolvent formalism; written as a series, the solution is known as the Liouville-Neumann series.

General theory

The general theory underlying the Fredholm equations is known as Fredholm theory. One of the principal results is that the kernel K is 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.

Applications

Fredholm equations arise naturally in the theory of signal processing, most notably as the famous spectral concentration problem popularized by David Slepian. They also commonly arise in linear forward modeling and inverse problems.

Liouville-Neumann series
Volterra integral equation

References

Integral Equations at EqWorld: The World of Mathematical Equations.
A.D. Polyanin and A.V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
Khvedelidze, B.V.; Litvinov, G.L. (2001), "Fredholm kernel", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
F. J. Simons, M. A. Wieczorek and F. A. Dahlen. Spatiospectral concentration on a sphere. SIAM Review, 2006, doi:10.1137/S0036144504445765
D. Slepian, "Some comments on Fourier Analysis, uncertainty and modeling", SIAM Review, 1983, Vol. 25, No. 3, 379-393.
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 19.1. Fredholm Equations of the Second Kind". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.

Mathematics Encyclopedia