Fredholm integral equation

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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.

1 Equation of the first kind
2 Equation of the second kind
3 General theory
4 See also
5 References
6 External links

[edit] Equation of the first kind

An inhomogeneous Fredholm equation of the first kind is written as:

$g(t)=\int_a^b K(t,s)f(s)\,ds$

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

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

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

In the general case, the Fredholm Integral Equation can be converted to an exactly equivalent Rao Integral Equation or Rao Transform. This is done by using a local form kernel L(t,s) defined by:

L (t, s) = K (t + s, t)

The equivalent Rao Integral Equation is:

$g(t) = \int_{t-b}^{t-a} L(t-s,s)\, f(t-s)\,ds$

Solution of the above equation is also the solution of the original general Fredholm Integral Equation. The equation above can be solved using the following steps. Substitute f(t-s) with its Taylor series expansion around t; interchange the order of integration and summation and simplify; using the resulting equation, derive expressions for the successive derivatives of g(t) with respect to t of order 1,2,3, ...; in the resulting system of equations, solve for f(t) by recursively substituting for the successive derivatives of f(t) begining from the first derivative. In practice, the Taylor series expansion of f(t-s) may need to be truncated at some order N. Solution of this equation is particularly useful in shift-variant image restoration or deblurring. Rao Integral Equations are a recent development.

[edit] Equation of the second kind

An inhomogeneous Fredholm equation of the second kind is given as

$f(t)= \lambda \phi(t) - \int_a^bK(t,s)\phi(s)\,ds$

Given the kernel K(t,s), and the function $f (t)$ , the problem is typically to find the function $φ(t)$ . 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. Another method of solving this equation is to use the equivalent Rao Integral Equation with the same kernel transformation and the same solution steps as above for the First Kind. The kernel transformation is $L (t, s) = K (t + s, t)$ and the corresponding Rao Integral Equation is:

$f(t)= \lambda \phi(t) - \int_{t-b}^{t-a} L(t-s,s)\phi(t-s)\,ds$ .

[edit] 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, known as the Fredholm 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.

[edit] See also

Liouville-Neumann series

[edit] 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
B.V. Khvedelidze, G.L. Litvinov, Fredholm kernel, (2001), SpringerLink Encyclopaedia of Mathematics
M. SubbaRao, Rao Transforms: Theory and Applications, self-published book, 120 pages, U.S. Copyright Registration No. TX 6-195-821, June 1, 2005, purchase at http://www.integralresearch.net.