Fredholm Integral Equation Article Index for
Fredholm 		Shopping
Integral 		Website Links For
Integral 		 

Information About

Fredholm Integral Equation
  CATEGORIES ABOUT FREDHOLM INTEGRAL EQUATION
   
fredholm theory
   
integral equations
   
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...




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 {Link without Title} (\omega)\over
\mathcal{F}_t {Link without Title} (\omega)}
ight]=\int_{-\infty}^\infty {\mathcal{F}_t {Link without Title} (\omega)\over
\mathcal{F}_t {Link without Title} (\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.


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 \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 , 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 Eigenvalue s that tend to 0.


SEE ALSO



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''