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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 kernel]]s and [[Fredholm operator]]s. The integral equation was studied by [[Ivar Fredholm]].
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| ==Equation of the first kind==
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| 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 equation|Volterra integral equation]] which has variable integral limits.
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| An [[inhomogeneous function|inhomogeneous]] Fredholm equation of the first kind is written as:
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| :<math>g(t)=\int_a^b K(t,s)f(s)\,\mathrm{d}s</math>
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| and the problem is, given the continuous [[kernel (integral equation)|kernel]] function ''K(t,s)'', and the function ''g(t)'', to find the function ''f(s)''.
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| If the kernel is a function only of the difference of its arguments, namely <math>K(t,s)=K(t-s)</math>, and the limits of integration are <math>\pm \infty</math>, 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
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| :<math>f(t) = \mathcal{F}_\omega^{-1}\left[
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| {\mathcal{F}_t[g(t)](\omega)\over
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| \mathcal{F}_t[K(t)](\omega)}
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| \right]=\int_{-\infty}^\infty {\mathcal{F}_t[g(t)](\omega)\over
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| \mathcal{F}_t[K(t)](\omega)}e^{2\pi i \omega t} \mathrm{d}\omega </math>
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| where <math>\mathcal{F}_t</math> and <math>\mathcal{F}_\omega^{-1}</math> are the direct and inverse [[Fourier transforms]] respectively.
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| ==Equation of the second kind==
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| An inhomogeneous Fredholm equation of the second kind is given as
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| :<math>\phi(t)= f(t) + \lambda \int_a^bK(t,s)\phi(s)\,\mathrm{d}s.</math>
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| Given the kernel ''K(t,s)'', and the function <math>f(t)</math>, the problem is typically to find the function <math>\phi(t)</math>. 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]].
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| ==General theory==
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| 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.
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| ==Applications==
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| 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 problem]]s.
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| ==See also==
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| * [[Liouville-Neumann series]]
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| * [[Volterra integral equation]]
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| ==References==
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| * [http://eqworld.ipmnet.ru/en/solutions/ie.htm Integral Equations] at EqWorld: The World of Mathematical Equations.
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| * A.D. Polyanin and A.V. Manzhirov, ''Handbook of Integral Equations'', CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
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| * {{springer|first1=B.V.|last1= Khvedelidze|first2= G.L. |last2=Litvinov |id=F/f041440|title=Fredholm kernel}}
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| * F. J. Simons, M. A. Wieczorek and F. A. Dahlen. ''Spatiospectral concentration on a sphere''. SIAM Review, 2006, {{doi|10.1137/S0036144504445765}}
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| * D. Slepian, "Some comments on Fourier Analysis, uncertainty and modeling", [http://scitation.aip.org/journals/doc/SIAMDL-home/jrnls/top.jsp?key=SIREAD SIAM Review], 1983, Vol. 25, No. 3, 379-393.
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| *{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 19.1. Fredholm Equations of the Second Kind | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=989}}
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| [[Category:Fredholm theory]]
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| [[Category:Integral equations]]
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