Equazione integrale di Fredholm

In matematica, l'equazione integrale di Fredholm è un'equazione integrale la cui soluzione è alla base della teoria di Fredholm, che studia gli operatori di Fredholm e i nuclei di Fredholm.

Equazione del primo tipo

Un'equazione di Fredholm non omogenea del primo tipo ha la forma:

g(t)=\int _{a}^{b}K(t,s)f(s)\,\mathrm {d} s

e la teoria di Fredholm studia come trovare la funzione $f(s)$ a partire dal nucleo integrale $K(t,s)$ e dalla funzione $g(t)$ . Le equazioni integrali di Fredholm sono caratterizzate dal fatto di avere estremi di integrazione costanti (a differenza dell'equazione integrale di Volterra, ad esempio).

Nel caso in cui $K(t,s)=K(t-s)$ e gli estremi di integrazione sono $\pm \infty$ , il membro alla destra può essere scritto come la convoluzione di $K$ e $f$ , in modo che la soluzione è data da:

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

dove ${\mathcal {F}}_{t}$ e ${\mathcal {F}}_{\omega }^{-1}$ sono rispettivamente la trasformata di Fourier e la sua antitrasformata.

Equazione del secondo tipo

Un'equazione di Fredholm non omogenea del secondo tipo ha la forma:

\phi (t)=f(t)+\lambda \int _{a}^{b}K(t,s)\phi (s)\,\mathrm {d} s

Dato un nucleo $K(t,s)$ e una funzione $f(t)$ , solitamente il problema è trovare $\phi (t)$ , spesso tramite l'uso del formalismo del risolvente.

Bibliografia

A.D. Polyanin and A.V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
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.
WH Press, SA Teukolsky, WT Vetterling e BP Flannery, Section 19.1. Fredholm Equations of the Second Kind, in Numerical Recipes: The Art of Scientific Computing, 3rd, New York, Cambridge University Press, 2007, ISBN 978-0-521-88068-8. URL consultato il 14 febbraio 2013 (archiviato dall'url originale l'11 agosto 2011).

Voci correlate

Collegamenti esterni

Integral Equations at EqWorld: The World of Mathematical Equations.
(EN) B.V. Khvedelidze, Fredholm equation, in Encyclopaedia of Mathematics, Springer e European Mathematical Society, 2002.

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