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July 2nd, 2017, 06:42 AM   #1
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Fredholm equation help

Hi there, I have been recently trying to solve the following integral equation:
\begin{eqnarray}
y(x)=f(x)+\int_{a}^{b}dt K(|x-t|)y(t)
\end{eqnarray}
where the kernel is symmetric and translational. I know there is a theory to solve this general Fredholm equations in finite intervals, but I hope the solution simplifies a lot when the kernel satisfies the above properties, can anyone help on this? I have been trying to find material on the internet but I have been unsuccesful , thanks a lot!
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July 2nd, 2017, 06:55 AM   #2
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The classic method is to apply Weiner-Hopf

https://www.google.co.uk/?gws_rd=ssl#q=weiner-hopf

Sneddon has a good derivation of applying this to Fredholm

The Use of Integral Transforms
Ian N Sneddon p 87 -91

A more modern book is

A textbook of Special Functions in Mathematics (Linear Integral Equations)

Pratap and Singh

This has lots of special cases and simplifications, including yours.
Thanks from nietzsche
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July 2nd, 2017, 07:00 AM   #3
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Quote:
Originally Posted by studiot View Post
The classic method is to apply Weiner-Hopf

https://www.google.co.uk/?gws_rd=ssl#q=weiner-hopf

Sneddon has a good derivation of applying this to Fredholm

The Use of Integral Transforms
Ian N Sneddon p 87 -91

A more modern book is

A textbook of Special Functions in Mathematics (Linear Integral Equations)

Pratap and Singh

This has lots of special cases and simplifications, including yours.

Thanks for the reply, but the Wiener-Hopf method is only applicable on the half line, that is, the interval $[0.+\infty)$ or $(-\infty,0]$, whereas I want to solve the equation in a finite interval in the positive semiaxis!!
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July 2nd, 2017, 07:31 AM   #4
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Quote:
Originally Posted by nietzsche View Post
Thanks for the reply, but the Wiener-Hopf method is only applicable on the half line, that is, the interval $[0.+\infty)$ or $(-\infty,0]$, whereas I want to solve the equation in a finite interval in the positive semiaxis!!
It is?
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July 2nd, 2017, 09:58 AM   #5
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Quote:
Originally Posted by studiot View Post
It is?
Yes, since the map represented there corresponds to the Fourier domain, not the actual domain of $x$ where the functions are defined. Precisely the WH method works with Fourier transforms because of the extension of the domain to the negative semiaxis.
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