My Math Forum Fredholm equation help

 Calculus Calculus Math Forum

 July 2nd, 2017, 06:42 AM #1 Member   Joined: Dec 2016 From: - Posts: 62 Thanks: 10 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!
 July 2nd, 2017, 06:55 AM #2 Senior Member   Joined: Jun 2015 From: England Posts: 796 Thanks: 233 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
July 2nd, 2017, 07:00 AM   #3
Member

Joined: Dec 2016
From: -

Posts: 62
Thanks: 10

Quote:
 Originally Posted by studiot 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!!

July 2nd, 2017, 07:31 AM   #4
Senior Member

Joined: Jun 2015
From: England

Posts: 796
Thanks: 233

Quote:
 Originally Posted by nietzsche 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?
Attached Images
 W_H.jpg (92.9 KB, 6 views)

July 2nd, 2017, 09:58 AM   #5
Member

Joined: Dec 2016
From: -

Posts: 62
Thanks: 10

Quote:
 Originally Posted by studiot 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.

 Tags equation, fredholm

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post DarkX132 Algebra 3 September 26th, 2014 10:15 PM PhizKid Differential Equations 0 February 24th, 2013 10:30 AM yiorgos Applied Math 7 July 12th, 2010 03:08 AM apalmer3 Real Analysis 1 July 2nd, 2009 02:21 AM jman5 Applied Math 0 March 8th, 2009 08:46 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top