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(xt)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: 845 Thanks: 253 
The classic method is to apply WeinerHopf https://www.google.co.uk/?gws_rd=ssl#q=weinerhopf 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. 
July 2nd, 2017, 07:00 AM  #3  
Member Joined: Dec 2016 From:  Posts: 62 Thanks: 10  Quote:
Thanks for the reply, but the WienerHopf 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: 845 Thanks: 253  It is?

July 2nd, 2017, 09:58 AM  #5 
Member Joined: Dec 2016 From:  Posts: 62 Thanks: 10  

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