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I came across an integral equation of the Form

[tex]x(t)=\int_\mathbb{R_+}{ds\, x(s)K(s,t)}[/tex]

wher K is some real function. (x is also real). Actually I will later need the higher dimensional case

[tex]x(t_1,\dots,t_n)=\int_\mathbb{R_+^n}{ds_1,\dots,ds_n\, x(s_1,\dots,s_n)K(s_1,\dots,s_n,t_1,\dots,t_n)}[/tex]

But it might be good to first learn the one dimensional case.

From my functional analysis course I remember integral equations of the form (of the Volterra type - ah memory comes back)

[tex]x(t)=\int_{-\infty}^t{ds\, x(s)K(s,t)}[/tex]

and I could transform this type on integral equation to the type I mentioned above (integration over R) via indicator functions which could then be absorbed into the kernel function K .... However I don't see how to do this the other way around.

I suppose there is plenty of literature on this kind of equations and I would appreciate it if you could point me to a particularly useful resource or provide some direct explanations and information on the methods to solve such equations

Thanks

-Pere

edit:

I should have looked on Wikipedia first......sorry .... seems to be a homogeneous Fredholm equation of the second kind .....

So there seem to be a couple of solving methods available like Integral Equation Neumann Series, Fourier Transformation ... I will see if something works for me, if not, I'll be back with the explicit formula of the kernel function

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# Integral equation

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