- #1

Pere Callahan

- 586

- 1

Hi,

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 equationsThanks

-Pereedit:

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

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 equationsThanks

-Pereedit:

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

Last edited: