Integral Equation: Solving Methods & Resources

[Pere Callahan]

Hi,

I came across an integral equation of the Form

$$x(t)=\int_\mathbb{R_+}{ds\, x(s)K(s,t)}$$

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

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

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)
$$x(t)=\int_{-\infty}^t{ds\, x(s)K(s,t)}$$

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

 

[John Creighto]

What are you solving for? Perhaps my background is laking and I don't understand enough about your problem but after reading wikpiedia it doesn't look like Fredholm's equation to me given that the non kernal function in the integral is of the same form as that on the left of the equals sign.

My nieve method of solving the above equation would be to expand x(s) and x(t) in series form, integrate and then equate terms.

 

[Pere Callahan]

I want to solve for the function x(t).

In the notation from Wikipedia (second paragraph on Fredholm equation of the second type) this x(t) corresponds to $$\varphi(t)$$, f(t) is identically zero in my case (homogeneous) and $$\lambda$$ is one. So I still think it is of the Fredholm type.

The notation there seems to be slightly confusing since in the first paragraph (Fredholm equation of the first kind) the function they solve for is f(t) while the inhomogeneity is denote by g(t) ...

Yes, expanding and solving is always worth a try

 

[snagy84]

pls help me solve this problem and i will be very glad

 

[Pere Callahan]

I'd like to bring this topic back up. I now have an explicit expression for the kernel function and also an idea as to how to solve it, but maybe I'm doing something wrong

I want to solve

$$f(x)=\frac{1}{\sqrt{8\lambda}}\int_{\mathbb{R}}{dy\, f(y)e^{-\sqrt{2\lambda}|x-\alpha y|}}$$
where $0<\alpha<1,\quad \lambda>0$, but I suppose the constants are not important.

What I tried is the following: I split the integral on the r.h.s into parts, one going from negative infinity to $x/\alpha$ and one going from $x/\alpha$ to positive infinity.

Then I differentiated the equation twice with respect to x and what i got is the following differential equation:

$$f''(x)+\frac{1}{\alpha}f(\frac{x}{\alpha})-2\lambda f(x)=0$$

However, I don't know how to fnd a solution to this kind of equation and the original integral equation doesn't look complicated enough to not allow for a simple solution (my feeling )

Any help (as always) appreciated

-Pere

 

[Pere Callahan]

So, what i wrote above seems to be right.

Assume I'm given some domain $D\subset \matbb{R}^2$, and want so solve the following integral equation

$$f(x)=\int_D{K(x,y)f(y)dy}\quad\forall x\in D$$

Assume further, that I'm unable to solve this exactly (which is for now indeed the case). Are there any approximate, iterative, numerical methods to get at least an idea of the solution?

Thanks
-Pere