Solving Integral Equation: \phi(s) from y(s)

[saltydog]

I'm interested in studying the following equation, solving for $\phi(s)$ given $y(s)$:

$$\int_0^s \frac{2r}{\sqrt{s^2-r^2}} \phi(r)dr=y(s)$$

or in more standard form:

$$\int_0^s K(s,r)\phi(r)=y(s)$$

This is how I think it should be approached:

The kernel,$K(s,r)$ is singular at $s=r$. Thus, the first step is to transform it to a finite kernel via composition with $\sqrt{\xi-r}$. Once this is done, then the resulting equation can be further transformed to a Volterra equation of the second kind which then can be solved by Picard's process of successive approximations. I realize effecting the integrations likely becomes intractable but I'd still like to determine the solution format.

Can anyone tell me if this is the correct approach to follow? I'll spend time with this approach and report here my progress.

Thanks,
Salty

 

[eljose]

Perhpas you can try converting the Volterra equation to a differential equation...

 

[saltydog]

Perhpas you can try converting the Volterra equation to a differential equation...

Thanks eljose. I don't think you can remove the other variable under the integral sign by differentiating since both s and r are under the radical. Thus, you can't convert it to a differential equation. Am I wrong?

Also, I've found that the kernel needs to be bounded to use the method above. The kernel stated above,$\frac{2t}{\sqrt{x+t}}$ is not bounded.

I'll take my own advice: when confronted with a difficult problem you can't solve (which I can't cus I ain't proud), put it up and work on simpler ones first. In that regard, I'll work on several examples of the generalized Abel equation:

$$\int_0^x{\frac{\phi(y)}{\sqrt{x-y}}dy=f(x)$$

The solution technique still involves a composition (convolution) and I know how to solve it and will report here some test cases for different values of $f(x)$. The method is quite elegant and perhaps others will think so too.

By chance, if any students or others are wondering why should integral equations be studied, there is a profound reason why: it's all in the history. Do you see why?

 

[saltydog]

I've asked that this thread be deleted. I really bit of more than I could chew and have "regrouped" in the form of a simpler equation that I've posted as "abel equation".

Salty