# Solving a finite Hilbert transform from 2D thin wing theory

I have a problem where I know $$w(x) =1$$ and want to find $$u(x)$$. It's a well-known problem I am recreating, from a 1972 paper. Note the paper doesn't solve the problem, the integral equation, it just cites its solution in older work.

Richard M. James On the remarkable accuracy of the vortex lattice method Comput. Methods Appl. Mech. Eng. 1, 59–79 (1972)

$$w(x)=\int_0^1 \frac{u(\text{xx})}{x-\text{xx}} \, d\text{xx}$$ $$u(1)=0$$

I also know the answer,

$$u(x)=\frac{1}{\pi}\sqrt{\frac{1-x}{x}}$$

I can work it backwards to verify (it takes a while)

u[x_] := 1/π Sqrt[(1 - x)/x]
Assuming[0 <= x <= 1, Integrate[u[xx]/(x - xx), {xx, 0, 1}, PrincipalValue -> True]]

1

But no joy trying to run it forwards. Tried the following, including wild spitballs:

DSolve[1 == Integrate[uu[xx]/(y - xx), {xx, 0, 1}], uu[xx], xx]

DSolve[{1 == Integrate[uu[xx]/(y - xx), {xx, 0, 1}], uu[1] == 0}, uu[xx], xx]

DSolve[1 == Integrate[uu[xx]/(y - xx), {xx, 0, 1}], uu[y], y]

DSolve[{1 == Integrate[uu[xx]/(y - xx), {xx, 0, 1}], uu[1] == 0}, uu[y], y]

DSolve[
1 == Integrate[uu[xx]/(x - xx), {xx, 0, 1}, PrincipalValue -> True],
uu[xx], xx]

Did a little digging here on Hilbert transforms, but this is a finite Hilbert transform. Didn't see any obvious questions that mirrored this one. Any ideas? I assume I am posing it wrong, as it doesn't seem like it'd be that hard to solve, based on the simplicity of the answer. But math can be surprising.

So can I solve this integral equation in symbolic form using Mathematica?

• Can you link to the paper, for reference? – J. M. is away Mar 29 at 14:58
• Updated to add the link, a 1972 paper. – MikeY Mar 29 at 15:10
• @MikeY At first glance, the article describes in detail the numerical solution method. What is the problem? – Alex Trounev Mar 29 at 19:36
• The numerical solution method approaches the idealized result, which is the equation referenced above. I just thought I would use Mathematica to confirm the result myself (and then do some other stuff) and was unable to do so. In short, am I using DSolve to solve the integral equation correctly? Wondered if it was my implementation, or something deeper. – MikeY Mar 29 at 19:39