Thanks, Alex. I have a few days to review your solution before the bounty expires. In the past, I had to deal with Tikhonov regularization and SVD. Since then, I have been left with the feeling that regularization is more of an art than a science. Could you explain the choice of regularizing functions? Have you tried other options? Why, for example, did you modify the BC and not the diffusion coefficients? Do you think your method will work when passing to a non-stationary problem? Why did you regularize the BC for chi=0 and chi=1? After all, this didn't seem to be a problem, did it?

First of all, this regularization works well for FF with exact coefficient only. For the model with F it is a good attempt to the final solution. Second, this is really art to find out the right function for regularization Mathematica FEM matrix. But nevertheless it works. Concerning nonstationary case we need to test my approach. Other options are solutions with using LDG method and wavelets method. Do you like these methods or FEM is ultimate?

What confuses me most about your solution is that the regularization led to a decrease in the maximum of the distribution function by hundreds of times compared to the solution without regularization. Similarly, I was previously confused by that the addition of the second derivative with respect to v caused this maximum to be increased hundreds of times. This is critically important for the stability criterion with respect to DСLС, one of the two main kinetic instabilities in an anisotropic plasma. Maybe we will continue chatting, otherwise I'm afraid that we will be sent there forcibly?

Sorry, Alex. I'm quite sure that the solution is wrong near $v_z=0$, i.e. at $\xi=0$. This is a plane of symmetry; it is entirely located inside the particle confinement region. At a low transverse velocity,$v_{\perp}$ there should be a maximum of the distribution function along the chord $v_{\perp}=const$. Compare my and your DensityPlot. Please, try to restore the Neumann zero condition on the boundaries of the solution area by the angle, at $\xi=0$ and $\xi=1$.

Sorry, Igor, maybe you are wrong about maximum location at $v_z=0$. As I remember the velocity distribution function has a maximum for v>0 dependent on temperature. Is temperature equals zero in your case?

No, I am insisting, Distribution function is not zero at $\xi=0$, there is not a leak of particles here except for point $u=\xi=0$. Loss cone is located at $\xi>\sqrt{1-1/M}$.

But in your model there is no any flux on the border $\xi =0$ and only very sharp function Source$ with maximum at u0 = 5, \[Xi]0 = Cos[45. Degree]. How you suppose to get something on the border $\xi =0$?

Alex, I mean that FF is peaked at $\xi=0$ for $u\lesssim 1$. Here DF approaches isotropic Maxwellian, at least for partially filled Loss Cone.

Do you mean Maxwell distribution $c u^2 e^{-u^2/(2 a^2)}$? Actually it comes to zero at $u\rightarrow 0$.

No, I expect $c e^{-u^2/(2 a^2)}$, that is the soluttion of

If you know FF on the border then why not to put DirichletCondition[FF[w,\xi]==E^(-((3^(2/3) (\[Pi]/2)^(1/3) w)/(2 \[Mu]^(1/3)))), \xi==0 ]?

I don't know amplitude of FF at w=0. Above, I took FF[0,[\Xi]]==1. More over I assume that there might be slight variation on $\xi$ even ar w=0.