The method of images does not work directly for the Helmholtz equation in this geometry. But maybe you can make it work approximately, by treating the additional constant term as a perturbation.

Hello! the fact is that yesterday, i found a soln of this problem (Dirichlet-Green fct for Helmholtz), but i don't really understand the way it is derived. See the pic:

The first formula, eq.(5.18), as it stands, looks quite suspicious (it looks like a solution of the homogeneous equation). There seems to be some information missing. While eq. (5.19) looks more correct. The parameter κ should be κn, determined for each n such that the Diriclet boundary condition is fulfilled. The solution can be derived by separation of variables, which at the end of the day amounts to writing the solution as a Fourier series in exp(in(θ-β)). The coefficient functions multiplying each exponential must satisfy an inhomogeneous Bessel equation of order n.

As you can see, the solution is quite messy, with an infinite number of terms. Hence, it is not clear that it is of any practical use. It may be much more efficient and accurate to solve the problem directly by some numerical method (or use some kind of perturbation expansion to gain a qualitative understanding).

My goal is to derive a numerical method (especially the numer fluxes) accurate on coarse grids, by locally solving the stationary eqn (see my "L-splines" project). This already gave good results for diffusion-dominant problems, https://www.researchgate.net/publication/312601843_A_two-dimensional_discretization_of_a_damped_heat_equation_using_Steklov-Poincare_operator in that there is no dimensional-splitting involved.

Yet, for transport-dominant convection-diffusion, the performance degrades.

I understood recently how to build numerical fluxes by means of Dirichlet-Green functions, and i checked the method for diffusion, using the Poisson kernel.

Starting from this formula, the modified Poisson kernel is quite simple, and this is the important quantity when deriving the numerical marching scheme.

Technical Report A two-dimensional discretization of a damped heat equation u...

There is a lot of info about the Green's function for a similar problem that is posed on the unit square  -- not on a disc --  in Franz, Sebastian; Kopteva, Natalia  "Green's function estimates for a singularly perturbed convection-diffusion problem". J. Differential Equations 252 (2012), no. 2, 1521–1545. This Green's function should in general have properties similar to yours. 

Thank you Martin! I was sure you would come up with valuable info ...

The great advantage of the disk is that it has smooth boundary, and then one can do trigonometric interpolation of discrete values on the circle.

Martin> for a similar problem that is posed on the unit square

For the unit square, the method of images can be used also for the Helmholtz equation (although there must in principle be an infinite number of images).

The Green-Dirichlet function in a rectangle appears to be easier to derive, i found it in several books. See e.g. this image:

Is this image from the same source as your previous one? If that is the case, you must store that source in a safe place where it can never be found again. Because the final formula is also obviously wrong (in addition to be incomplete). By inspection you immediately see that the Dirichlet boundary condition is not satisfied for the purported gn.

The correct expression for gn can be found by a couple of lines of calculation. First, it must satisfy the homogeneous x-equation for all x != ξ, satisfy the boundary conditions at x=0 and x=a, and be continuous at x=ξ. This determines the solution to the form

gn(x,  ξ)= Nn sinh(hn x))  with x = max(x, ξ).

The normalisation constant Nn is determined by the requirement that the x-derivative of gn  must jump by 1 at x = ξ. This determines 1/Nn = hn sinh(hn a), modulo a sign which depends on the precise definition of the Green function.

How can an author who is unaware of such elementary facts be trusted on more complicated issues?

Dear Kåre: all the sources are different, involving both books and papers. It's true that gn(x,  ξ) doesn't satisfy Dirichlet BC's, so that i don't really see how the x-boundaries can be satisfied (the y-boundaries are handled by means of the terms sin(\nu y), with \nu depends on n).

My impression at this stage is that this area of application is replete with "results" of poor quality ... and such a situation asks for a supplementary working effort to check everything which was published before.

I think one reason errors in such complicated analytic expressions are never uncovered,  may be because they are never used in practical applications. In a sense, it makes no difference if they are not even wrong, or just plain wrong. To be fair, such series expansions, when correct, may be worth while if one only wants to evaluate the result at a few points.

I applaud your efforts. One thing is to uncover actual errors, equally important is to investigate how efficient and accurate such formulas are in practise, even when they are mathematically correct.

Good, we agree on this last point.

According to your comments, i propose to check one last source, namely one coming from applications and published in some IEEE journal.

The Green fct for a disk is the one written in eqns. (2)-(3): do you think it might be correct ? Indeed, it seems different to the one i showed earlier in the conversation.

Thanks!

The derivation is a bit longish, with many computational steps, which necessarily increases the probability of errors. The result is not obviously wrong, it will take some time too check if it is right (by a more direct method).

Added: This looks like a clear not-even-wrong result. Because I cannot find any mention of which boundary conditions are supposed to be implemented. Which leads to the philosophical problem: Is it possible for the solution of a unspecified problem to be right or wrong?

To be fair: Their equation (3) implicitly indicates that they are imposing Neuman boundary conditions. But, if they are not even aware that this is a point to be precise about, how can one trust the rest?

It's true that the paper is too much telegraphic-style, with far-from-optimal notations. I was hoping that it would be reasonable, but now i even begin to have my own doubts.

A last possibility comes from a specialized book which proceeds rather directly by starting from the free-space Green fct (on which seemingly everybody agrees) and then correct it by means of a homogeneous soln so that the sum vanishes on the circle:

Laurent> A last possibility 

The solution satisfies the Dirichlet boundary condition at r=a, and the normalisation constant is correct in its essential part. There is a purely numerical constant, and a possible power of k0a which is not obvious to me. But since everything else is correct, the full result looks trustworthy (and can easily be checked numerically on some simple cases).

Added (with correction): A comparison with eq. (2)  in the IEEE paper, show much agreement when one takes into account the change from Dirichlet to Neumann boundary conditions. There are some difference in details, which can be attributed to a difference in the range of the sum. And there is an overall normalisation constant, which can depend on how the Green function is defined.  

Thanks. This screen capture comes from the following book,

https://books.google.it/books/about/Green_s_Functions_with_Applications.html

I attach a few pages of the book, where this case appears to be treated.

BTW, it seems that Duffy's expression (numbered (5.3.33) in the attached PDF) matches the one displayed in the very first image i posted within this conversation, which was numbered (5.19), at least when restricting it to its real part. This is an encouraging signal, because we're seeking coherence between all these sources in the literature.

Eq. (5.3.33) in Duffy is almost correct:

To arrive at a cleaner expression, which I am all for, the sum over n runs from minus infinity to plus infinity, with equal contributions from n and -n (thereby avoiding an ugly special case for n=0). However, in that connection it has been overlooked that all Bessel functions of negative orders are not defined identical to the corresponding ones of positive orders. So, if one interprets n as |n| on all Bessel functions in (5.3.33), the result is correct. 

xx The same thing is overlooked in eq. (5.3.35), which in addition is a factor 2 too large. xx

Correction: It was I who overlooked the fact that only non-negative n is summed over in (5.3.35), which both explains the factor of 2, and avoids the pitfall with Bessel functions with negative orders.

The derivation of these results is really extremely simple. Except for a stumbling stone in the form of a minus sign which refused to go away. In the end this could be located to a sign error^* in Abramowitz and Stegun (or its modernised DLMF NIST version).

^*) What is world coming to, when even Abramowitz and Stegun can't be trusted!

Working Paper Amazing Relations HOWTO

Fantastic cat! Here, we love them, too ...

https://28b687da-a-62cb3a1a-s-sites.googlegroups.com/site/laurentgossecnr/like-don-t-like/SAM_0131r.jpg

A typo in Abramowitz-Stegun ... incredible !! I'll look at the calculation tomorrow!

I was in contact with DLMF NIST about the Wronski relation. It turns out that they define the Wronski determinant with a different sign than I thought, as

W{f, g} = (f g' - f 'g)

With this definition their final expression is correct. The problem was that they also have an intermediate expression which corresponds to the opposite sign. And I used this expression to (wrongly) deduce their convention.  The same thing in Abramowitz and Stegun.

Ok. In any case, i want to write down the derivation of the Green function, and the resulting modified Poisson kernels, for the convection-diffusion operator. I'll submit the PDF to you in case you might want to see the complete thing. From the data of these kernels, the derivation of a "truly multi-dimensional" numerical scheme is rather straightforward. However, it's quite easy to make small errors ...