# Obtaining the Green's function for a 2D Poisson equation ( in polar coordinates)

I am trying to solve the following BVP within an annular region of radii $$r_1$$, and $$r_2$$ : $$\begin{cases} \nabla^2u=f\\ u(r_1) = p\\ u(r_2) = q \end{cases}$$ If we define an auxiliary problem in terms of Green's function as $$\begin{cases} \nabla^2G=\delta^2(r-r')\\ G(r_1) = 0 \\ G(r_2) = 0\\ \end{cases}$$ We have the solution of u (as given by Green's identities as) $$\DeclareMathOperator{\Dm}{\operatorname{d}\!} u = \oint u \frac{\partial G}{\partial n} \Dm S + \int Gf \Dm V \tag{Eqn. A}$$ How do I proceed to obtain the form of the Green's function ?

I understand that G for a finite boundary problem is done by superposition :

$$G = G_{Freespace} + G_{Homogeneous}$$

From my little searching I found that, $$G_{Freespace} = Aln(r-r')$$, and $$G_{Homogeneous}=a_0 + a_nr^ncos(n\phi) + b_nr^nsin(n\phi)$$

However, the expected solution ( from a paper) that I am seeking is of the form:

$$2\pi G=H_0 (r_1,r_2) + \sum_{n=1}^{\infty} H_n(r_1,r_2) cos(n(\phi_1 - \phi_2))$$

The form of $$H_0$$ and $$H_n$$ are given in the attachment below.

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How do I obtain the solution above from the problem posed ?

"The Green's function" is a (very common) grammatical error that Clifford Truesdell I think once called an atrocity to the English language. (Complains a German... ;) ) But you're excused, the paper you cite does it too.^^ Edit: My brain has betrayed me, I thought to have read this in Truesdell's "An iditiot's fugitive essays on Science", but it's actually a footnote in Rohrlich's "Classical Charged Particles", Chap. 4-7. Full quote:

The commonly used expressions "the Green's function" and "a Green's function" represent an atrocity to the English language. I doubt that those who use them ever refer to "a Shakespeare's sonnet".

More on-topic: You're halfway there! Usual technique is to use the free-space and homogeneous contributions you give (the latter is missing a sum!) and try to match the boundary conditions. Here you need to fulfill $$G=0$$ at $$r_1$$ and $$r_2$$. Have fun comparing coefficients and using trigonometric identities. I might have a closer look later myself.

Edit: Some more info to get into the solution process. The solution of problems like this in cylindrical coordinates is a common problem, and you could have a look at J.D. Jackson: Classical Electrodynamics, Chapter 3.11; Panofsky&Phillips: Classical Electricity and Magnetism, Chapter 4-9f or W.A. Strauss: Partial Differential Equations, Chapter 10.2, for example.

To find the Green function as the sum of the free-space and homogeneous conribution, let's start with the free-space contribution: It reads $$G_f\left(\vec{r},\vec{r}'\right) = - 2 \pi \ln\left(\frac{\left|\vec{r}-\vec{r}'\right|}{r_0}\right)$$ with an arbitrary constant $$r_0$$ (if this arbitrariness weirds you out, have a look in Panofsky&Phillips, but it's not that important). Now to express that in cylindrical coordinates with radius $$r$$/$$r'$$ and polar angle $$\phi$$/$$\phi'$$ ($$r_1$$, $$r_2$$ etc. in the paper, which in your notation are the constant inner and outer radii) $$G_f\left(\vec{r},\vec{r}'\right) = - 2 \pi \ln\left(\frac{\sqrt{r^2+\left(r'\right)^2-2rr'\cos\left(\phi-\phi'\right)}}{r_0}\right)$$ by the law of cosines. But we need to do some work on this to obtain the form in the paper. J.D. Jackson might help you here a bit.

For the homogeneous part we use your ansatz. This stems from a separation ansatz (see the cited references) and contains the sine and cosine contributions (which are eigenfunctions of the second-derivative-operator on a circle) and powers of the radius $$r$$ with coefficients later to be matched to fullfill the radial part of the Poisson equation and the boundary conditions. Generally we could start off with something like $$G_h\left(\vec{r},\vec{r}'\right) = \sum_n r^n\left(a_n \sin\left(n\phi\right) + b_n \cos\left(n\phi\right)\right) \sum_m r'^m\left(a'_m \sin\left(m\phi'\right) + b'_m \cos\left(m\phi'\right)\right)\,,$$ which looks bad, but can be simplified immediately by using symmetries of the problem.

Invariance under rotation tells us right away that the dependence on $$\phi$$ and $$\phi'$$ can only be a dependence on their difference, so we'd expect something like $$... \left(a_n \sin\left(n\left(\phi-\phi'\right)\right) + b_n \cos\left(n\left(\phi-\phi'\right)\right)\right)\,,$$ but then further we can use that $$G\left(\vec{r},\vec{r}'\right) = G\left(\vec{r}',\vec{r}\right)$$ and we're only left with the cosines. You see the solution in the paper come along, right? Still, way to go.

• Thanks. What I did not understand is how $\phi_2$ came up in the homogeneous part. Some pointers towards the solution could really help me understand. Commented Apr 3, 2022 at 2:35
• I'm curious, why would "Green's function" be a grammatical error?
– lcv
Commented Apr 3, 2022 at 9:35
• @lcv Not sure (maybe my calling it a grammatical error is actually an error^^), but I've added a reference. Commented Apr 3, 2022 at 11:04
• Thank you very much @kricheli. That gives me some very good pointers to work on. Commented Apr 3, 2022 at 16:51
• @kricheli Not surprisingly there are some posts on the matter. See e.g., nature.com/articles/nphys411 or english.stackexchange.com/questions/140529/… . However none of those posts provide a definite answer.
– lcv
Commented Apr 4, 2022 at 11:52