Green’s Functions for Neumann Boundary Conditions

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

please find the attached file. 

Comments for author File: Comments.pdf

Author Response

I would like to thank the reviewer for a careful review of my paper, and respond to the Comments here.

1. I have added the book by Evans.
2. I have added a section comparing the Neumann and Dirichlet 3D GF's
3. I now use f for a general function, but use $\phi$ for the electric potential.
4. u and v are for respective eigenfunctions.
5. Symmetry fails in a specific case, but I don't think for a physical reason because it is so easy to restore.
6. I have strengthened the Abstract a bit.  I usually try for concise abstracts without too many promises.
7. I do not discuss the many other possibilities, to keep to my main point, comparison of the Neumann and Dirichlet cases. 
8. The domain of $\bf r$ fo Fourier series and integrals is a long standing point of contention for Physicists and Mathematicians.  I don't want to get into it here. My point of view is that readers will understand, since I write down Fourier series, the $\bf r$ I use is sufficient for them.  

The draft would benefit from citing additional mathematical sources (e.g., Evans PDEs,
Strauss, or functional analysis texts) where Green’s functions are formally defined. Consider
including historical works or modern PDE literature that treat Neumann problems more
rigorously.
While a worked 1D example is given, the 3D section remains mostly formal. An explicit
3D example (e.g., Laplacian in a cube or sphere) would greatly strengthen the pedagogical
impact. Including a short example from electrostatics or heat conduction with Neumann
boundaries would help demonstrate the practical significance.
Some notation could be made more consistent, e.g., switching between f (x), ϕ(r), and
u(x) may confuse a reader. A uniform notation would improve readability. Equation num-
bering should be more carefully aligned with references in the text.
Symmetry Discussion
The proof that Neumann Green’s functions are not necessarily symmetric is interesting,
but the pedagogical explanation could be expanded. A short intuitive explanation (why
symmetry fails physically) would make this point stronger.
The abstract is concise but could emphasize more clearly what the new contribution of
this paper is compared to textbook treatments. The summary could briefly reflect on possible
extensions (e.g., mixed boundary conditions, Robin boundary conditions, or applications to
PDEs beyond the Laplacian form).
In the beginning please define the domian of r. The derivation in Section 2 relies heavily
on function expansions. It would be useful to comment briefly on the functional-analytic
framework (e.g., Hilbert space setting, completeness of Neumann eigenfunctions), to make
the presentation rigorous.
The manuscript addresses an important and under-discussed topic and presents it in
a pedagogical way. However, in its current form it reads more like an extended lecture
note rather than a polished research article. With improvements in references, clarity of
exposition, and the addition of illustrative examples, it could become a strong contribution
to the literature.
Suggested Decision: Minor Revision

Reviewer 2 Report

Comments and Suggestions for Authors

Please find the attached file

Comments for author File: Comments.pdf

Author Response

A report on the paper “Green's functions for Neumann boundary conditions"
Title: Green's functions for Neumann boundary conditions
Author: Jerrold Franklin
Journal: Mathematics
Manuscript ID: mathematics-38883628
In this article, the author derived an appropriate Neumann Green’s function with
Neumann boundary conditions and properties incorporated. Also, he gave some essential
differences between the Neumann Green’s function and the Dirichlet Green’s function
which is generally treated in Math Physics or Electromagnetism texts for the solution of a
partial differential equation in the form
ℒ??(?)=∇ ⋅[?(?)∇?(?)]=?(?).
The results in this paper are interesting and clearly presented, but the introduction
section should include all relevant references and provide enough background and
information to understand the literature of this research. Thus, I recommend it for
publication in your journal after minor version.
Some comments:
1. Some typos are found in this paper, please check the whole paper carefully.
2. Some details should be added. For example, eq. (31) is not clear how to obtain it.
3. What are the conditions on ?,?,?,???,?.
4. The introduction part should include more relevant references and provide enough
background information to understand the literature of this research.
5. Please explain how eq. (2) gives eq. (4).
6. The author should define ?∗ in equation (11).
7. Please explain in detail why the Neumann boundary condition for the solution ϕ(r)
must satisfy (7).
8. It is good to give the findings for the general form ?(?)≠0.
9. Please check equation (46), ∇(?)→∇?(?).
10. The author found 6 differences, not 7. Please check (6), page 10.
11. Some recent references should be added.

----------------------------------------------------------------------

1. I am trying to fix all typos.
2. I have included the two equations that lead to Eq.(31), now (33).
3.  I don't know what "the conditions on ?,?,?,???,?" means?
4.  I have added one math reference on Partial Differential Equations. I know of 20 physics books I could add for Green's function. I think the four books I listed are sufficient. Actually I expect that any reader will use their own books that they have seen in courses.

5. I added, "implementing Eqs. (1) and (3) leads to" Eq. (4).

6. ?∗ in equation (11) is the complex conjugate of u in Eq. (9), which I have no
7. w defined just before the equation.

8. The Neumann boundary condition follows the constraint by applying the divergence theorem to Eq.(1).  I have added two sentences with more detail.

9. Including q(r} would add complication to every equation, making it harder to see the differences between Dirichlet and Neumann. Actually, I don't know of any textbook that includes q(r).
10. corrected
11. corrected
12. See 4.