An Analytic Solution for 2D Heat Conduction Problems with General Dirichlet Boundary Conditions

Round 1

Reviewer 1 Report

The Authors presented the transient heat conduction for a rectangular plate with the space-time dependent Dirichlet boundary conditions. Generally, this equation has been solved in a great many publications and in many different ways, including using Green's function.

In my opinion, the authors should very clearly emphasise the applicability of the presented solution and indicate what is new in relation to other publications.

There are many editorial errors in the article, and the drawings are of very poor quality.

Author Response

Dear Referee #1:

The point-by-point answers to the reviewer’s comments are as follows:

Point 1: The Authors presented the transient heat conduction for a rectangular plate with the space-time dependent Dirichlet boundary conditions. Generally, this equation has been solved in a great many publications and in many different ways, including using Green's function. In my opinion, the authors should very clearly emphasis the applicability of the presented solution and indicate what is new in relation to other publications.

Author’s Reply: Thanks for the reviewer’s suggestion. The applicability of the study presented is emphasized in the item (1) of the last paragraph of Section I and in the last paragraph of Section V. The correctness of the solution in this study is verified by comparing with the results of Young et al. [27]. To the best of the authors’ knowledge, the other cases in this paper have never been presented in past studies. These new findings are described in the item (2) of the last paragraph of Section I and in the items (1)-(3) of Section V.

Point 2: There are many editorial errors in the article, and the drawings are of very poor quality.

Author’s Reply: Thanks for the reviewer’s suggestion. The editorial errors have been checked carefully and corrected throughout the article. And the drawings have been revised to improve the quality.

Reviewer 2 Report

REPORT on the article “An analytic solution for the heat conduction of a plate with general Dirichlet boundary conditions”

Journal: Axioms (ISSN 2075-1680)

Manuscript ID: axioms-2288496

Authors: Heng-Pin Hsu , Te-Wen Tu , Jer-Rong Chang

This paper  deals  with a closed form solution for transient heat conduction of a rectangular plate with the general Dirichlet boundary conditions. Here, the boundary conditions of the four edges of the plate are specified as the general case of space-time dependence. First of all, the physical system is decomposed into two one-dimensional subsystems, and each subsystem can be solved by the proposed method combining the shifting function method with the eigenfunction expansion theorem. Two numerical examples are provided  to investigate the analytical solution of the plate with the space-time dependent boundary conditions and so on.

The topic of the paper is an attractive area of  researches.

The similarity rate is 28%. It is a normal rate. See, the attached PDF file.

However, the following points can be considered for the revision.

Point 1: The dimensionless form of physical system can be obtained clearly. It can be shown.

Point 2: There are more punctuations errors, writing and grammatically mistakes in the entire article. The entire article needs a carefully checking and necessary corrections can be done.

Point 3: The novelty and contributions of the article can be compared with literature in the items 1)-3).

Point 4: The following papers , which are related to the PDEs , can be added to the references of this manuscript   to update them.

- An analytical technique for solving new computational of the modified Zakharov-Kuznetsov equation arising in electrical engineering. J. Appl. Comput. Mech. 7(2) (2021) 715-726.

- New solitary wave structures to the (2+1)-dimensional KD and KP equations with spatio-temporal dispersion. Journal of King Saud University – Science. 32 (2020) 3400–3409.

I would like to suggest the acceptation of the article  in “Axioms”  if the suggestions above can be done.

Comments for author File: Comments.pdf

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

Review of the Axioms-2288496

 AUTHORS:  Heng-Pin Hsu, Te-Wen Tu and Jer-Rong Chang

 TITLE:  An Analytic Solution for the Heat Conduction of a Plate with General Dirichlet Boundary Conditions

 Firstly:

The authors claim that the main goal of the work is to obtain an analytical solution of heat conduction in a rectangular plate. However, the presented equations, and in particular the boundary conditions, do not describe heat transport in a plate, but in an infinite bar with a rectangular cross-section.  

So the authors do not know what problem they are solving.

 Secondly:

The authors introduced the dimensionless formulation of the problem by means of equations (8). In this formulation, the rectangle has been transformed into a square with corners at (X,Y) =(0.0), (0.1), (1.0) (1.1).

For the problem formulated in this way, the authors obtained a general solution (equation (45)) in the form of two series of functions. These series consist of the expressions in braces multiplied by the sine function. The values of these sine functions at the corners of the square are zero. This means that, regardless of the initial and boundary conditions, the temperature in all corners is always zero.

This means that solution (45) is wrong.

 Thirdly:

The solution method proposed by the authors using superposition is not a scientific novelty. Therefore, even if the solution were correct, the article would not deserve to be published.

 Recapitulation

The article is not suitable for publication.

Author Response

Dear Referee #3:

The point-by-point answers to the reviewer’s comments are as follows:

Point 1: The authors claim that the main goal of the work is to obtain an analytical solution of heat conduction in a rectangular plate. However, the presented equations, and in particular the boundary conditions, do not describe heat transport in a plate, but in an infinite bar with a rectangular cross-section. So the authors do not know what problem they are solving.

Author’s Reply: Thanks for the reviewer’s valuable suggestion. Based on this valuable comment, the title of the manuscript is changed from “An Analytic Solution for the Heat Conduction of a Plate with General Dirichlet Boundary Conditions” to “An Analytic Solution for 2D Heat Conduction Problems with General Dirichlet Boundary Conditions”. All statements in the manuscript concerning “heat conduction in a plate” have been corrected. A diagram of “an infinite bar with a rectangular cross-section” has also been added to Figure 1(a).

Point 2: The authors introduced the dimensionless formulation of the problem by means of equations (8). In this formulation, the rectangle has been transformed into a square with corners at (X,Y) =(0,0), (0,1), (1,0) (1,1). For the problem formulated in this way, the authors obtained a general solution (equation (45)) in the form of two series of functions. These series consist of the expressions in braces multiplied by the sine function. The values of these sine functions at the corners of the square are zero. This means that, regardless of the initial and boundary conditions, the temperature in all corners is always zero. This means that solution (45) is wrong.

Author’s Reply: Thanks for the reviewer’s suggestion. It was described in the last paragraph of Section III that “From the above derivation process, it can be seen that the assumptions in Equations (29) and (A5) have restrictions on the boundary conditions and initial condition, that is, these values at the four corners of the rectangular region should be zero. If the values of the boundary conditions and initial condition at the four corners of the rectangular region are not zero, they should be zeroed first.” This limitation could be overcome by transforming the temperature function before using the method proposed in this paper. In the near future, a method that works for the case of non-zero values at the four corners will be proposed. These descriptions have been added to the last paragraph of Section V.

Point 3: The solution method proposed by the authors using superposition is not a scientific novelty. Therefore, even if the solution were correct, the article would not deserve to be published.

Author’s Reply: Thanks for the reviewer’s suggestion. Based on this comment, "The novelty and contributions of this paper are as follows" in the last paragraph of Section 1 is revised to "The contributions of this paper are as follows".

Author Response File: Author Response.doc

Round 2

Reviewer 1 Report

N/A

Reviewer 2 Report

REPORT on the REVISED article “An analytic solution for the heat conduction of a plate with general Dirichlet boundary conditions”

Journal: Axioms (ISSN 2075-1680)

Manuscript ID: axioms-2288496

Authors: Heng-Pin Hsu , Te-Wen Tu , Jer-Rong Chang

FORMER SUGGESTIONS and PRESENT RESPONS

Point 1: The dimensionless form of physical system can be obtained clearly. It can be shown.

Answer: The dimensionless form of physical system has been shown accordingly in the revised article.

Point 2: There are more punctuations errors, writing and grammatically mistakes in the entire article. The entire article needs a carefully checking and necessary corrections can be done.

Answer: In the revised article, various punctuations errors, writing and grammatical mistakes have been corrected.

Point 3: The novelty and contributions of the article can be compared with literature in the items 1)-3).

Answer: On thepages 9, 10, the novelty and contributions of the article in the items (1)-(3) have been compared with literature and in the revised article three items are modified into two items.

Point 4: The following papers , which are related to the PDEs , can be added to the references of this manuscript   to update them.

Answer: They have been added accordingly.

By virtue of the answers above, I would like to suggest the acceptation of the article  in “Axioms”.