An Algorithm for Construction of the Asymptotic Approximation of a Stable Stationary Solution to a Diffusion Equation System with a Discontinuous Source Function
Round 1
Reviewer 1 Report
In this paper, authors presented an algorithm for constructing the asymptotic approximation of a stable stationary solution to diffusion equations system. Analysis and detailed mathematical derivations are presented. The idea is interesting and has academic reference significance. But I have some comments as follows:
1) First of all, in the Introduction, authors should clearly highlight the novelty and contribution of the work, preferably with a bulleted or enumerated list.
2) The derivation and calculation process of the entire paper is very complex and cumbersome, and few readers are expected to be fully interested in understanding every detail. It is best to provide a relatively simple concrete example problem to illustrate the application effect of the proposed algorithm.
3) It is recommended to use necessary graphs or tables to summarize the core elements of the proposed algorithm, so as to clarify the conditions, basic steps, and output results of the algorithm.
4) Some main Mathematical notation and uncommon symbols should be listed in a table at the beginning of Section 2.
5) In the final section, an overall conclusion of the entire work should be provided.
The English expression is clear and can basically be understood. But the expression of some technical statements is worth further optimization.
Author Response
The authors would like to thank the respected Reviewer for reading our work and for valuable comments.
1) First of all, in the Introduction, authors should clearly highlight the novelty and contribution of the work, preferably with a bulleted or enumerated list.
That was added.
2) The derivation and calculation process of the entire paper is very complex and cumbersome, and few readers are expected to be fully interested in understanding every detail. It is best to provide a relatively simple concrete example problem to illustrate the application effect of the proposed algorithm.
3) It is recommended to use necessary graphs or tables to summarize the core elements of the proposed algorithm, so as to clarify the conditions, basic steps, and output results of the algorithm.
2)-3) The Example was added to the text including the analytical explanations, numerical solution and explanatory illustration.
4) Some main Mathematical notation and uncommon symbols should be listed in a table at the beginning of Section 2.
We have organized the paragraph with the notations, (170, (18).
5) In the final section, an overall conclusion of the entire work should be provided.
The “Conclusions” section was added to the text
Reviewer 2 Report
This paper presented an algorithm for constructing the asymptotic approximation of a stable stationary solution to diffusion equations system in a two-dimensional domain with a smooth boundary and a discontinuous source function. Existence and stability theorems are proven by using the upper and lower solutions which are constructed as modifications of the asymptotic approximation of the solution. The paper is fruitful and useful. However, the following comments are concerned to be addressed before accepting the paper.
1. The paper contains a large amount of theoretical analysis and proof. However, there is a lack of numerical examples to validate the algorithm.
2. In recent years, some simple and accurate numerical methods have been proposed for solving diffusion equations. To introduce the readers a complete picture of this dynamic development, I suggest that the revised paper should cite the following references: https://doi.org/10.4208/aamm.OA-2019-0269; https://doi.org/10.1016/j.aml.2020.106308.
3. Finally, the English language should be thoroughly checked and polished.
English language should be thoroughly checked and polished.
Author Response
The authors would like to thank the respected Reviewer for reading our work and for valuable comments.
- The paper contains a large amount of theoretical analysis and proof. However, there is a lack of numerical examples to validate the algorithm.
The Example was added to the text including the analytical explanations, numerical solution and explanatory illustration.
- In recent years, some simple and accurate numerical methods have been proposed for solving diffusion equations. To introduce the readers a complete picture of this dynamic development, I suggest that the revised paper should cite the following references: https://doi.org/10.4208/aamm.OA-2019-0269; https://doi.org/10.1016/j.aml.2020.106308.
We have cited these papers
- Finally, the English language should be thoroughly checked and polished.
We have made English correction
Round 2
Reviewer 1 Report
The authors have made a great work addressing all my concerns. I recommend the paper to be accepted.
The English expression is clear and can basically be understood.
