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Peer-Review Record

Analytical and Numerical Results for the Transient Diffusion Equation with Diffusion Coefficient Depending on Both Space and Time

Algorithms 2023, 16(4), 184; https://doi.org/10.3390/a16040184
by Mahmoud Saleh 1, Endre Kovács 1,* and Imre Ferenc Barna 2
Reviewer 1: Anonymous
Algorithms 2023, 16(4), 184; https://doi.org/10.3390/a16040184
Submission received: 17 February 2023 / Revised: 17 March 2023 / Accepted: 25 March 2023 / Published: 28 March 2023
(This article belongs to the Section Algorithms for Multidisciplinary Applications)

Round 1

Reviewer 1 Report

The paper under review deals with a linear parabolic PDE. The case when diffusion (heat conduction) coefficient depends on both time and spatial variable is considered. A new exact self-similar solution is constructed. A lot of numerical solutions algorithms are proposed, mainly constructed by the authors. Complex computational experiments were carried out to estimate numerical solution accuracy with respect to the exact solutions, as well as convergence time. Various adaptive step-size controllers were constructed and compared. From the analysis of the calculation results, some of the algorithms were selected as most suitable.

 In general, the paper is a completed study containing interesting mathematical results. Formulation of the problem is clear. All the applied methods are sufficiently described and cited. The results are discussed correctly and illustrated sufficiently.

 However, a few drawbacks should be noted.

1. There is a mistype in equation (2), see file attached.

2. Line 142:  D=k/(cr) if c and r are constant, however, they are depend on x in equation (7). Constant c and r are considered two lines below.

 3. In Section 4, there are subsections 4.2 and 4.3, but 4.1 is absent.

 4. Figs. 7, 10, 13: it is hard to match the lines in the legend to graphs. Some lines from the legend are undetectable in the figure.

 5. Line 395. Mistype: “In Fig. 7, we plotted…”; must be “In Fig. 10, we plotted …”.

 The paper is worth publishing after making corrections to eliminate the above-mentioned drawbacks.

Comments for author File: Comments.pdf

Author Response

Response to Reviewer 1.

We are grateful to the reviewer for her/his precious time devoted to reviewing our manuscript and the positive attitude. The point-by-point responses are the following.

 

1. There is a mistype in equation (2), see file attached.”

The reviewer is absolutely right. We have now corrected that typo, as well as extended this paragraph to become easier to understand.

 

“2. Line 142:  D=k/(cr) if c and r are constant, however, they are depend on x in equation (7). Constant c and r are considered two lines below.”

We are grateful for the competent reviewer to notice that hidden mistake. Now we have moved the D=k/(c*ro) relation just after Eq. (1), where no space dependency presents.

 

“ 3. In Section 4, there are subsections 4.2 and 4.3, but 4.1 is absent.”

We have now fixed this typo, thanks to the reviewer.

 

“ 4. Figs. 7, 10, 13: it is hard to match the lines in the legend to graphs. Some lines from the legend are undetectable in the figure.”

We have now reproduced these 3 figures and now these lines are more distinguishable.

 

“ 5. Line 395. Mistype: “In Fig. 7, we plotted…”; must be “In Fig. 10, we plotted …”.

We have now fixed this typo, thanks to the reviewer.

 

Reviewer 2 Report

The paper deals with the linear diffusion equation. The authors construct an exact self-similar solution for the one-dimensional case. Besides, many numerical algorithms are proposed, analyzed, and compared.

The paper is well-written, the motivation is convincing, the results are clearly described. I have some comments that should be corrected.

Comments

1.       There seems to be a mistake in Eq. (2) since h should be differentiated as a complex function.

2.       The authors should avoid multiple citing like [24-31] and [30-38]. If the results are very close, there is no need to mention all papers. Otherwise, describe them separately.

3.       Figures are best converted into vector format. Please, pay attention to a font size and the font itself. It should be as in the main text.

4.       It is better to use the Latex template since some formulas are hard-to-read. If possible, avoid using fractions like a/b for indexed variables (for example, line 298).

5.       Please, format the references with respect to the template.

6. In Section 4, paragraph 4.1 is missed.

Author Response

Response to Reviewer 2.

The paper deals with the linear diffusion equation. The authors construct an exact self-similar solution for the one-dimensional case. Besides, many numerical algorithms are proposed, analyzed, and compared.

The paper is well-written, the motivation is convincing, the results are clearly described. I have some comments that should be corrected.

We are grateful to the reviewer for the positive attitude towards our manuscript. We try to do our best to further improve the manuscript during the revision process.

 

“1.    There seems to be a mistake in Eq. (2) since h should be differentiated as a complex function.

The reviewer is absolutely right. We have now corrected that typo, as well as extended this paragraph to become easier to understand.

 

“2.    The authors should avoid multiple citing like [24-31] and [30-38]. If the results are very close, there is no need to mention all papers. Otherwise, describe them separately.

We have now reduced the bulk citations and at the same time, the proportion of self-citations.

 

“3.    Figures are best converted into vector format. Please, pay attention to a font size and the font itself. It should be as in the main text.”

Thanks to the reviewer for the advice. We are continuously trying to develop in these technical issues as well.

 

“4.    It is better to use the Latex template since some formulas are hard-to-read. If possible, avoid using fractions like a/b for indexed variables (for example, line 298).

Thanks to the reviewer for these advices. Next time we will try to follow them.

 

“5.    Please, format the references with respect to the template.

The references are generated by Mendeley. We tried to do our best to set it to follow the template.

 

“6.  In Section 4, paragraph 4.1 is missed.”

We have now fixed this typo, thanks to the reviewer.

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