A Space-Time Legendre-Petrov-Galerkin Method for Third-Order Differential Equations
Round 1
Reviewer 1 Report
Review manuscript ID: axioms-2191566
Type of manuscript: Article
Title: A space-time Legendre-Petrov-Galerkin method for third-order differential equations
Journal: Axioms
Publisher: MDPI Basel Switzerland
Authors: Siqin Tang, Hong Li*
Date: 20 January 2023
This article describes a numerical technique, that is, a space-time Legendre-Petrov-Galerkin technique, to solve a third-order partial differential equation that is closely related to the well-known classical Korteweg-de Vries (KdV) equation. Although the authors did not specify the function f on the right-hand side of (1.1) in their proposed model, the KdV equation admits f as a nonlinear term u*u_x. This remark appears to have been absent in the manuscript. Observe also that Mikhail Kovalyov from Alberta Canada and E. van Groesen have worked extensively on the KdV type of model in their long-time career as applied mathematicians. However, their contributions appear to be absent from the list of references. The other remarks are as follows:
* Avoid starting a new sentence with "And".
* Wrong use of articles: a, an, and the.
* KdV was not written correctly. This should be KdV, not KDV.
* Please check whether tau can be written in a Greek letter instead of the Latin alphabet.
The following is an improved abstract.
In this study, the Legendre-Petrov-Galerkin method is considered in both space and time to solve the third-order differential equations. In the theoretical analysis, error estimates in the weighted space-time norms are obtained for fully discrete schemes. Numerical experiments were conducted for linear and nonlinear problems, and spectral accuracy was derived for both space and time. Moreover, the numerical results are compared with those computed by other numerical methods to confirm the efficiency of the proposed method.
The following is an improved introduction:
Spectral methods [1–5] play an increasingly important role in numerical methods for solving partial differential equations (PDEs). In most applications of time-dependent problems [6,7], spectral methods are applied in space combined with time discretization using finite-difference methods. However, such a combination leads to a mismatch in the accuracy in the space and time of the fully discrete scheme. In recent years, some researchers have proposed space-time spectral methods [8–13] for time-dependent problems to attain high-order accuracy in space and time simultaneously.
Please improve the remainder of this manuscript.
Author Response
Dear Professor,
We highly appreciate the review feedback and we have carefully revised the paper addressing the concerns raised in the reviews. Please see the attachment for the response. We solicit your further consideration and look forward your kind further suggestions if there are some more.
Sincerely
Author Response File: 
Author Response.pdf
Reviewer 2 Report
In the reviewed work, the authors explore a third-order partial differential equations with nonperiodic boundary conditions and study the important problem concerning multi-domain forms in space or multi-interval forms in time of space-time Legendre-Petrov-Galerkin methods in the forthcoming study, leading not only to reduce the scale of the problems but also reach the better accuracy. In the present investigations, the authors develop and improve essential results given in [18,22,23,24]. The purpose of this paper is to investigate exponential convergence both in space and time, by means of L2-error estimates, numerical experiments and numerical results. The authors present a rigorous proof of error estimates in the weighted space-time norms.
In the paper presents four fundamental contributions:
1) Detailed analysis of convergence in L2-norm for the fully discrete schemes;
2) Error estimates in the weighted space-time norms are obtained for the fully discrete scheme;
3) Implementation of methods by choice of proper test and trial functions in both space and time.
4) Numerical results are compared with those obtained by other ([18, 21, 22]) to demonstrate the accuracy and efficiency of the methods for the third-order partial differential equations. Numerical examples including nonlinear problems are presented.
In my point of view, this work is very well ordered and well written. The introduction is detailed enough to give a good historical overview. The exposition is clear and correct.
I believe that the submitted article should be published.
Below, the authors can find some corrections or comments that can be useful for the improvement of the paper.
Remarks:
- Some of typos, in the form of spaces between word and punctuation sign - this is desirable to eliminate. Some of the typos I noticed are shown in the attachment.
- For readers convenience, if the authors consider, a DOI numbers of the article and ISBN of the books may be included in the references. Also, the journal titles and book titles should be in Italic.
Comments for author File: Comments.pdf
Author Response
Dear Professor,
We highly appreciate the review feedback and we have carefully revised the paper addressing the concerns raised in the reviews. Please see the attachment for the response. We solicit your further consideration and look forward your kind further suggestions if there are some more.
Sincerely
Author Response File: Author Response.pdf
Reviewer 3 Report
A space-time Legendre-Petrov-Galerkin method for third-order
differential equations
In this article, Legendre-Petrov-Galerkin methods are considered in both space and time to solve the third-order differential equations. In theoretical analysis, error estimates in the weighted space-time norms are obtained for the fully discrete schemes. Numerical experiments are given for
linear and nonlinear problems and spectral accuracy is derived for both space and time. Moreover, some numerical results are compared with those computed by other numerical methods to confirm
the efficiency of our methods.
1. The abstract must be more in detail.
2. Improve the introduction section specially the in fractional calculus point of view. some recent published papers.
3. Give reference to the Equations.
4. Many complicated symbols are introduced and the paper is looks difficult for the readers.
5. Add nomenclature in order to easy for readers.
6. Improve the figures quality.
7. Discussed graphs in physical sense.
Author Response
Dear Professor,
We highly appreciate the review feedback and we have carefully revised the paper addressing the concerns raised in the reviews. Please see the attachment for the response. We solicit your further consideration and look forward your kind further suggestions if there are some more.
Sincerely
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Second review manuscript ID: axioms-2191566
Type of manuscript: Article
Title: A space-time Legendre-Petrov-Galerkin method for third-order differential equations
Journal: Axioms
Publisher: MDPI Basel Switzerland
Authors: Siqin Tang, Hong Li*
Date: 6 February 2023
Thank you for revising your manuscript. The following provides additional remarks.
* Ref. [20] typo: KortewegCde and the journal title in lowercase.
* Ref. [21] typo: The authors are E. van Groesen and Andonowati. The authors did not check the paper carefully. The journal title is also lowercase.
* Section 2 on weak formulation is too short. What are alpha and beta? What is the meaning of the circled cross product sign?
* Definition 3.3 is misleading. The authors started with Let ..., but did not continue with then ... . The structure is theorem-like instead of definition-like. This is confusing.
* Typo: right-hand
* What is the meaning of the equal sign with a dot above it?
* Does star denote dual? Unclear!
* Combine two inequalities.
* Lemmas 3.1 and 3.2.
* What is LPG?
* Some notations should not be in italics, e.g., vec, sin, sech, etc.
* Figure captions are incomplete and require more information.
* Figures are too small, especially the axes labeling.
* What is LPM?
* English is still poor and requires further improvement.
Author Response
Dear Reviewer,
Thank you again for taking the time to review the manuscript! Thank you for your constructive comments and we have made modifications according to the suggestions seriously. Please see the response.
Response to Reviewer 1 Comments
Point 1: Ref. [20] typo: KortewegCde and the journal title in lowercase.
Response 1: Thank you for your careful review. We have corrected the typo and the journal title in uppercase.
Point 2: Ref. [21] typo: The authors are E. van Groesen and Andonowati. The authors did not check the paper carefully. The journal title is also lowercase.
Response 2: Thank you for your careful review. We have corrected the typo and the journal title in uppercase.
Point 3: Section 2 on weak formulation is too short. What are alpha and beta? What is the meaning of the circled cross product sign?
Response 3: Thank the expert for your valuable suggestions.
- We have supplemented the content in Section 2.
- Alpha and beta can be understood as the parameters in the weight function given in the paper. The value is a constant and we have added the explanation in the paper.
- We have added the meaning of the circled cross product sign in the paper, please see Definition2.1.
Point 4: Definition 3.3 is misleading. The authors started with Let ..., but did not continue with then ... . The structure is theorem-like instead of definition-like. This is confusing.
Response 4: Thank you for your careful review. We have added one word “where” to make the sentence complete and removed the indication of “Definition 3.3”.
Point 5: Typo: right-hand
Response 5: Thank you for your careful review. We have corrected the typo.
Point 6: What is the meaning of the equal sign with a dot above it?
Response 6: Thank the expert for your valuable suggestions. “the equal sign with a dot above it” symbol has no special meaning, it only means equal sign. So to avoid any confusion among readers, we have removed it.
Point 7: Does star denote dual? Unclear!
Response 7: Thank you very much for your advice. Following the advice given, we have added a
remark in Section 2. Please see Remark 1 on page 3.
Point 8: Combine two inequalities. Lemmas 3.1 and 3.2.
Response 8: Thank the expert for your valuable suggestions. Two inequalities in Lemmas 3.1 and 3.2 are about spatial and temporal directions, respectively. In addition, the definition of the operators in two lemmas and the prerequisite conditions that the inequality needs to satisfy are also different. For these reasons, we considered that it would be preferable to present the two inequalities separately.
Point 9: What is LPG?
Response 9: Thank the expert for your valuable suggestions. The “LPG” in Table1 and Table2 for Example 1 is abbreviation of “Legendre-Petrov-Galerkin” and that is the way it's written in the reference[22] . To avoid any confusion among readers, We have put it another way both in tables and main text.
Point 10: Some notations should not be in italics, e.g., vec, sin, sech, etc.
Response 10: Thank the expert for your valuable suggestions. We have modified them.
Point 11:Figure captions are incomplete and require more information.
Response 11: Thank the expert for your valuable suggestions. We have supplemented the figure captions in more detail.
Point 12: Figures are too small, especially the axes labeling.
Response 12: Thank the expert for your valuable suggestions. We have expanded the figures appropriately.
Point 13: What is LPM?
Response 13: Thank the expert for your valuable suggestions. The “LPM” in Table3 and Table4 for Example 2 is abbreviation of “ Legendre pseudo-spectral method” and that is the way it's written in the reference[26] . To avoid any confusion among readers, We have put it another way both in tables and main text.
Point 14: English is still poor and requires further improvement.
Response 14: Thank the expert for your valuable suggestions. We have further improved the English expression of the article.
Reviewer 3 Report
Accept
Author Response
Dear Reviewer,
Thank you again for taking the time to review the manuscript! We have made appropriate improvements to the English expression and writing of the article.
Round 3
Reviewer 1 Report
Third review manuscript ID: axioms-2191566
Type of manuscript: Article
Title: A space-time Legendre-Petrov-Galerkin method for third-order differential equations
Journal: Axioms
Publisher: MDPI Basel Switzerland
Authors: Siqin Tang, Hong Li*
Date: 14 February 2023
This study employs a space-time spectral method for approximating third-order differential equations with non-periodic boundary conditions. While the use of Legendre-Petrov-Galerkin discretization in both space and time is an interesting approach, the limited scope of the study is a drawback. The focus is solely on non-periodic boundary conditions, so it is unclear how the method would perform with other types of boundary conditions. Additionally, the theoretical analysis only provides proof of error estimates for the fully discrete scheme, but it is unclear what limitations or assumptions were made in this proof. Furthermore, the comparison of the numerical results to those computed by other methods is not thorough or convincing enough to confirm the efficiency of the proposed method.
