Besse relaxation difference scheme for a nonlinear integro-differential equation

Xinya Peng; Leiwei Li; Jia Zhang

doi:10.1371/journal.pone.0327515

Peer Review History

Original SubmissionJanuary 20, 2025
18 Apr 2025 Decision Letter - Nikos Kavallaris, Editor PONE-D-25-03186Besse Relaxation difference Scheme for a Nonlinear Integro-Differential EquationPLOS ONE Dear Dr. Zhang, Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process. Please submit your revised manuscript by Jun 01 2025 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file. Please include the following items when submitting your revised manuscript: A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'. A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'. An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'. If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter. If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: https://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols. Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols. We look forward to receiving your revised manuscript. Kind regards, Nikos Kavallaris, Ph.D Academic Editor PLOS ONE Journal Requirements: 1. When submitting your revision, we need you to address these additional requirements. Please ensure that your manuscript meets PLOS ONE's style requirements, including those for file naming. The PLOS ONE style templates can be found at https://journals.plos.org/plosone/s/file?id=wjVg/PLOSOne_formatting_sample_main_body.pdf and https://journals.plos.org/plosone/s/file?id=ba62/PLOSOne_formatting_sample_title_authors_affiliations.pdf 2. Thank you for stating in your Funding Statement: This research was supported by the Hunan Youth Fund Project under Girant [20221140879],Excellent Youth Project of Hunan Provincial Department of Education [22B0254] and Central South University of Forestry and Technology introduced Taents and scientific Research Start-up Fund Project [2021YJ0057]. Please provide an amended statement that declares all the funding or sources of support (whether external or internal to your organization) received during this study, as detailed online in our guide for authors at http://journals.plos.org/plosone/s/submit-now. Please also include the statement “There was no additional external funding received for this study.” in your updated Funding Statement. Please include your amended Funding Statement within your cover letter. We will change the online submission form on your behalf. 3. We note that your Data Availability Statement is currently as follows: All relevant data are within the manuscript and its Supporting Information files. Please confirm at this time whether or not your submission contains all raw data required to replicate the results of your study. Authors must share the “minimal data set” for their submission. PLOS defines the minimal data set to consist of the data required to replicate all study findings reported in the article, as well as related metadata and methods (https://journals.plos.org/plosone/s/data-availability#loc-minimal-data-set-definition). For example, authors should submit the following data: - The values behind the means, standard deviations and other measures reported;- The values used to build graphs;- The points extracted from images for analysis. Authors do not need to submit their entire data set if only a portion of the data was used in the reported study. If your submission does not contain these data, please either upload them as Supporting Information files or deposit them to a stable, public repository and provide us with the relevant URLs, DOIs, or accession numbers. For a list of recommended repositories, please see https://journals.plos.org/plosone/s/recommended-repositories. If there are ethical or legal restrictions on sharing a de-identified data set, please explain them in detail (e.g., data contain potentially sensitive information, data are owned by a third-party organization, etc.) and who has imposed them (e.g., an ethics committee). Please also provide contact information for a data access committee, ethics committee, or other institutional body to which data requests may be sent. If data are owned by a third party, please indicate how others may request data access. Comments from PLOS Editorial Office: We note that one or more reviewers has recommended that you cite specific previously published works. As always, we recommend that you please review and evaluate the requested works to determine whether they are relevant and should be cited. It is not a requirement to cite these works. We appreciate your attention to this request. [Note: HTML markup is below. Please do not edit.] Reviewers' comments: Reviewer's Responses to Questions Comments to the Author 1. Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. Reviewer #1: Yes Reviewer #2: Yes ******** 2. Has the statistical analysis been performed appropriately and rigorously? Reviewer #1: Yes Reviewer #2: N/A ****** 3. Have the authors made all data underlying the findings in their manuscript fully available? The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified. Reviewer #1: Yes Reviewer #2: Yes ****** 4. Is the manuscript presented in an intelligible fashion and written in standard English? PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here. Reviewer #1: Yes Reviewer #2: Yes ****** 5. Review Comments to the Author Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters) Reviewer #1: Each and every comments must be addresed regarding convergence stability and example having unbdd derivative with recent modified works for complicated problems. This must be addressed in the revised version without which it can not be processed further. I will recheck it Reviewer #2: The authors present a Besse type finite difference relaxation scheme for approximating solutions of a nonlinear integer-differential equation modelling systems with memory and nonlocal effects. Some comments are in order: 1) p2. Line 22, “.. Georgios and Zouraris...” Is that 2 different authors? From [11] it seems is just one author 2) p2, lines 30-31, the context and wording is not clear. 3) In the introduction the authors should clarify, maybe in a separate paragraph, what are they novelties introduced in this paper, and how their approach compares to the other methods in the literature for this problem, in terms of accuracy and computational cost. 4) p7, line 138, In Example 1 what are the sizes of the space and time intervals? The same question for Example 2 in p.9, line 163. 5) The errors and rates are reported using the maximum norm. It would be beneficial if they are also reported in the discrete L^2 norm. 6) The authors, in Tables 1 and 2 reported the required CPU time. How does this time compares, if a non-relaxation method is used to approximate solutions of model (1)? A comparison table would very beneficial. 7) Can the authors comment on the extension and applicability of their approach in 2-space dimensions with non rectangular domains ? ****** 6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files. If you choose “no”, your identity will remain anonymous but your review may still be made public. Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy. Reviewer #1: No Reviewer #2: No ******** [NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.] While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step. Attachments Attachment Submitted filename: Review_PONE-D-25-03186.pdf https://doi.org/10.1371/journal.pone.0327515.r001
Revision 1
14 May 2025 Author Response To editor:the authors received no additional external funding for this study To Reviewer #1. Each and every comments must be addressed regarding convergence stability and example having unbdd derivative with recent modified works for complicated problems. This must be addressed in the revised version without which it cannot be processed further. I will recheck it. Reply: We sincerely appreciate your insightful comments, which have significantly strengthened the rigor and clarity of our work. We have made substantial revisions to our manuscript based on your valuable feedback. Firstly, we have added detailed proofs of stability and convergence for both the Besse Relaxation Difference Scheme and the Besse Relaxation Compact Difference Scheme, which now appear in Sections 5 and 6 of the revised manuscript (Pages 6–10, Lines 145–258). Meanwhile, we have briefly described this part of the work in the section of Abstract. Secondly, We have expanded the Introduction section to include recent methods from the literature that address complex challenges in fractional partial differential equations, including a discussion of these methods’ advantages and disadvantages, in order to clarify our motivation for using the Besse relaxation difference scheme (specifically, see Page 1, Lines 4–12, and Page 2, Lines 22–47). Thirdly, we have added a new comparative experiment that adopts recently improved methods. Specifically, we have also conducted a comprehensive comparison between our proposed Besse Relaxation Difference Scheme and the two-term relaxation Crank–Nicolson method in terms of CPU time, revealing a 45% improvement at second order and a 35% improvement at fourth order (see Pages 13–15, Lines 297–307). The detailed comparison results are presented in Tables 1 and 2 below. Finally, we have added a novel numerical example focusing on unbounded derivatives, presented in Section 6 (see Pages 15–16, Lines 318–330). These additions effectively validate the applicability of our method to real-world fractional order problems. Similarly, we have briefly described this part of the work in the section of Abstract. To Reviewer #2. The authors present a Besse type finite difference relaxation scheme for approximating solutions of a nonlinear integer-differential equation modelling systems with memory and nonlocal effects. Some comments are in order. Reviewer Comment 2.1 — p2. Line 22, “. . . Georgios and Zoularis. . . ”,Is that 2 different authors? From [11] it seems it is just one author. Reply:Thank you for pointing out the issue regarding “Georgios and Zoularis”. In accordance with your suggestion, we have corrected “Georgios and Zoularis” to “Zouraris” and highlighted this amendment in red in the fourth paragraph of the Introduction. Meanwhile, we have thoroughly reviewed the entire manuscript to avoid the recurrence of similar issues. Reviewer Comment 2.2 — p2, lines 30–31, the context and wording is not clear. Reply:Thank you for pointing out that the wording in lines 30–31 was unclear. Based on your suggestion, we have revised this sentence as follows: Before:Although Besse relaxation difference schemes can improve computational efficiency using the simplification of nonlinear term handling, they are explicit methods and usually perform weak accuracy and stability in complex boundary conditions or nonlinear problems. Changed: However, given that the second-order Besse relaxation difference scheme is still limited to second-order spatial accuracy, there is an urgent need for compact variants with higher algebraic complexity to further improve computational accuracy when solving complex partial differential equations [16]. Reviewer Comment 2.3 — In the introduction the authors should clarify, maybe in a separate paragraph, what are they novelties introduced in this paper, and how their approach compares to the other methods in the literature for this problem, in terms of accuracy and computational cost. Reply:We greatly appreciate this valuable suggestion. In response, we have revised the section of Introduction. Specifically, a detailed explanation of the innovative aspects of our study has been revised and presented on Pages 2, Lines 48–63. In short, our contributions are as follows: (1) Two novel time–space discretization schemes, namely the Besse relaxation and Besse relaxation compact schemes, are proposed to enhance stability and improve accuracy. (2) Rigorous stability and convergence analyses confirm that the schemes are unconditionally stable and convergent. (3) Numerical experiments on equations with singular solutions, smooth solutions, and unbounded derivatives demonstrate the robustness and adaptability of our approach. In addition, we have further clarified how our approach compares to existing methods in the literature with respect to accuracy and computational cost (see Pages 2, Lines 22–47 for more details). Reviewer Comment 2.4 — p7, line 138, In Example 1 what are the sizes of the space and time intervals? The same question for Example 2 in p.9, line 163. Reply:Thank you for asking about the sizes of the space and time intervals in Examples 1, 2. In both examples, we have chosen a temporal grid number N = 4096 and a spatial grid number J = 1024. Specifically, the time interval is [0, T], with ∆t = T /N, and the spatial domain is [xmin, xmax], with ∆x = (xmax −xmin)/J.Meanwhile, following the suggestions from other reviewers, we have added Example 3. In Example 3, we likewise use the same temporal and spatial grid settings. In addition, we have clarified these details in the revised manuscript (see Pages 11, Lines 275–276). Reviewer Comment 2.5 — The errors and rates are reported using the maximum norm. It would be beneficial if they are also reported in the discrete L 2 norm. Reply:Thank you for your valuable suggestion. In accordance with your request, we have included an error analysis under the discrete L^2 norm for Examples 1 and 2 to enhance the credibility of our results (see Pages 11–15, Lines 277–317 for more details). The partial detailed comparison results are presented in Tables 3 and 4 below. Clearly, our proposed method can still achieve the expected order of convergence under the discrete L^2 norm. Reviewer Comment 2.6 — The authors, in Tables 1 and 2 reported the required CPU time. How does this time compares, if a non-relaxation method is used to approximate solutions of model (1)? A comparison table would very beneficial. Reply:Thank you very much for your valuable feedback. In accordance with your recommendation, we applied the non-relaxation Crank – Nicolson method to solve Example 1. The results indicate that, compared with non-relaxation method, our proposed approach achieves approximately a 45% reduction in CPU time at second-order accuracy and about a 35% reduction at fourth-order accuracy, as detailed in the comparison presented on Pages 13–15 ,Lines 297–307. The detailed comparison results are presented in Tables 5 and 6 below. Reviewer Comment 2.7 — Can the authors comment on the extension and applicability of their approach in 2-space dimensions with non rectangular domains? Reply:We sincerely appreciate the reviewer’s interest in extending our fourth-order Besse relaxation compact difference scheme to two-dimensional settings and non-rectangular domains. Based on your feedback, we have revised the section of Conclusion. The detailed modifications can be found on Pages 17, Lines 338–351. In short, the potential for extending and applying our method in two-dimensional space and non-rectangular domains can be summarized as follows: (1) In terms of adaptability, the compact stencil naturally applies in each spatial direction while preserving high-order accuracy, and the temporal Besse relaxation remains effective for multi-dimensional fractional systems in 2-space dimensions. Meanwhile, for more complex geometries, we can employ boundary-fitted coordinates or unstructured meshes, incorporating polynomial reconstructions or ghost point methods to maintain stability and convergence. (2) In terms of generalization, we plan to investigate a wider range of mesh topologies, validate our approach on benchmark problems, and refine implementation details (including parallelization) for large-scale simulations. We trust that this explanation clarifies our method’s adaptability and thank the reviewer for prompting us to elaborate. Attachments Attachment Submitted filename: Response to reviewers.pdf https://doi.org/10.1371/journal.pone.0327515.r002
17 Jun 2025 Decision Letter - Nikos Kavallaris, Editor Besse Relaxation difference Scheme for a Nonlinear Integro-Differential Equation PONE-D-25-03186R1 Dear Dr. Zhang, We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements. Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication. An invoice will be generated when your article is formally accepted. Please note, if your institution has a publishing partnership with PLOS and your article meets the relevant criteria, all or part of your publication costs will be covered. Please make sure your user information is up-to-date by logging into Editorial Manager at Editorial Manager® and clicking the ‘Update My Information' link at the top of the page. If you have any questions relating to publication charges, please contact our Author Billing department directly at authorbilling@plos.org. If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org. Kind regards, Nikos Kavallaris, Ph.D Academic Editor PLOS ONE Additional Editor Comments (optional): Reviewers' comments: Reviewer's Responses to Questions Comments to the Author 1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation. Reviewer #2: All comments have been addressed Reviewer #3: All comments have been addressed ******** 2. Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. Reviewer #2: Yes Reviewer #3: Yes ****** 3. Has the statistical analysis been performed appropriately and rigorously? Reviewer #2: N/A Reviewer #3: Yes ****** 4. Have the authors made all data underlying the findings in their manuscript fully available? The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified. Reviewer #2: Yes Reviewer #3: Yes ****** 5. Is the manuscript presented in an intelligible fashion and written in standard English? PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here. Reviewer #2: Yes Reviewer #3: Yes ****** 6. Review Comments to the Author Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters) Reviewer #2: (No Response) Reviewer #3: This revised manuscript has investigated the stability and the error of the presented numerical method, thus it is suitable for publication. ****** 7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files. If you choose “no”, your identity will remain anonymous but your review may still be made public. Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy. Reviewer #2: No Reviewer #3: No ******** https://doi.org/10.1371/journal.pone.0327515.r003
Formally Accepted
Acceptance Letter - Nikos Kavallaris, Editor PONE-D-25-03186R1 PLOS ONE Dear Dr. Zhang, I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now being handed over to our production team. At this stage, our production department will prepare your paper for publication. This includes ensuring the following: * All references, tables, and figures are properly cited * All relevant supporting information is included in the manuscript submission, * There are no issues that prevent the paper from being properly typeset You will receive further instructions from the production team, including instructions on how to review your proof when it is ready. Please keep in mind that we are working through a large volume of accepted articles, so please give us a few days to review your paper and let you know the next and final steps. Lastly, if your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org. You will receive an invoice from PLOS for your publication fee after your manuscript has reached the completed accept phase. If you receive an email requesting payment before acceptance or for any other service, this may be a phishing scheme. Learn how to identify phishing emails and protect your accounts at https://explore.plos.org/phishing. If we can help with anything else, please email us at customercare@plos.org. Thank you for submitting your work to PLOS ONE and supporting open access. Kind regards, PLOS ONE Editorial Office Staff on behalf of Dr. Nikos Kavallaris Academic Editor PLOS ONE https://doi.org/10.1371/journal.pone.0327515.r004

Open letter on the publication of peer review reports

PLOS recognizes the benefits of transparency in the peer review process. Therefore, we enable the publication of all of the content of peer review and author responses alongside final, published articles. Reviewers remain anonymous, unless they choose to reveal their names.

We encourage other journals to join us in this initiative. We hope that our action inspires the community, including researchers, research funders, and research institutions, to recognize the benefits of published peer review reports for all parts of the research system.

Learn more at ASAPbio .