Spectral technique with convergence analysis for solving one and two-dimensional mixed Volterra-Fredholm integral equation

A. Z. Amin; A. K. Amin; M. A. Abdelkawy; A. A. Alluhaybi; I. Hashim

doi:10.1371/journal.pone.0283746

Peer Review History

Original SubmissionJanuary 26, 2023
8 Feb 2023 Decision Letter - Mahmoud A. Zaky, Editor PONE-D-23-02279Spectral technique with convergence analysis for solving multi-dimensional mixed Volterra-Fredholm integral equationPLOS ONE Dear Dr. Abdelkawy, 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 Mar 25 2023 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. 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We will update your Data Availability statement to reflect the information you provide in your cover letter. 4. PLOS requires an ORCID iD for the corresponding author in Editorial Manager on papers submitted after December 6th, 2016. Please ensure that you have an ORCID iD and that it is validated in Editorial Manager. To do this, go to ‘Update my Information’ (in the upper left-hand corner of the main menu), and click on the Fetch/Validate link next to the ORCID field. This will take you to the ORCID site and allow you to create a new iD or authenticate a pre-existing iD in Editorial Manager. Please see the following video for instructions on linking an ORCID iD to your Editorial Manager account: https://www.youtube.com/watch?v=_xcclfuvtxQ Additional Editor Comments: 1- The language should be polished. 2- In the title "multi-dimensional" but there are only one- and two dimensions problems. What about the high dimensions? 3- The novelty of the paper should be discussed in comparison with proposed high dimensional spectral methods for the same problems in literature. 4-Please add references for lemmas in section 5.1. 5- Please follow "?" in the manuscript and add missed references. 6 The list of references should be updated and its style should be unified. [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: Partly Reviewer #2: Yes ******** 2. Has the statistical analysis been performed appropriately and rigorously? Reviewer #1: N/A Reviewer #2: Yes ****** 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: No 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: Herein, the authors offered a numerical scheme based on shifted Jacobi-Gauss collocation method for handling the mixed Volterra-Fredholm integral equations. The method transform the equation into a system of algebraic equations. The present algorithm is extended to solve the two-dimensional case. Convergence analysis for the present method is discussed. Various numerical examples are approached to demonstrate the applicability of the technique. I have read the text thoroughly, it needs a careful revision before the possibility of its potential publication in PONE, the following comments should be meticulously addressed, a point-by-point rebuttal letter is a must: 1. It is not good to boost citations by citing Refs via bulk criteria, see [1, 2, 3, 4, 5, 6, 7, 8], [24, 24, 25, 26, 27, 28], [29, 30, 31, 32, 33, 34, 35, 36, 37]. Please cite only relevant work; also use the cite package to present Refs as [1-4], not [1,2,3,4]. 2. The main motivation of the method should be polished within the introductory section. 3. All properties in Section 2 are not new, it is not fair to do not cite a seminal text for these properties, The text of Shen et al. 2011, https://link.springer.com/book/10.1007/978-3-540-71041-7 is welcome. 4. For the two models in 3.1 and 3.2, please cite a Ref in which you picked these models from, also mention the regularity conditions on all functions parameters that ascertain the existence and uniqueness of the solutions, see for instance 10.22075/IJNAA.2015.179 5. The convergence rate of the unknown expansion coefficients was previously studied in depth in https://ijnao.um.ac.ir/article_25278.html; https://doi.org/10.1016/j.camwa.2019.03.011; https://doi.org/10.1007/s40314-018-0633-3. Please mention this at the beginning of Section 5. 6. Some Equations in Page 7 do not fit with the margins of the paper, please adopt. 7. Compare your work with https://doi.org/10.1007/s40065-019-0243-y. 8. Mention the platform used to write the codes as well as the specifications of the machine used to debug these codes. 9. Comment on future extensions in this direction in the conclusion section. 10. Proofread the whole manuscript to get rid of any typos or grammatical errors. sss Reviewer #2: The paper discusses the analysis and the numerical solution of mixed Volterra-Fredholm integral equations (MV-F-IEs). The numerical approach is based on shifted Jacobi-Gauss collocation (SJ-G-C) method. The proposed technique with shifted Jacobi-Gauss (SJ-G) nodes is applied to diminish the MV-F-IEs to a system of algebraic equations that are easily solved. The present algorithm is extended to solve the two-dimensional MV-F-IEs. Convergence analysis for the present method is discussed and confirms the exponential convergence of the spectral algorithm. Various numerical examples are approached to demonstrate the power and accuracy of the technique. The work is meaningful and has sufficient novelty, although the same technique seems to be applied to other models by the co-authors. However, the analysis is technical and concrete, and the results are fascinating. The overall structure seems to be logical; definitions are given most of the time. Thus, I suggest the acceptance of the paper for publication. Before that, the authors should consider the following minor comments. For the minor comments, see the attached file. ****** 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: Yes: Youssri Hassan Youssri Reviewer #2: Yes: A. K. Omran ******** [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. <quillbot-extension-portal></quillbot-extension-portal> Attachments Attachment Submitted filename: Review_report_PONE-D-23-02279.pdf https://doi.org/10.1371/journal.pone.0283746.r001
Revision 1
8 Mar 2023 Author Response Dear respected Editor We verified certain that all tables were cited in the text, as seen below. Table 6.1 mentioned in Lines 212 and 218. Table 6.2 mentioned in Line 229. Table 6.3 mentioned in Line234. Table 6.4 mentioned in Line 240. Responses to referees I) With respect to the fi\frst referee (Reviewer: 1): 1. We carefully checked our manuscript and corrected the errors through- out the paper. 2. The main motivation of the method added in the introductory sec- tion. 3. We checked the references list and inserted additional references. 4. We have reviewed and corrected all of the reviewer's required changes. 5. All additions were highlighted in red color. II) With respect to the second referee (Reviewer: 2): The main advantages of the present study are: 1. We carefully checked our manuscript and corrected the errors through- out the paper. 2. The main motivation of the method added in the introductory sec- tion. 3. We inserted references for the properties mentioned in Section 2. 4. For the two models in 3.1 and 3.2, We inserted suitable references. 5. We compared our work with https://doi.org/10.1007/s40065-019-0243- y.(ex3). 6. We mentioned the platform used to write the codes as well as the speci\fcations of the machine used to debug these codes. 7. We cited only relevant work; also used the cite package. 8. we inserted a Comment on future extensions in this direction in the conclusion section. 9. All additions were highlighted in red color. II) With respect to the Editor: 1. We carefully checked our manuscript and corrected the errors through- out the paper. 2. The main motivation of the method added in the introductory sec- tion. 3. We replaced "multi-dimensional" by "one- and two-dimensional"in the title. 4. We inserted a comparison with proposed high dimensional spectral methods for the same problems in literature. 5. We added references for lemmas in section 5.1. 6. We checked the references list and inserted additional references. 7. The reference list has been updated, and the style has been uni\fed. 8. All additions were highlighted in red color. Attachments Attachment Submitted filename: Responses.pdf https://doi.org/10.1371/journal.pone.0283746.r002
16 Mar 2023 Decision Letter - Mahmoud A. Zaky, Editor Spectral technique with convergence analysis for solving one and two-dimensional mixed Volterra-Fredholm integral equation PONE-D-23-02279R1 Dear Dr. Abdelkawy, 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 for payment will follow shortly after the formal acceptance. To ensure an efficient process, please log into Editorial Manager at http://www.editorialmanager.com/pone/, click the 'Update My Information' link at the top of the page, and double check that your user information is up-to-date. If you have any billing related questions, 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, Mahmoud A. Zaky Academic Editor PLOS ONE ********** <quillbot-extension-portal></quillbot-extension-portal> https://doi.org/10.1371/journal.pone.0283746.r003
Formally Accepted
28 Mar 2023 Acceptance Letter - Mahmoud A. Zaky, Editor PONE-D-23-02279R1 Spectral technique with convergence analysis for solving one and two-dimensional mixed Volterra-Fredholm integral equation Dear Dr. Abdelkawy: I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department. 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. If we can help with anything else, please email us at plosone@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. Mahmoud A. Zaky Academic Editor PLOS ONE https://doi.org/10.1371/journal.pone.0283746.r004

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