Regularization of Nonlinear Volterra Integral Equations of the First Kind with Smooth Data
Round 1
Reviewer 1 Report
Comments and Suggestions for Authors-
The manuscript needs clearer explanation of the novelty compared to existing regularization methods.
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The mathematical derivations lack sufficient detail and should be expanded for better readability.
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The description of the subdomain method and compatibility condition requires more rigorous justification.
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Numerical examples or simulations are missing and should be added to validate the theoretical results.
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The conclusions are brief and need stronger discussion of practical implications and limitations.
Author Response
For research article
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Response to Reviewer 1 Comments
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1. Summary |
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We sincerely thank the Reviewer1 for the careful reading of our manuscript “Regularization of nonlinear Volterra integral equations of the first kind with smooth data” and for the constructive comments. Below we provide point-by-point responses, and we indicate how the manuscript has been revised accordingly. All changes are incorporated in the revised version.
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2. Questions for General Evaluation |
Reviewer’s Evaluation |
Response and Revisions |
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Does the introduction provide sufficient background and include all relevant references? |
Yes/Can be improved/Must be improved/Not applicable |
In the revised version we expanded the Introduction to provide a more detailed background on Volterra integral equations of the third kind. We added recent references [9–11, 20–21], which cover related results on regularization and numerical methods. The novelty of our approach compared to existing works is also emphasized. |
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Are all the cited references relevant to the research? |
Yes/Can be improved/Must be improved/Not applicable |
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Is the research design appropriate? |
Yes/Can be improved/Must be improved/Not applicable |
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Are the methods adequately described? |
Yes/Can be improved/Must be improved/Not applicable |
The methods are described in sufficient detail to allow reproduction of the results. In the revised manuscript, we clarified the assumptions (conditions (a)–(e)), expanded the explanation of the regularization operator, and provided additional details on the iterative numerical scheme, including the initial guess, stopping criterion, and the use of Newton corrections. These clarifications ensure that the methodology is transparent and reproducible. |
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Are the results clearly presented? |
Yes/Can be improved/Must be improved/Not applicable |
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Are the conclusions supported by the results? |
Yes/Can be improved/Must be improved/Not applicable
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The conclusions are fully supported by the analytical results and numerical experiments. In particular, the error estimates derived in Theorems 1 and 2 are confirmed by the computational data in Section 3, and the Discussion highlights the agreement between theory and practice. We also revised the Conclusion to emphasize only those findings that are directly validated by our results. |
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Are all figures and tables clear and well-presented? |
Yes/Can be improved/Must be improved/Not applicable
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Additional clarifications As additional clarifications, we note that all figures have been improved in resolution and their captions expanded. The reference list has been carefully checked for consistency, and duplicates have been removed. All changes in the revised manuscript are highlighted in red for the reviewers’ convenience. |
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Reviewer 2 Report
Comments and Suggestions for AuthorsPlease see the attachment.
Comments for author File: 
Comments.pdf
Author Response
Response to Reviewer 2 Comments
- Summary
We sincerely thank Reviewer 2 for the thorough reading of our manuscript and for the constructive and detailed comments. The suggestions have been extremely helpful in improving the clarity, precision, and presentation of the paper. We carefully revised the manuscript, and below we provide a point-by-point response to each comment.
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2. Quality of English Language |
Response |
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The English could be improved to more clearly express the research. |
We thank the reviewer for this remark. In the revised version, the manuscript has been carefully proofread and polished to improve the clarity of English expression. Redundant phrases were shortened, technical terms were unified, and minor grammatical issues were corrected. We believe that the revised text now presents the research more clearly and is suitable for publication. |
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3. Point-by-point response to Comments and Suggestions for Authors |
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Comments 1: The introduction of the paper fails to highlight the innovation points: (i) The authors should clearly compare the proposed method with existing studies to demonstrate its novelty. (ii) The authors should elaborate on the references cited in the paper, and the introduction to relevant research findings from recent years is lacking. (iii) The model
the paper considered is simple. The authors should be more explicit about whether the innovation lies in the results or the improvement in the conditions. It is recommended that the authors consider adopting more complex models, such as those involving generalized fractional integral operators. Response 1: We appreciate this valuable remark. In the revised introduction, we have:
Comments 2: In p2, line 59, the expression of may be missing a symbol.
Response 2: We thank the reviewer for noticing this issue. In the previous version, a space was missing between two expressions. Please see line 68.
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Comments 3: The definition of the space “ ” should be given for readers’ convenience. |
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Response 3: is the space of continuous functions of , that have continuous derivatives with respect to the first argument . Please see lines 71-73.
Comments 4: In p2, line 62 and line 63, the definiton of “D”should be distinguished. Response 4: By applying the operator , where I is the identity operator, and - (the differentiation operator), equation (1) is transformed into a Volterra integral equation of the third kind. Please see line 75.
Comments 5: In p6, line 135, the authors should check whether the formula is correct. Response 5: We thank the reviewer for noticing this issue. Please see lines 153-154.
Comments 6: In p10, line 192, the authors should check whether the formula is correct. Response 6: We thank the reviewer for noticing this issue. Please see lines 207-208. Comments 7: The conclusion section is excessively redundant and should focus on highlighting the key points. Response 7: We thank the reviewer for this suggestion. In the revised version, the Conclusion section has been shortened and restructured. Repetitive material was removed, and the section now highlights only the key contributions of the paper: the construction of the Lavrentiev-type regularization operator, the proofs of uniqueness and uniform convergence, and the consistency of the numerical experiments with the theoretical error bounds. Please see section Conclusions.
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4. Response to Comments on the Quality of English Language |
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Response 1: The revised text has been carefully proofread and polished for clarity, readability, and academic style. 5. Additional clarifications |
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As additional clarifications, we note that: · All figures were improved in resolution and formatting; captions were expanded for better readability. · Numerical tables were reformatted with consistent notation and alignment. · The reference list was carefully checked for consistency, and duplicates were removed. · All changes in the revised manuscript are highlighted in red for the reviewers’ convenience.
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Reviewer 3 Report
Comments and Suggestions for AuthorsPlease find the attached file of the review report
Comments for author File: Comments.pdf
Author Response
Response to Reviewer 3 Comments
- Summary
We would like to sincerely thank Reviewer 3 for the careful reading of our manuscript and for the constructive comments and suggestions. The feedback was very helpful in improving the clarity and quality of the paper. We have carefully considered all the remarks and revised the manuscript accordingly. Below we provide a point-by-point response to each of the reviewer’s comments, with explanations of the changes made in the revised version.
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2. Questions for General Evaluation |
Reviewer’s Evaluation |
Response and Revisions |
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Does the introduction provide sufficient background and include all relevant references? |
Yes/Can be improved/Must be improved/Not applicable |
In the revised version we expanded the Introduction to provide a more comprehensive background on Volterra integral equations of the third kind. We also added several recent and relevant references (Nemati et al. 2021; Usta 2021; Ma and Huang 2021; Hashemizadeh 2021) to strengthen the literature review and highlight the novelty of our contribution. The Introduction now clearly situates the work within the context of current research. |
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Is the research design appropriate? |
Yes/Can be improved/Must be improved/Not applicable |
We thank the reviewer for this valuable remark. In the revised version, the research design has been clarified and improved. We specified the assumptions more explicitly (conditions (a)–(e)), provided a clearer explanation of the Lavrentiev-type regularization operator, and added details of the iterative numerical scheme (initial guess, stopping criterion, and Newton corrections). These changes make the design of the study more transparent and reproducible. |
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Are the methods adequately described? |
Yes/Can be improved/Must be improved/Not applicable |
We thank the reviewer for pointing this out. In the revised manuscript, the description of the methods has been substantially improved. We clarified the assumptions (conditions (a)–(e)), provided the missing definitions of the domain D and the space and explained the role of the regularization operator in more detail. In the numerical section, we added information on the discretization procedure, the initial guess, the stopping criterion for Picard iterations, and the application of Newton corrections. These improvements ensure that the methods are now described comprehensively and reproducibly. |
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Are the results clearly presented? |
Yes/Can be improved/Must be improved/Not applicable |
We thank the reviewer for this helpful remark. In the revised manuscript, we improved the presentation of the results. We clarified the structure of Section 3, reformatted the table of numerical errors with consistent notation, and added explanatory text to highlight the agreement between theoretical estimates and computational data. The captions of the figures were expanded for better readability, and references to the corresponding theorems were included. We believe that these changes make the results clearer and easier to follow. |
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Are the conclusions supported by the results? |
Yes/Can be improved/Must be improved/Not applicable |
We thank the reviewer for this important remark. In the revised manuscript, the Conclusions section has been carefully revised to ensure that all statements are directly supported by the analytical results and numerical experiments. Redundant or overly general claims were removed, and the conclusions now emphasize only the findings that are explicitly confirmed by Theorems 1–2 and the computational data presented in Section 3. We believe that the revised version provides a concise and well-supported summary of the work. |
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Are all figures and tables clear and well-presented? |
Yes/Can be improved/Must be improved/Not applicable |
We thank the reviewer for this remark. In the revised version, the figures have been improved in resolution and formatting, and the captions have been expanded to provide clearer explanations. The numerical results in Table 1 have been reformatted with consistent notation and alignment for better readability. We believe that these improvements make all figures and tables clearer and more informative. |
- Point-by-point response to Comments and Suggestions for Authors
Comments 1: The uniqueness of the solution is mentioned in the abstract but is not a main result of Theorems 1 or 2, which focus on convergence. Please show the uniqueness of solution under the given conditions in the analytical section.
Response 1: We appreciate this suggestion. In the revised manuscript, we have explicitly added Lemma 1 in the analytical section, where the uniqueness of the solution is proven using the Lipschitz condition and Gronwall’s inequality. This addresses the reviewer’s concern.
Comments 2: In the proof of Theorem 1, the function is defined. Could a brief intuitive explanation of what this function represents?
Response 2: In what follows, we replace the notation by its explicit form , which represents the integrating factor generated by the resolvent method and accumulates the effect of the kernel regularization along the trajectory from to .
Comments 3: In line numbers 140 to 142, the reference [26] is cited twice; please show the evaluation steps of all the functions.
Response 3: We have revised the proof to show the evaluation steps in more detail and removed redundancy. Now the reference [26] is cited appropriately, with explicit derivations given.
Comments 4: The exposition of the numerical method is succinct. Could you furnish additional information? For instance:
- What was the preliminary estimate for the Picard iterations?
- What was the precise convergence criterion?
- What is the usual number of Picard iterations required prior to switching to Newton’s technique for modest ε?
Response 4: The initial guess for the Picard iterations was chosen as , which is a standard and convenient choice ensuring simplicity and convergence without loss of generality. The convergence criterion was set as
Typically, for moderate values of the Picard method converged in about three iterations. For smaller values of one additional local Newton step was applied to guarantee stability and accuracy.
Comments 5: On Page 13, the error on [0, b₁] is described as A₁ε + B₁/ε ². The 1/ε² term would grow infinitely large as ε→0, which is the opposite of your data and theory. Is this a typo? Did you mean A₁ε + B₁ε^γ (with γ=1/2) or perhaps A₁ε + B₁ε ²?. Please explain the detail
Response 5: We thank the reviewer for noticing this. We carefully checked the text, and confirm that the expression in the revised manuscript is correctly written as . which is consistent with the theoretical results and numerical data. The erroneous form with does not appear in the revised version.
Comments 6: Please include a section of Discussion that elaborates on your results in detail, and also clarify the precise implications of these figures. The conclusion section should be separate.
Response 6: We have followed this recommendation. The manuscript now contains a separate Section 4 (Discussion), where the results are analyzed in detail and compared with numerical findings. The Conclusions section is presented separately as Section 5.
Comments 7: Please improve the introduction section by incorporating additional Volterra integral equations relevant to real-world applications.
Response 7: We have revised the introduction and expanded it with references to applications in rheology, heat conduction, viscoelastic materials, control systems with memory, plasma dynamics, and population biology. This improvement clarifies the motivation and relevance of the study. Please see section Introduction.
Comments 8: The English language needs to be improved, and there are some typing mistakes.
Response 8: The manuscript has been carefully revised to improve grammar and style. Typographical mistakes have been corrected, and the text has been proofread for clarity and readability.
Round 2
Reviewer 2 Report
Comments and Suggestions for AuthorsThe authors have completed the revisions.
Reviewer 3 Report
Comments and Suggestions for AuthorsThe authors have significantly improved the manuscript; thus, I recommend accepting it for publication.
