Qualitative Analysis for Solving a Fractional Integro-Differential Equation of Hyperbolic Type with Numerical Treatment Using the Lerch Matrix Collocation Method
and Mai Taha 2Round 1
Reviewer 1 Report
In this manuscript, the authors present a new approach for solving a nonlinear hyperbolic fractional integro-differential equation (NHFIDEq); fixed-point theorems are used to provide the existence and uniqueness of a solution. Also, by utilizing the Lerch matrix collocation (LMC) method, a numerical representation of the solution of the given NHFIDEq is obtained, with the error analysis and the convergence analysis of the problem.
Related to the manuscript's content, it is necessary:
- to be pointed out more clearly what novelty this method brings in connection with other methods existing in the literature.
- some relations must be reviewed for better clarity (e.g., the relation (8)).
- to be corrected on some aspects related to the drafting.
- to include additional comments on the application of these results (e.g., the comparisons between the exact solutions and numerical solutions for different values of Lerch parameters lambda and time t).
Some text editing corrections should be applied.
Author Response
We appreciate the time and effort that you and the other reviewers dedicated to providing feedback on our manuscript and are grateful for the insightful comments and valuable improvements to our paper. We have incorporated most of the suggestions made by you. Please open the Pdf file below, which contains a response form discussed point by point with your comments.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Dear Authors,
Please correct the paper according to the remarques.
Sincerely.
Comments for author File: Comments.pdf
Author Response
We appreciate the time and effort that you and the other reviewers dedicated to providing feedback on our manuscript and are grateful for the insightful comments and valuable improvements to our paper. We have incorporated most of the suggestions made by you. Please open the Pdf file below, which contains a response form discussed point by point with your comments.
Author Response File: Author Response.pdf
Reviewer 3 Report
The paper deals with an important task. It has a scientific novelty. It has a logical structure. The proposed approach is logical. The introduction and experimental sections must be improved.
Positive sides:
1. The introduction examines the state of the problem area and the advantages of previously studied approaches.
2. The solution existence and uniqueness of the model (1), (2) was investigated.
3. The truncated Lerch series form was used for numerical investigation of the model (1), (2).
4. The numerical results and residual error analysis were presented.
5. The validation of the obtained results was conducted.
Suggestions:
1. Introduction section should be extended using more clearly the motivation of this paper.
2. In Tables 2 and 3, it was appropriate to present the error values for a larger variation of the N parameters, in order to be able to estimate the trends more effectively.
3. Not clear are fig. 4 and 6. Since the nature of the change in the solution function shown in Fig. 1 and 2 at t=0.9 visually does not correspond to what is shown in Fig. 4 and 6.
4. It would be appropriate to expand the overview section with a review of methods for finding the solution for the mathematical models based on the use of fractional order derivatives, such as approximation using splines, using the finite element method and others. For example, pay attention to publications DOI: 10.3390/a15020069, DOI: 10.3390/e22111328 and DOI: 10.1007/978-3-031-04812-8_9
5. Conclusion section should be extended using:
- limitations of the proposed approach;
- prospects for future research.
Author Response
We appreciate the time and effort that you and the other reviewers dedicated to providing feedback on our manuscript and are grateful for the insightful comments and valuable improvements to our paper. We have incorporated most of the suggestions made by you. Please open the Pdf file below, which contains a response form discussed point by point with your comments.
Author Response File: Author Response.pdf
Round 2
Reviewer 3 Report
The authors took into account the recommendations.
