Review Reports - A New Technique for Solving a Nonlinear Integro-Differential Equation with Fractional Order in Complex Space

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Title: A new technique for solving a non-linear integro-differential equation with fractional order in complex space

The current study aims to analyze the nonlinear fractional integro-differential equation. The authors studied the existence and uniqueness of the solution for the said equation. Further, a novel numerical approach is used to combine the rationalized Haar wavelet method with the Bernoulli method. For the validity of the equation, the authors give some examples at the end of the paper. The paper seems very interesting and I recommend the paper for publication after the minor comments. Some useful comments for the authors are given as:

1. The authors are advised to remove the abbreviations from the abstract.

2. The authors should change the word “utilised” to utilized.

3. The authors are advised to use fractional integro-differential equations or only the abbreviation (NFIDEq) no need to write both in every place.

4. The authors should improve the fractional part of the paper and cite the following recent work in the introduction and compare it with their paper:

a) Analysis of age wise fractional order problems for the Covid-19 under non-singular kernel of Mittag-Leffler law

b) Investigation of financial bubble mathematical model under fractal-fractional Caputo derivative

c) Dynamical properties of a meminductor chaotic system with fractal–fractional power law operator

d) Dynamical analysis of a discrete-time SIR epidemic model

5. The authors are advised to improve the English of the paper.

6. Can these methods apply to the disease model?

7. What is the main objective of this paper by using the current method?

8. On page 3 correct the sentence of line 95.

9. There is no discussion of the figures.

10. The authors are advised to improve the conclusion section and add the future direction.

11. References should be uniform.

Comments on the Quality of English Language

Average.

Author Response

We appreciate the time and effort that you 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 word file below, which contains a response form discussed point by point with your comments.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors In this paper, the authors try to find the solution of the non-linear fractional integrodifferential equation NFIDEq by using both analytical and numerical approaches. They demonstrate the existence and uniqueness of a solution under appropriate conditions with the help of Banach fixed-point theorems. The validation of the presented method was performed by utilizing a novel strategy that involves implementing the RHM method with the application of the Bernoulli polynomial technique. By doing so, the solution of the given NFIDEq is transformed into a representation that corresponds to a linear system of algebraic equations. A comparison between the presented method and the RHM method has been conducted using two numerical examples. Results show that the proposed method achieve a smaller absolute error.

From my point of view, I recommend the publication of this work in "Fractal Fract" after taking the following minor notes in consideration:

1- There are some typos in the paper. So, it is recommended to review the English language in the whole paper. See, for example:
- The subtitle 4.1 (Line 200)
- Line 130 in Section 3 (Before studding...)

2- Cite a reference for the information stated on lines 67-68 (Beylkin...)

3- The CPU time is a very important factor in proving the efficacy of the method. It is suggested to mention the CPU time needed to reach the desired solution for the two examples used in the comparison between the presented method and the RHM method.

4- In the introduction section, the authors stated that there are many applications for the NFIDE in applied sciences. Is there any real-life application that can be added to the examples to support the applicability of the proposed method? As in many papers, the examples usually are been selected among the presented methods, and in many cases, if we apply other examples, the methods fail to achieve the desired solution.

Author Response

We appreciate the time and effort that you 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 Word file below, which contains a response form discussed point by point with your comments.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The manuscript presents interesting results. The introduction is fine, and the authors should address the folloing issues:

1) Better explain the used values of N in example 1 and 2. Why are they different?

2) If possible, compare to a classical/conventional numerical technique used to obtain the solution.

Author Response

We appreciate the time and effort that you 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 Word file below, which contains a response form discussed point by point with your comments.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

See the attachment.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Author Response

We appreciate the time and effort that you 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 Word file below, which contains a response form discussed point by point with your comments.

Author Response File: Author Response.pdf

Reviewer 5 Report

Comments and Suggestions for Authors

Page 2, equation (1) : What is the use of ζ(s,t)?
Page 3, Definition 4 : It’s written “Bernoulli paynomials”
Section 5, Example 1 & Example 2 : Is K(s,t) the same as k(s,t)?
Section 5, The numerical results are not discussed. The numerical solutions presented in tables 1 & 2 and in the following graphs and figures are not explained and analyzed so that conclusions could be obtained.

Comments on the Quality of English Language

None

Author Response

We appreciate the time and effort that you 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 Word file below, which contains a response form discussed point by point with your comments.