Investigating a Generalized Fractional Quadratic Integral Equation

Round 1

Reviewer 1 Report

Fractalfract-1655514

Title: Investigating a generalized Hilfer-type fractional quadratic integer equation

Authors: Basim N. Aboodh, Saleh S. Redhwan

An investigation of the analytical and approximate solutions of a fractional quadratic integral equation is presented in the frame of the generalized Hilfer fractional integral operator applying Banach fixed point theorem, Adomian decomposition method and Picard method.

The paper is interesting but needs to be improved in several aspects:

- The abstract is somehow short. Should be more detailed.

- The introduction, namely the introduction to fractional calculus, should be extended. Relevant references should be included and discussed.

- The English writing should be seriously improved. If you consider needed, please ask for the help of a colleague or someone more familiar with the English language. This point needs particular attention, you can find several sentences initiated in a lowercase letter, even the title of a reference. Randomly in a sentence you can find an uppercase letter. There are sentences without verbs and others with 2 verbs. Probably in some sentences there is some text missing. This situation impoverishes the article presentation.

- You can find several acronyms that are not defined. I can highlight RL, that should be presented as Riemann-Liouville.

- In page 6 you refer to hypotheses (ii) and (iii). There are no hypotheses with this reference in the text.

- Some of the errors in the text are typos that should be corrected. You should also look carefully to the equations, as some should also be corrected.

- The article must have a conclusion.

- You should emphasize what are your novel contributions presented in the article.

Author Response

Respected dear reviewer 
Please find the attached PDF (Response to the Reviewer)

Author Response File: Author Response.pdf

Reviewer 2 Report

Report on “INVESTIGATING A GENERALIZED HILFER-TYPE FRACTIONAL
QUADRATIC INTEGER EQUATION” by B.N. Aboodh and S.S. Redhwan, submitted for possible publication in Fractal and Fractional.

In the paper under review, the authors investigated the analytical and approximate solutions for a class of fractional quadratic integral equations involved with Hilfer fractional integral operator. They used Banach fixed point technique, Adomian decomposition method and Picard Method For the main results.

The results look interesting and applicable. However, the authors are requested to fix the following issues.

1/ The posed problem and the used techniques are known and standard in the literature, hence It is needed to reformulate the contribution and motivation based on the most recent works.

2/ Definition 2.1.

Inside the integral, “F(\Theta, x(\Theta))” is appearing as a constant with respect to the “v”! update.

3/ Where is the use of Lemmas 2.3 and 2.4?

4/ Lemma 3.1, include a proof or set a reference.

5/ The subsection “3.1. Uniqueness result:”, it should be “3.1. Existence and uniqueness of solutions”

6/ detail more the chosen example to illustrative the three main results, and replace “i=1,2,\cdots,” by “i=1,2,\ldots,”

Author Response

Respected dear reviewer 
Please find the attached PDF (Response to the Reviewer).

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Some aspects still have to be tackled, e.g.,

-The acronyms in page 1, line 22 are not defined,

-The English writing can be improved in page 2 line 54, 55 and 57, page 5 line 85, page 6 line 87, page 8 line 97, page 9 line 101, page 10 line 114,

-In page 4 line 80 there is one capital letter in the middle of the sentence,

-Correct the formatting in page 2 line 36, page 4 line 81,

-Correct the misspelled words page 2 “approxmite”, page 6 “hyoptheses”.

Author Response

we would like to thank the dear referee for his/her efforts in reviewing this paper
Please find the attached file (Response to reviewer 3 PDF) 

Author Response File: Author Response.pdf