Numerical Solution of Time-Fractional Emden–Fowler-Type Equations Using the Rational Homotopy Perturbation Method

Round 1

Reviewer 1 Report

In this paper the authors a rational homotopy perturbation method to solve the time-fractional Emden-Fowler-type equations. The method is based on a series of power series in rational form. The existence and uniqueness of the equation are proved by the Banach fixed-point theorem. In addition, we approximate the term h(z) by a polynomial of a suitable degree and then solve the system using the proposed method and obtain an approximate symmetric solution. Two numerical examples are investigated using this proposed approach. The effectiveness of the proposed approach is checked by representing graphs of exact and approximate solutions.

I recommended significant revision. The observations are:

1.     Add the significance of fractional derivative in the abstract.

2.     Improve the introduction section. I recommend improving by considering the results “Analysis of fractional multi-dimensional Navier–Stokes equation”, “Fractional analysis of coupled burgers equations within yang Caputo-Fabrizio operator”, “An Efficient Technique of Fractional-Order Physical Models Involving ρ-Laplace Transform”, “Fractional system of Korteweg-De Vries equations via Elzaki transform”.

3.     Give reference to the Equations.

4.     Compare your work with another method (optional).

5.     Discuss the graphs physically.

Author Response

First of all, we are thankful to you for your comments 

we tried to justify comments. 

1) Add the significance of fractional derivative in the abstract.

Added 

2) Improve the introduction section. I recommend improving by considering the results “Analysis of fractional multi-dimensional Navier–Stokes equation”, “Fractional analysis of coupled burgers equations within yang Caputo-Fabrizio operator”, “An Efficient Technique of Fractional-Order Physical Models Involving ρ-Laplace Transform”, “Fractional system of Korteweg-De Vries equations via Elzaki transform”.

Improved. Added the suggested references 

3) Give reference to the Equations.

Added 

4) Compare your work with another method (optional).

We have used examples of [34] and improved or corrected the solution of that paper.  

5) Discuss the graphs physically.

Graphs are simply representations of the solution of equations. Basically, we solved arbitrary equations. 

Reviewer 2 Report

1 what is the differences  HOMOTOPY PERTURBATION METHOD and RATIONAL HOMOTOPY PERTURBATION METHOD.

2 What is the advantage of this method, why you use this method. 

3 write paper in journal templete and references according to the journal rules.

4 Write one example step by step to show clearly difference between your method and homotopy method.

5 Add one example nonlinear and compare with previous work 

6 Add the labels of figures x-axis and y-axis

7 Write graphical disscusion and modified all figures.

Author Response

First of all, we are thankful to the reviewer for his/her constructive comments.

We have tried to answer are comments as much as possible 

1 what is the differences  HOMOTOPY PERTURBATION METHOD and RATIONAL HOMOTOPY PERTURBATION METHOD.

We have discussed differences in the introduction section. Highlighted in RED fonts in this section. 

2 What is the advantage of this method, why you use this method? 

As shown in the results. The solutions obtained by Rational Homotopy Perturbation Method are almost the same.  

3 write paper in journal templete and references according to the journal rules.

Done 

4 Write one example step by step to show clearly the difference between your method and the homotopy method.

We found that the RHPM method is more reliable as compared to homotopy as we found an accurate solution see table. Solution by the HPM method may find in the available literature see [34].  

5 Add one example nonlinear and compare with previous work 

We have proposed the scheme for nonlinear equations which can be easily applied to nonlinear problems. However, we did not find any nonlinear example for an exact solution to verify. We are in the construction of nonlinear example with exact solution for verification. 

6 Add the labels of figures x-axis and y-axis

7 Write graphical disscusion and modified all figures.

Round 2

Reviewer 1 Report

Accept in present form

Reviewer 2 Report

Accepted