Applications of the Tarig Transform and Hyers–Ulam Stability to Linear Differential Equations
, K. Alagesan 1,†, Vediyappan Govindan 2,†, Salman Saleem 3,*,†, A. Al-Zubaidi 3,† and C. Vimala 4,†Round 1
Reviewer 1 Report
In this paper, the authors considered the Tarig transform for homogeneous and nonhomogeneous linear differential equations. By using the Tarig transform, the conditions for Hyers-Ulam stability of the addressed differential equations were obtained. Some pplications to general fractional order linear differential equations were given. The main results of this paper are interesting. Therefore, I can recommend it accepted if the authors can make some reivisions as follows.
1) The motivations and contributions of this paper should be clarified in the introduction part.
2) Some remarks should be added comparing the existing results with this paper.
3) The authors should give the legend of figures.
Author Response
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Reviewer 2 Report
In this paper the authors used Tarig transform for solving non-homogeneous and homogeneous linear differential equations. The paper is interesting but need major revision. I suggest to consider the following comments and suggestion in the revised version:
1- In the abstract, line 2, they said "Using this unique integral transform". I suggest to indicate the reason, because we can use other transform such as "Laplace transform" or "Elzaki transform".
2- Page 2, Line 36, they have mentioned that "v_i(z)" are constant. So they can us "v_i" instead "v_i(z)".
3- Proof of "Proposition 1." is available in other publication. I suggest to cite them and delete the proof.
4- In view of (29), I suggest to add another example when $1<\mu$.
The English can be improved.
Author Response
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Reviewer 3 Report
1. The introduction should be improved and it must be give a clear and explicit indications of the study and it is importance, therefore, it should explain what was presented on this study. 2. The authors should be referring to the new results such as thermos and definitions and must cite the old one. 3. Please check the manuscript carefully for typos and grammar errors. 4. Please check the citation format and punctuation of the references. 5. Punctuations should be followed at the end of the equations and this is very importance. 6. The authors mentioned in abstract, ((we discusses the Tarig transform for homogeneous and non2 homogeneous linear differential equations)). Based on this sentences there is one example tested on this manuscript in form of remark Eq. (33) and in fractional form. The examples or the applications or the tested examples should be more clear and at least two examples. 7. The authors use ((In this section, we look at a few fractional differential equation applications of the Tarig transform technique)). they didn’t mention fractional differential equations in the abstract, so that they should explain exactly what they used on this manuscript. 8. The abstract should be more clear and describe the presented work carefully.
Comments for author File: Comments.pdf
Minor editing of English language required
Author Response
Attached the report
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Round 2
Reviewer 2 Report
The revised looks fine. I suggest to accept it.
Reviewer 3 Report
there is no any comments
Minor editing
