Finite Series of Distributional Solutions for Certain Linear Differential Equations

Round 1

Reviewer 1 Report

In introduction,should give some describtion about the backgroud of the considered equations (5)(6) for potential applications.

Author Response

In introduction, should give some description about the background of the considered equations (5) and (6) for potential applications.
Response: We have added related background to the considered equation (5) in the introduction part.

Author Response File: Author Response.pdf

Reviewer 2 Report

The paper is written quite well, and so the publication of this excellent paper in Axioms is strongly recommended after minor revision; please see the attachment.

Comments for author File: Comments.pdf

Author Response

I have revised accordingly to all comments.

Author Response File: Author Response.pdf

Reviewer 3 Report

Results are good and essential for current interest of nonlinear science. However, I think that for the manuscript to be more complete the references should be added:

 

1. Emad A. Az-Zo’bi et. al. Numeric-Analytic Solutions for Nonlinear Oscillators via the Modified Multi-Stage Decomposition Method. Mathematics 2019, 7, 550; doi:10.3390/math7060550
2. O. González-Gaxiola, J. A. Santiago and J. Ruiz de Chávez, Solution for the Nonlinear Relativistic Harmonic Oscillator via Laplace-Adomian Decomposition Method. Int. J. Appl. Comput. Math (2017) 3:2627–2638; DOI 10.1007/s40819-016-0267-3

Author Response

Results are good and essential for current interest of nonlinear science. However, I think that for the manuscript to be more complete the references should be added. Response: I have added more complete the references in the section Conclusions by the references [24] and [25].

Reviewer 4 Report

The paper is devoted to the construction of distributional solutions of linear nonstationary differential equations with singularities in coefficients. The article is well-written and offers a few example. Nevetheless, I have some minor comments:

1. It is necessary to start the article from the results on the existence os the solutions and to say clearly on the conditions where the solutions exsit in classical or distributional sence.
2. It is desirable to present some numerical examples.
3. Some comparison analysis is required. For example to compare the solutions for the example 5 with ones obtained with the Runge-Kutta method (within some time intervals)
4. If we used Your method for the equation without singularities. What is the comparison with the classical solutions? Please analyse it and demonstrate with the help of an example. Can we compare it with Your results? 
5. The conclusion should be extended

Author Response

I have revised accordingly to all comments.

Author Response File: Author Response.pdf

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.