Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis
Round 1
Reviewer 1 Report
I have read the whole paper very carefully. Generally speaking, I reject this paper because
I found the following paper for the same author :
-Tohru Morita and Ken-ichi Sato, Solution of Inhomogeneous Fractional Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis, Mathematics, Mathematics 2021, 9(16), 1944; https://doi.org/10.3390/math9161944
I noted that this paper is more general than the submitted paper.
Author Response
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Author Response.pdf
Reviewer 2 Report
Manuscript ID appliedmath-1705994
Review report:
Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Greens Function, in Nonstandard Analysis
By: Tohru Morita
Reviewer comments
The author presented by Morita and Sato in Mathematics 2017; 5, 62: 124, 2021;
9. 1944: 124, on the problem of obtaining the particular solution of an inhomogeneous differential equation with polynomial coefficients in terms of the Greens
function. In the first one, the problem is treated in distribution theory, and in the
second paper, the formulation is given on the basis of nonstandard analysis, where
fractional derivative of degree, which is a complex number added by an infinitesimal
number, is used. In the present paper, a simple recipe based on nonstandard analysis, which is closely related with distribution theory, is presented, and is applied to
Kummers, the hypergeometric and the Hermite differential equation.
A MINOR revision is required to make the manuscript worth publishing.
Abstract should be rewritten and extended so that it can reflect the overall
contain of the paper.
The novelty and contributions of the paper can be written in a more emphasized way.
What is the applicability of the study in terms of real world problems?
A professional proofreading revision is strongly required.
Future research direction must be shown in conclusion.
Include related references and remove unnecessary references.
https://doi.org/10.1186/s13662-021-03340-w,
https://doi.org/10.1016/j.aej.2020.07.014,
https://doi.org/10.1016/j.rinp.2021.103888
After revisions I recommend strongly this paper for publication.
Comments for author File: Comments.pdf
Author Response
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Reviewer 3 Report
Its a very good paper, well written lots of results. The introduction
might benifit from some additional information about these kinds of
derivatives.
The main idea is to solve certain pdes using Greens functions and nonstandard analysis. topic seems quite original to me. paper adds to understanding current work on fractional derivatives and nonstandard analysis. It is well written, but for me hard to read. The conclusions seem to consistent with everything else.Author Response
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Reviewer 4 Report
The paper is devoted to differential equations with polynomial coefficients. The author considers the problem of obtaining particular solutions of such equations in terms of the Green’s function. Using methods of nonstandard analysis constructed solutions of various differential equations with polynomial coefficients, in particular, Kummer’s and hypergeometric differential equations. The proposed method is applied for fractional differential equations and Hermite differential equations.
The paper is well-written and relevant. All proofs are correct and complete.
Just one comment which may be corrected by proofreading: Line 195, should be Differential instead of Differentional.
Author Response
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