A Crank–Nicolson Compact Difference Method for Time-Fractional Damped Plate Vibration Equations

Round 1

Reviewer 1 Report

In the paper the authors consider in section 2 equation (1), using (1) equation (2) and  (15) are introduced. It is not clear whether equation (1) is new or not. The authors must clarify this and in case it is not new a reference to the equation must be provided. another suggestion is that Lemma 3 is referenced from [26] but  not directly. Both Lemma 1 and Lemma 2 in [26] are phrased differently from what is obtained in the current manuscript, I suggest that the authors should rephrase  Lemma 3 accordingly. Then a full stop should be placed at the end of the Lemma. In section 3 and 4 stability analysis and  convergence analysis of the results are presented, respectively. In section 3, Theorem 1 is presented with a correct proof and section 4, Theorem 2 is presented with proof. This is followed by  three numerical examples to justify the results presented in the earlier sections, namely, section 2 - section 4.

Also, see the attached for more comments that will assist on the presentation of the paper. I suggest that some background information/notations from other references be included instead of the reader having to search for such information.

Comments for author File: Comments.pdf

Author Response

1. For the equation (1) in section 2 , the definition has been further refined in the paper.
2. After careful study and inspection, the problem of "Lemma 3" in the original text has been solved, and it has been revised in the next submitted paper.
3. For the comments in the appendix below, which have been revised in the paper. The details are as follows.

• "Δu" and the compact difference operators "A"_x and "A_y" have been defined in the paper.  
• Equation (1) in Section 2 has given the definition of fractional derivative and defines "1<α<2".
• "u∈C³ [t_0,t_n]" means that the third derivative of ''u'' is continuous in [t_0,t_n].

Reviewer 2 Report

Manuscript ID: axioms-1903115

Manuscript Title: A Crank-Nicolson Compact Difference Method for Time-fractional Damped Plate Vibration Equations

Authors:  Cailian Wu , Congcong Wei , Zhe Yin , Ailing Zhu 

 

 

Dear    Editor,

I have attached below the report.

Best wishes,

 

 

 

 

 

The authors discussed the Crank-Nicolson compact difference method for the time-fractional damped plate vibration problems. They introduced the time-fractional damped plate vibration equations,  the second-order space derivative and the first-order time derivative to convert fourth-order differential equations into second-order differential equation systems. I have some comments and questions that the authors should do it, so that the paper will be looks better:

·      Explain your original contribution.

·      Proofreading review is required for the paper.

·      Discuss the advantages of the suggested methods over other existing methods.

·      Write motivation for the carried research work in introduction.

·      The English structure of the article, including punctuation, semicolon, and other structures, must be carefully reviewed.

·      The paper is not formatted in accordance with the journal requirement.

·      Writing the captions on the figures in detail is required.

·      References are adequate; however, other related works such as given below may also be cited in the introduction

https://doi.org/10.1016/j.chaos.2022.112394

https://doi.org/10.3390/math10071089

doi: 10.3934/math.2022364

After revisions I recommend  this paper for publication in  AXIOMS .

 

 

 

 

Author Response

    Thank you for your valuable comments on my paper! Here are my answers to your questions.

1. My answer (given in the introduction) to the questions about  "original contribution" and "write motivition"  is: 

    To date, the authors have found no literatures that use the compact difference method to numerically simulate time-fractional damped plate vibration equations. Therefore, this paper seeks to solve time-fractional damped plate vibration equations by introducing the second-order space derivative and the first-order time derivative to convert fourth-order differential equations into second-order differential equation systems.

    Numerical cases are provided to validate the convergence order and feasibility of the given difference format. This paper not only discusses the compact difference method for the damped plate vibration equations with time-fractional derivatives, but also simulates the influence of damping coefficient on the amplitude of vibration. This paper provides a novel effective numerical method for solving damped plate vibration problems, which enlarges the toolkit for simulating damped plate vibration problems and contributes to a more thorough and complete system of theories for numerically solving time-fractional fourth-order differential equations.

    2. Advantages of the compact difference method (given in the introduction):

    The compact difference method, which can further accelerate the convergence of the general difference methods following a similar idea as the efficient numerical methods developed in [12] accelerate most existing approaches, is a high-accuracy numerical method that uses relatively fewer nodes arranged in a compact way.

    3. The English structure of the article, including punctuation, semicolon and other structures, has been carefully checked and revised.

    4.  Captions have been written on the graphs in detail.

    5.  The introduction has been revised, further enriching the introduction and references.