Review Reports

Comments 1: The function f(u) = u^ 3 − u does not satisfy the Lipschitz condition (3)  for all values of the argument.

Response 1:  Thank you very much for pointing this out. I have already deleted this property from the manuscript. And on page 2, lines 26-27, I have added: f(u)=u³-u satisfied the following properties |f(u)|≤|u|³+|u|.

Comments 2: It is necessary to correctly define the finite element space S_h:

 indicate with respect to which scalar product the orthonormal basis is chosen;

clarify whether the number M is related to the degree of the polynomials or to the dimension of the space S_h.

Response 2: Thank you for pointing this out. On page 3, lines100-101, we add the conditions for the orthonormal basis to be satisfied and clarify that M is the dimension of the space S_h.

Comments 3: Explain what the authors mean by f(u^{n-2}) for n=1 in (9) and everywhere below.

Response 3: Thank you for pointing this out.  In this paper, for all f(u^{n-2}), n ranges from 2 to N.

Comments 4: The proof of inequality (24) is very strange (involving U^0) and relies on the  global Lipschitz property of the function f, which does not exist.  As for further calculations involving the function f, we can use its Lipschitz property on a bounded set. But the boundedness of solutions is a result of Theorem 2, the proof of which, in my opinion, is incorrect.

Response 4: Thank you for pointing this out.  I have made a modification on page 6, lines 145-149 of the manuscript. Using the property |f(u)|≤|u|³+|u|, I re-demonstrated this process.

Comments 5: The same questions (1- 4) arise when reading the published article by Chang, X.; Li, H. The Reduced-Dimension Method for Crank-Nicolson mixed final element solution factor vectors of the extended Fisher-Kolmogorov equation. Axioms. 2024, 13, 710. Questions and comments that arose while reading this article

Response 4: Thank you for pointing this out. The answers and modifications to the above four questions are also applicable to this article: Chang, X.; Li, H. The Reduced-Dimension Method for Crank-Nicolson mixed final element solution factor vectors of the extended Fisher-Kolmogorov equation. Axioms. 2024, 13, 710.

Comments 1: After reviewing the manuscript, I consider that its content is not within the scope of the journal Fractal and Fractional. The article focuses on developing and analyzing a numerical method based on the Crank-Nicolson scheme to solve fourth-order parabolic equations with variable coefficients. Although it contributes to dimension reduction by proper orthogonal analysis (POD), its main focus lies on numerical methods for partial differential equations, with no clear relation to topics of fractals, fractional dynamics, or fractional models that characterize the journal's focus.

Therefore, I suggest that the manuscript be considered for a journal that is more aligned with numerical methods and analysis of partial differential equations.

Response 1:   

Dear Reviewer,

Thank you for your thoughtful feedback.  We appreciate your comments and would like to address your concerns regarding the scope of our manuscript.

Regarding your comment about the scope of the journal Fractal and Fractional, we respectfully clarify that our manuscript aligns well with the journal's stated scope, particularly in the areas of analysis of PDEs and numerical analysis. According to the journal's official scope, Fractal and Fractional covers a wide range of topics in mathematics, including but not limited to:

l Analysis of PDEs

l Numerical analysis

l General mathematics

l Geometry

l Differential geometry

l Classical analysis and ODEs

l Fractional-order differential and integral equations

l Dynamical systems, bifurcation and chaos

l Topology

l Functional analysis

l Mathematical analysis

l Complex analysis

l Mathematical physics

l Number theory

l Probability and stochastic analysis

l Quantum algebra

l Statistics, data analysis, and time series analysis

l Operators theory, integral and differential operators, integral transforms

l Mathematical finance

Our manuscript focuses on the numerical solution of a fourth-order parabolic partial differential equation (PDE) using a POD-based reduced-dimension method combined with the Crank-Nicolson mixed finite element method. This work belongs to the fields of Analysis of PDEs and Numerical analysis, which are explicitly listed within the journal's scope.

Furthermore, in order to fit the theme of the journal Fractal and Fractional,  we have added a numerical experiment of time-fractional fourth-order parabolic equation on lines 367-396 , page 20 in the manuscript. Numerical results show that our reduced-dimension method can also be applied to solve the fractional equations. In addition, we have added the content related to time fractional order in the Abstract (lines 10-13, page 1),  Introduction (lines 76-80, page 2) and Conclusion (lines 416-426, page 24).  

We hope this clarification addresses your concern regarding the scope of our manuscript. We would be grateful if you could reconsider our submission in light of this explanation.

Thank you once again for your time and constructive feedback.

Comments 1: Please add some calculation value in abstract 

Response 1: Thank you very much for this suggestion. I have added some calculation value in abstract on lines 14-18, page 1. The details are as follows:

Under a spatial discretization grid $40 \times 40$, the traditional CNMFE method requires $2 \times 41^2$ degrees of freedom at each time step, while the RDCNMFE method reduces the degrees of freedom to $2 \times 6$ through POD technology. The numerical results show that the RDCNMFE method is nearly $10$ times faster than the traditional method. This clearly demonstrates the significant advantage of the RDCNMFE method in saving computational resources.

Comments 2: Please define variable for all equation

Response 2: Thank you for your comment. We have carefully reviewed the manuscript and confirmed that all variables in the equations are explicitly defined in the paper.  

Comments 3:  I suggest author add comparative result with other literature 

Response 3: Thank you very much for this suggestion. The main objective of this paper is to highlight the advantages of the POD-based reduced-dimension Crank-Nicolson mixed finite element (CNMFE) method . In this study, we employ the mixed finite element method for numerical solution, with the time discretization based on the Crank-Nicolson (CN) scheme. The reduced-dimension scheme is developed from the traditional CNMFE scheme using POD technology. To fully demonstrate the advantages of the reduced-dimension method, we conducted a detailed theoretical analysis of both the traditional and reduced-dimension schemes, and designed numerical experiments based on these two schemes. The comparison of numerical results before and after dimensionality reduction clearly demonstrates the effectiveness of the proposed method, making further comparisons with other methods unnecessary.

Comments 4: check typo and grammatical error

Response 4: Thank you for your feedback. We have thoroughly reviewed the manuscript and made revisions to address typo and grammatical errors.