An Efficient Localized Meshless Method Based on the Space–Time Gaussian RBF for High-Dimensional Space Fractional Wave and Damped Equations

Round 1

Reviewer 1 Report

The authors present the RBF method for solving fractional wave 
and damped equations.
Numerical results show convergence of the scheme; stability of the scheme
is investigated through adding of noise of increasing size.
I suggest to study the following points before considering publication:

1. Discuss the rate of convergence (error / time of execution 
or number of degrees of freedom). This could add a better view of the 
caracteristics of the method.

2. The results are given for a regular distribution (figure 1), for which
plenty of methods exist. It would be interesting to know if the method still
 works for an irregular distribution (typically the mesh points of a
 general unstructrured mesh). The conditon number could be seriously affected.

Author Response

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Author Response File: Author Response.docx

Reviewer 2 Report

In this paper, the authors investigated an efficient localized meshless method based on the space-time Gaussian radial basis functions has been discussed. The authors aim to deal with the left Reimann-Liouville space fractional derivatives wave and damped wave equation in high-dimensional space. These significant problems as anomalous models could arise in several research fields of science, engineering, and technology. Since the explicit solution to such equations often does not exist, the numerical approach to solve this problem is fascinating. According to the authors, they propose a novel scheme using the space-time radial basis function with advantages in time discretization. Moreover, this approach makes the (n + 1)-dimensional spatial-temporal computational domain for n-dimensional problems. Therefore, the local feature, as a remarkable and efficient property, leads to the sparse coefficient matrix, which could reduce the computational costs in high-dimensional problems. Some benchmark problems for wave models, both wave and damped, have been considered highlighting the proposed method performances in terms of accuracy, efficiency, and speed-up. The obtained experimental results show the computational capabilities and advantages of the presented algorithm. Before, I can recommend this paper for publication, it should be revised subject to the following suggestions.

 

1. Radial basis functions are used in the definition of the numerical method but no theoretical and background information are given. Section 2 starts with description of the methodology while radial basis functions theory should be discussed beforehand (see Gregory Fasshauer book, 2007, "Meshfree approximation methods …").
2. What is the shape parameter “c"? What are the effects of this free parameter on the solution accuracy? How is the value of the shape parameter determined? Is it done through brute force or obtained through some well-defined procedure?
3. In the paper, parameter c is introduced and then defined in next section for numerical purposes. Literature is rich of methods for a proper choice. The values of c assumed in the numerical section is arbitrary and does not appear to be motivated. The introductory section requires a major emphasis on the selection of shape parameter c whole identification as known is a challenging issue for scientists.
4. The current introduction section is very good. But the authors should include some of the suggested references related to global and local meshless methods based on radial basis functions.

Numerical solution of time-fractional coupled Korteweg–de Vries and Klein–Gordon equations by local meshless method; Gaussian radial basis functions method for linear and nonlinear convection–diffusion models in physical phenomena; Numerical solution of two-term time-fractional PDE models arising in mathematical physics using local meshless method

5. The methods developed will be applicable to other applications – this should be made clearly (abstract or introduction)
6. The other definitions of the fractional derivative operator may be given in the text.
7. More description on Examples should be given.
8. Please also check each reference, there are some typos.

  

9. The Conclusions section is not satisfactory; it should be further improved. Please emphasize the main novelty of this paper and the significance of the results in the conclusion. 

 

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

I thank the authors to have added the two studies. The rate of convergence seems not easy to interpret, as the values are quite different depending on the parameters/cases

Reviewer 2 Report

I have re-reviewed the rebuttal report as well as the revised manuscript. All my review comments are well implemented and the revised manuscript is now in a much better shape. I thereby recommend the revised paper to be accepted in the Journal.