Numerical Solution of Fractional Elliptic Problems with Inhomogeneous Boundary Conditions

Round 1

Reviewer 1 Report

Please look at the reviewer report enclosed.

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

The definition of the fractional Laplacian on the whole space Rⁿ is easily understood through the Fourier transform. The challenge however represents the case when this equation is posed in a bounded domain and proper boundary conditions are needed for the correctness of the corresponding problem. It is worth note that the case of inhomogeneous Dirichlet boundary data has been neglected up to the last years. The reason is that imposing nonzero boundary conditions in the nonlocal setting is nontrivial. The presented article deals with the numerical solution of such problems. The topic is relevant and the paper contains interesting theoretical and experimental results.

The suggestion is to accept the manuscript for publication after due revision in order to answer the following minor questions and comments:

1. See r. 26-27: “Indeed, it seems to be hard to model this non-local phenomenon if we only have boundary data.”

Some further explanations are recommended at this point.

2. The authors need to improve the first two introductory sections by paying additional attention to the recent development of the numerical methods for the studied class of problems.
3. Please add the missing citations related to some of the results presented in Section 2.
4. The assumptions of Theorem 1 and the following results exclude from consideration the case  α> 3/4.

Some comments on this limitation are recommended to better understand the results presented.

5. See r. 162-164: “Note that in (18), the error estimation …, so we expect superconvergence ….”

The common understanding of “superconvergence” is different. Here the expectations are for a higher convergence rate.

6. See r. 168-169: “At the same time, for a relatively fine mesh, the computation with improper integrals becomes inaccurate, which results oscillations in u_h and deteriorates the accuracy of the reconstructed approximation …”

Authors should explain better this observation. Is it possible to improve the accuracy of the computation with improper integrals?

7. In (7) the right hand side is f(x)=0. Can you add (say, in Section 4) some comments on the case of nonzero right hand side?  

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

In the related manuscript, they have derived some original and good results. They have also given a numerical experiment to confirm these results.

The English of this paper is acceptable, but should be improved. All results are correct.

Accordingly, this manuscript deserves for publication in this reputed journal, Fractal Fract (ISSN 2504-3110), after the following corrections.

This paper will be recommended for publication after some revisions.

1. There are some grammar errors and punctuation. So, the author have to check this manuscript word by word for grammar errors and punctuation. For example;

• End of the equation (12), “.” must be “,”
• There is a “.” end of the equation under the line 96.
• The second “for” must be deleted in the line 51.
• etc.

2. The authors should obviously indicate what is the new in this manuscript.
3. To improve the manuscript, they can support their results with more numerical experiments and they can give more explanations of the figures.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf