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Volume 11
Issue 5
10.3390/axioms11050192
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Review
Peer-Review Record

Continuous-Stage Runge–Kutta Approximation to Differential Problems

Axioms 2022, 11(5), 192; https://doi.org/10.3390/axioms11050192
by Pierluigi Amodio 1, Luigi Brugnano 2,* and Felice Iavernaro 1
Reviewer 1: Anonymous
Reviewer 2: Anonymous
Axioms 2022, 11(5), 192; https://doi.org/10.3390/axioms11050192
Submission received: 21 March 2022 / Revised: 12 April 2022 / Accepted: 18 April 2022 / Published: 21 April 2022

Round 1

Reviewer 1 Report

The article reviews previous work of the authors on Hamiltonian boundary value methods in the framework of continuous stage Runge-Kutta methods. While the presentation is mathematically sound, the only novelty appears to be the application to higher order differential equations; since applications to second order problems have already been considered in [2] and the application to higher order differential equations does not contain any new ideas, I think that the manuscript does not reach the level of novelty to justify publication in its present form. However, a careful numerical study of HBVM for higher order problems also, maybe, a comparison to symplectic methods in the case of second order differential equations (which admit a Hamiltonian formulation) could make the article a valuable contribution to the domain.

 

A few minor points concerning notation:

  • Equation (2), a and b are functions but a(c,tau), b(c) are real numbers. Please delete the arguments in (2). Moreover, target spaces should be denoted as sets, so the target set 1 should read {1}.

  • Introducing the target set {1} of b in (2) makes (8) superfluous. Did you mean to have a target set [0,\infty) in (2)?

  • No comma needed after the table in (16).

  • It would be good to mention that the tensor symbol denotes the Kronecker product in (22) unless this is intentionally avoided because of the presence of infinite matrices.

Author Response

See the attached file, thank you.

Author Response File: Author Response.pdf

Reviewer 2 Report

Rather than using footnotes, I would prefer to see explicit assumptions for a particular section of the paper (e.g., the assumption of analyticity of the function f - see also Section 3). That said, it does increase the readability of the paper. As for me, the interesting question is whether and to what extent this assumption can be weakened, but this is obviously beyond the scope of the paper. AC functions are sometimes essential in analytic considerations of this type of question.

Author Response

See the attached file, thanks you.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Thanks to the authors for considering my suggestions and clarification that the article is a review article. I think the article is now ready for publication.

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