Analysis of a Nonlinear ψ-Hilfer Fractional Integro-Differential Equation Describing Cantilever Beam Model with Nonlinear Boundary Conditions
, Weerawat Sudsutad 2,*,†, Chatthai Thaiprayoon 3,4,†, Jutarat Kongson 3,4,*,† and Jehad Alzabut 5,6,†Round 1
Reviewer 1 Report
The paper is perfectly good mathematics, but the title and introduction are misleading. From them the reader would think that the paper is about an analysis of a novel model for a cantilever beam. But, the problem addressed appears to be simply abstract mathematics. This seems to be born out in the Examples section. An example with some obvious physical significance would be a welcome addition. As the paper is now written, it is simply a nice contribution to pure mathematics concerning the well posedness and stability of a complicated model. The impact of the paper would be much greater if there were a conclusion about beam theory that was new and relevant.
Author Response
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Author Response File: 
Author Response.pdf
Reviewer 2 Report
The paper under review deals with existence and uniqueness results to a kind of fractional integro-differential equations with nonlinear boundary condition given in (4) on page 2. Indeed the uniqueness result in Theorem 1 is based on the fixed point theorem of Banach and its proof is done in several steps. In addition, using the fixed point theorem of Schaefer, an existence result is proved in Theorem 2. Moreover, some stability results have been obtained in Section 4 and the applicability of their results is stated in Section 5 via some numerical examples.
Even though the paper is quite technical, it is well written. As far as I know the results are new, correct and of some interest. The literature references also provide a good insight into the topic. In my opinion the paper deserves publication in its current form.
Author Response
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Reviewer 3 Report
The following major remarks appear:
- some important fractional beam models are missing in the introduction e.g. Meccanica, 53(4):1115–1130, 2018.;
International Journal of Mechanical Sciences, 201, 2021.;
2020 Journal of the Mechanical Behavior of Materials 29(1):9-18;
Archive of Applied Mechanics,86(6):1133–1145, 2016.;
Journal of Engineering Mechanics, 143(5), 2017.;
European Journal of Mechanics, A/Solids Open AccessVolume 54, Pages 243 - 25119 August 2015;
International Journal of Non-Linear Mechanics 125(1):103529 ;
International Journal of Mechanical Sciences Volume 186 15 November 2020 Article number 105902;
- "one-length elastic beam" -> "plane elastic beam" or "2D elastic beam" (please check whole paper)
- in Eq 3 define the meaning of \omega
- Eq. 1-3 can not be just obtained just by exchanging operator to fractional (energy considerations are needed please follow International Journal of Mechanical Sciences Volume 186 15 November 2020 Article number 105902)
- Eq. 4 does not represent the beam model it is just a fractional differential equation
- in Eq. 4 not only 'left' operator should be used (mechanical space in symmetric you can not say that 'left' spacial dimension of the beam is important)
- all examples (especially graphical) must include classical solutions (order 4)
Author Response
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Author Response File: Author Response.pdf
Round 2
Reviewer 3 Report
Thank you.
