Sequential Riemann–Liouville and Hadamard–Caputo Fractional Differential Systems with Nonlocal Coupled Fractional Integral Boundary Conditions

Round 1

Reviewer 1 Report

Please see the attached file.

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

The authors studied a sequential fractional differential system containing mixed Riemann-Liouville and Hadamard-Caputo fractional derivatives, 
subjected to nonlocal coupled fractional integral boundary conditions. Existence and uniqueness are established via the Banach contraction mapping principle and Leray-Schauder alternative. The results are confirmed by numerical examples.

The article is of interest to specialists in fractional equations and their applications.

The question of substantiating the used constructions of fractional derivatives is open and not clarified by the authors.

Perhaps the following references can help.

Tarasov VE. Fractional dynamics: applications of fractional calculus to dynamics of particles, fields, and media. Springer 2011.

Uchaikin V, Sibatov R. Fractional kinetics in space: Anomalous transport models. World Scintific 2018.

If possible, refer to these works.

Author Response

Please see the attached.

Author Response File: Author Response.pdf

Reviewer 3 Report

After having assessed the suitability for publication of the Manuscript ID: axioms-1300758, having the title "Sequential Riemann-Liouville and Hadamard-Caputo Fractional Differential Systems with Nonlocal Coupled Fractional Integral Boundary Conditions", I have distinguished several elements that from my point of view should be made less confused and more comprehensible by the authors in view of improving the quality of the manuscript. Therefore, I have devised and wrote a series of comments to the authors of the manuscript under review.

The authors have conducted a research targeting the existence and uniqueness of solutions for a sequential fractional differential system containing Riemann-Liouville and Hadamard-Caputo fractional derivatives, with nonlocal coupled fractional integral boundary conditions. In this purpose, the authors make use of Banach contraction mapping principle and Leray-Schauder alternative. Afterwards, in order to illustrate their obtained results, the authors present an example.

The article under review will be improved if the authors address the following aspects in the text of the manuscript and reflect them clearly point-by-point within the coverletter:

1. The main weak point. In my opinion, the main weak point of the Manuscript ID: axioms-1300758 consists in the fact that even if the manuscript is interesting, it still remains an aspect that must be emphasized and strengthened more by the authors. The manuscript approaches similar problems tackled in a previous paper, published by a research team comprised of 4 authors (from which 2 are also authors of the manuscript under review), namely the paper "Asawasamrit, S.; Ntouyas, S.K.; Tariboon, J.; Nithiarayaphaks, W. Coupled Systems of Sequential Caputo and Hadamard Fractional Differential Equations with Coupled Separated Boundary Conditions. Symmetry 2018, 10, 701. https://doi.org/10.3390/sym10120701", cited as [19]. Consequently, the authors have succeeded in establishing a precedence in their line of research. Therefore, I consider that the manuscript under review will benefit a lot if the authors refer the previous paper and highlight how the timeline of their research has evolved from the findings of their previous study to the present research results reported in the current manuscript. In my opinion, presenting a detailed comparison of the approach and the results from the paper under review with the ones from the cited paper [19], is a very important, relevant and mandatory aspect. In this context, I would like the authors of the Manuscript ID: axioms-1300758 to highlight clearly, by writing in the paper what are the main differences between their conducted study and the previous one and the reason why they have used a large amount of text from the previous paper, without rephrasing and citing it. I would like the authors of the Manuscript ID: axioms-1300758 to rephrase and cite appropriately the large amount of text from the previous paper that they have used in the manuscript, as this citation does not appear in all the cases where the excerpts of text from the previous manuscript have been used. Moreover, the authors must provide a series of details on the practical applicability of their results with regard to situations where such problems can be encountered and how they can be addressed using the results and findings of their study.
2. The "Abstract". The manuscript will benefit if the authors provide a structured abstract, that covers the following aspects: the background (in which the authors should place the issue that the manuscript addresses in a broad context and highlight the purpose of the study), the methods used to solve the identified issue (that should be briefly described), a summary of the article's main findings followed by the main conclusions or interpretations. In the abstract the authors must also declare and briefly justify the novelty of their work. The authors must present in a clearer manner the above-mentioned aspects: the background, the methods, the main findings, the conclusions, as in the actual form of the manuscript, the abstract offers information related only to some of these aspects and even so, their delimitation is unclear.
3. The "Introduction" section: the state of art, the gap in the current state of knowledge and highlights. In this section, at Lines 9-13, the authors state: "Fractional differential equations has played a very important role in almost all branches of applied sciences because are considered a valuable tool to model many real world problems. For details and applications we refer the reader to monographs [1]-[9]. The study of coupled systems of fractional differential equations is important as such systems appear in various problems in applied sciences, see [10]-[14]. Fractional differential equations have several kinds of fractional derivatives, such as Riemann-Liouville fractional derivative, Caputo fractional derivative, Hadamard fractional derivative end so on. In the literature there are many papers studying existence and uniqueness results for boundary value problems and coupled systems of fractional differential equations and used mixed types of fractional derivatives, see [15]-[24]." I consider that it is not appropriate for the manuscript to cover 24 (like the authors have did) or even more scientific works in 10 lines of text just for the sake of obtaining an appropriate size of the References section. In this section, the authors must introduce a presentation of the current state of the research field by reviewing it carefully and by citing key publications. The authors have cited some valuable works from the current state of art, works that are relevant to their study, but the literature review conducted within the "Introduction" section should be be performed in a critical manner. After having extended and performed the literature survey in an appropriate manner, the authors will be able to pinpoint an exact deficiency, an unsolved problem that still exists in the current body of knowledge that their study addresses. This aspect will improve the manuscript under review on multiple plans, as the identified deficiency, the identified unsolved problem will offer great opportunities to highlight and prove the contribution, the advancement that the conducted research has brought to the existing state of knowledge. It will benefit the paper if in the final part of the "Introduction" section, the authors present the main contributions of their paper, eventually synthetized within a bulleted list.
4. Presenting the devised approach. In order to help the readers better understand the methodology of the conducted research, the authors should devise a flowchart when presenting their approach, a flowchart that depicts the steps that the authors have processed in developing their research and most important of all, the final target. This flowchart will facilitate the understanding of the proposed approach and at the same time will make the article more interesting and help promote it to the readers.
5. The detailed explanations. The manuscript consists mainly in equations, definitions, lemmas and theorems, lacking explanations, interpretations and in particular details regarding the meaning, the purpose, the usefulness of the research developed within the paper. The authors must take into account that this is a scientific article that should contain not only mathematical formalism (which indeed is of paramount importance), but also detailed explanations. Therefore, the authors must explain and analyze in detail all the results that have been inserted within the manuscript as equations, because it is not suitable to put the reader in the situation of interpreting, analyzing, continuing or refining the study from the manuscript under review.
6. The example from Section 3. In the actual form of the manuscript, in Section 3, "Main results", at Line 119, the authors state "Next, we present examples to illustrate our results.", and afterwards, at the following line and up to the Line 141, is presented only one example, namely Example 1. In this context, I consider as an improper expression the plural form of the noun "examples". In addition to this, the authors should discuss more the presented example and explain in the article if only a single example is sufficient in order to validate the usefulness of the proposed approach. The authors should provide within the manuscript more details regarding the chosen example, if it has a practical applicability, if the chosen example models a practical problem and what are the reasons for choosing this example instead of others. Is it simpler, more relevant, more useful than other examples that can be chosen?
7. Discussing the obtained results. When presenting and discussing their obtained results, in order to validate the contribution and usefulness of the conducted study, the authors should compare their research design, proposed approach, registered experimental results with the ones obtained by other valuable scientific works. It will benefit the paper to examine, to debate the registered results and the way how they can be regarded with prospects to preceding studies that have targeted similar or related goals. Unfortunately, in the current form of the manuscript under review this discussion, this debate does not exist. The authors must address these important aspects in order to be able to pinpoint exactly the clear advancement that their conducted study from the manuscript has made to the current body of knowledge.
8. The advantages and disadvantages of the proposed approach. The authors should underline both the advantages and disadvantages of their proposed approach when compared with other valuable studies from the current state of art. When discussing their obtained results, the authors should emphasize not only the novel aspects and strong points of their developed method, but also to point out objectively the existing limits of their method, possible circumstances that will hinder their method’s effectiveness and state clear and accurate directions they will pursue in their future research activities in order to extend the current research and overcome the existing limitations.
9. Insights. The paper will benefit if the authors make a step further, beyond their approach and provide an insight when discussing their obtained results regarding what they consider to be, based on the obtained results, the most important benefits of the research conducted within the manuscript, taking also into account its practical applicability.
10. The "Conclusions" section. Firstly, in the actual form of the manuscript, the title of this section in misspelled as "Conlusion". Secondly, I have noticed that the authors have intended to entitle the section "Conclusion" but obviously, this section depicts more than one conclusion. Therefore, the title "Conclusions" is more suitable (just like the official Axioms MDPI Journal's Template recommends, https://www.mdpi.com/journal/axioms/instructions ). In the current form of the manuscript, the "Conclusions" section is almost identical with the "Abstract" of the paper. It will benefit the manuscript if the authors devise a proper "Conclusions" section in which they state the most important outcome of their work. The authors should not restate (like they did in the current form of the manuscript) what they have done or what the article does, they should focus instead on what they have discovered and most important on what their findings mean.

Author Response

Please see the attached.

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

After having assessed the suitability for publication of the revised version of the Manuscript ID: axioms-1300758, having the title "Sequential Riemann-Liouville and Hadamard-Caputo Fractional Differential Systems with Nonlocal Coupled Fractional Integral Boundary Conditions", I can conclude that the authors have addressed the most important signaled issues, therefore improving the manuscript.