Effective Method for Solving Different Types of Nonlinear Fractional Burgers’ Equations
Round 1
Reviewer 1 Report
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Reviewer 2 Report
A new method to solve partial differential equations (PDEs) called as fractional
reduced differential transform method (FRDTM) is proposed in the article.
The paper gives a good numerical examples and description of the method. However, the novelty in the introduction should be further justified.
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This paper proposed a new method to solve partial differential equations (PDEs) called as fractional reduced differential transform method (FRDTM).
The authors tested the proposed method to solve nonlinear fractional Burgers’ equations in one, two-coupled and three 3 dimensions.
It is relevant to the journal’s scope and interesting. It is written well, clear and easy to read. No grammar errors were found. The conclusions are consistent with the evidence presented. It addresses the main question posed.
However, the introduction needs to bring out the novelty in more detail. The difference of this work with existing work should be given in detail.
Minor:
Caption of Table5: space between ofv4.
Spaces needed in the caption of Table 4. 1andR.
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Reviewer 3 Report
This paper is concerned with applying the fractional reduced differential transform method (FRDTM) to several variations of Burgers's equation, and plotting and tabulating the results. My comments are mostly minor (e.g. small English fixes), but there are a few important issues which should be clarified. If all of these can be resolved, then the paper can be published.
IMPORTANT ISSUES
- The introduction contains a very thorough and well-written literature review. Just a few sentences should be added to clarify the novelty of this paper: references [26] and [27] already applied the FRDTM to some fractional Burgers's equations, so what is the difference between their work and the work done here?
- Section 3 must be clarified to explain the method in a correct way. It seems that the authors are confusing the non-fractional RDTM with the fractional FRDTM, sometimes using alpha and sometimes not. Equations (4) and (8) involve no fractional powers of t, but equations (5), (6), (7) involve the fractional α power. The first displayed equation is misleading since the function w is in general not separable in this way. It would be better to say that, motivated by components of the form x1i1 x2i2 ... xnin tj, we write the general solution function w as an infinite linear combination of such components to obtain the second half of equation (4). The results displayed in Table 1 also seem mostly wrong, since there is no mention of α in most of the W functions. In particular, the last three rows are surely wrong by basic calculation. It could also be mentioned that the transform is essentially a generating function.
- The results displayed in Tables in Section 4 should be clarified. It seems that in each case, the exact solution and corresponding error term are only calculated in the non-fractional setting α=1. This should be stated clearly, as it means that the efficiency and accuracy of the fractional method is not actually studied in this paper. For this reason, maybe it would be clearer to split each Table into two parts, or clearly separate the α=1 columns from the fractional columns.
- The title and abstract are misleading as they suggest that the FRDTM is proposed for the first time in this paper, while in fact it already exists in the literature. In the abstract, the first sentence could be changed to "In this study, the fractional reduced differential transform method (FRDTM) for solving partial differential equations (PDEs) is used." and "proposed" should be removed in the third sentence. The title could be changed to e.g. "Applying the fractional reduced differential transform method to different types of nonlinear fractional Burgers' equations".
- In section 2, Ross's proposed criteria for fractional derivatives are mentioned. These are not the only proposals for fractional calculus criteria; a complete account should also mention other proposals such as the following:
Hilfer, R.; Luchko, Y. Desiderata for Fractional Derivatives and Integrals. Mathematics 2019, 7, 149.
Baleanu, D.; Fernandez, A. On Fractional Operators and Their Classifications. Mathematics 2019, 7, 830.
MINOR FIXES
- Abstract. Line 1: change "called as" to "called the". Line 3: changed "on iterative" to "on an iterative". Line 4: change "three 3" to "three". Line 5: change "compared" to "compare".
- Introduction. Line 27: change "Differential" to "The differential". Lines 29 and 31: change "Oturance" to "Oturanc". Line 29: change "called reduced" to "called the reduced".
- Section 2. Equation (1): change z to x. Also consider mentioning the extension of the gamma function to negative arguments. Line 72: change "equation is" to "equations with the Caputo derivative are", and change "derivative at" to "derivatives at". Equation (2): consider using \displaystyle in each of the two cases to make the expressions display more nicely.
- Section 3. Title: remove the citations here. First line: change "definitions and the properties" to "definitions and properties". Line 84: change "then FRDTM" to "then the FRDTM". Line 85: change "gives approximate" to "gives an approximate".
- Section 4. Line 98: change "solution comparing with the exact solution, table 2," to "solution, which is compared with the exact solution in Table 2:" and change "solution of non-fractional" to "solution of the non-fractional". Line 99: change "shows that the" to "shows the". Line 108: change "Eq. 17 subject to 18" to "Eq. (17) subject to (18)". Line 127: change "is depend" to "depends". Line 128: change "that need" to "and therefore needs". Throughout this section: be careful with spaces between numbers; there are many things like "0.8and0.7" which should be "0.8 and 0.7".
- Section 5. Line 140 and 145 and 148: "Example 4" should be something else, as there is no Example 4. Line 141: change "notable small" to "notably small". Line 145: change "seventh term of FRDTM is" to "seven terms of the FRDTM are". Line 148: "(21)-dimensional" should be something else. Line 151: change "with there" to "and the".
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