Numerical Simulation Based on Interpolation Technique for Multi-Term Time-Fractional Convection–Diffusion Equations
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe study needs further major modifications, first of all the introductory information is too long and need to short cut,
second the presentation needs to improve
third the theorem 1 and theorem 2 are considered main contribution of the present study but they are straightforward of the existing literature, see [32], [33], [34], and [35].
Fourth, the numerical examples are simple and related to each other. So, need to provide more realistic numerical examples.
Author Response
Please see attachment.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsThe paper deals with the numerical solution of multi-term time-fractional convection-diffusion equations.
The proposed numerical method is described in detail, using a barycentric rational interpolation collocation method.
I have basically two issues with the paper.
First the method is claimed in the introduction to have a "good robustness to irregular data".
- The test cases only deal with very regular data, and more precisely with simple very regular solutions to which the forcing term has been adapted. This allows of course to match the regularity hypotheses of the theorems, but not real life applications.
- Moreover the test cases are described (parameter choice...) but not really commented and the reader is supposed to draw his own conclusions from the various tables. Therefore more precise comments and a test case in a less regular case would be welcome.
- In the theorems the needed regularity is given in a non conventional way, with only the regularity of the higher derivatives. It of course yields the regularity for the function $u$, but it is more usual to give the functional space in which $u$ is supposed to be. In the same vein, in Theorem 1 for example, you should first say that $u$ is the solution to (1) and then that its time and space derivatives are supposed to have a given regularity.
- $u$ and $R_{t,m} u$ are equal at $t=0$, but is it obvious that it also holds for $u_t$ and that there is no second contribution to the estimate in (22). And it seems that the estimate (23) needs one more degree of regularity as it is supposed to have.
Second, notations are a bit difficult to follow and the description of the method is sometimes lengthy.
- The maximum values for $i$, $j$, $k$ and $r$ could be e.g. simply $I$, $J$, $K$ and $R$ instead of $n$, $m$, $s$,and $p$.
This would be more simple and would free a few letters to serve as integers (such as $n$, $m$ and $p$) when needed, instead of $a$, $b$, $g$, $e$\dots which are not usual notations for integers.
- Section 2.1 is lengthy since the two equations on $r(x)$ including (4) and (5) are essentially the same, and (7) is only the transcription in the time domain.
- Naming $\beta$ as $\beta_0$ would allow not to duplicate certain notations and make shorter some equations, e.g. (15), (16)...
- In the proof of Theorem 2, I would not name the time derivatives with $\zeta$ but with $t$. It means derivative with respect to the second variable (which as been first named as $t$) and it is computed at $(x,\zeta)$.
- The signification of the $\doteq$ (dotted equal) symbol is unclear (I even asked a few colleagues with various mathematical backgrounds). It should be explained when first used.
- Also equation (16) is redundant with the previous equation (15) for all $x$ and $t$ and (17).
A few small last remarks,
- equation (6) is not really an equality but a continuous limit as $x \to x_a$.
- The is a confusion between $B_2$ and $B_3$ on top of page 9
Apart from these remarks, the paper is clearly in the scope of the Fractal and Franctional journal, and with small improvements as noted above, it could be published in this journal.
Author Response
Please see attachment.
Author Response File: Author Response.pdf
