Solving a System of One-Dimensional Hyperbolic Delay Differential Equations Using the Method of Lines and Runge-Kutta Methods

Round 1

Reviewer 1 Report

The manuscript presents interesting results regarding the study of a numerical scheme aimed at the solution of differential equations containing some delay in the independent variable.

After addressing the following issues, the manuscript deserves to be published:

1) The introduction needs to be carefully reexamined in order to reference actual real-world applications of the mathematical formalism considered in the mansucript. An important example, would be the dynamic behavior of heat exchangers, widely used in the chemical industry. The reference MORBIDELLI, M.; VARMA, A. Mathematical Methods in Chemical Engineering, 1st. Ed. Oxford University Press: Oxford, 1997, presents interesting examples.

2) The authors should adequately present the novelty aspects and contributions of the manuscript.

3) The authors should present the description of the computer used (type of processor and RAM memory) as well as the computing time.

4) Is it possible for the authors to compare the results of the simulation of the example (Table 1) using the proposed methodology to the solution of the same example obtained by another previously reported method in order to show the advantages of the studied numerical scheme?

5) Is it possible for the authors to considered a simpler example with an analytical solution in order to be able to be compare to the numerical solution? A simpler example, that could be solved, for example, using 
Laplace Transform.

Author Response

Review comments attached 

Author Response File: Author Response.pdf

Reviewer 2 Report

This paper considers a system of one-dimensional hyperbolic/transport delay differential equations (HDDEs) subject to appropriate initial and boundary conditions. However, the problem considered in the article does not belong to classical delay equations because there is no delay in time variable. The problem under consideration has only one step away in spatial variable, so its title is misleading.

It is almost impossible to read the article because authors use undefined notations for domain "D" and delta with plus and minus.


Remarks: Domain "D" was not explicitly defined. When Theorems 1 and 2 are formulated, it is not clear in what domain they hold. It is hard to believe that Theorem 3 is valid because there is no proof for it and as it is formulated it is not clear in what domain it is valid. Also, for second derivatives to be continuous and bounded, there should be more corner conditions to be imposed.

Figures 6 and 12 are almost invisible.

Examples 1 and 2 do not demonstrate the numerical method in use: there is no explicit algorithm for how solutions are approximated for negative x; for positive x, there is RK4. Moreover, these examples are almost identical and do not provide new information, so one of them can be dropped.

Also initial functions are taken from the Schwartz space and do not reflect possible discontinuities discussed in the article.

Suggestions: remove discontinuous part of the article and concentrate on material supported by existing examples. Include description of developed algorithm (references 25, 26 are not sufficient). It is strongly recommended to include the standard wave equation and demonstrate how the "delay" problem works in this case. 

Author Response

Revision comments are attached 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The modified version can be accepted.