A New Higher-Order Convergence Laplace–Fourier Method for Linear Neutral Delay Differential Equations

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The ideas explored in the paper are interesting in the field of delay  differential systems. However, it seems to us that the suggested approximations on the key differential equation have to be justified with more details and rigor.

Eqn. 2: Which is its analytical relation  of the history function with the initial  conditions of (1); and can it be displayed explicitly?.

Is it possible that zero-pole cancellations take place in (2) for potential relations between the initial conditions of (1), which would depend on the interval [-tau, 0],  and it parameters a, b, c?. Furthermore , the admissible class of functions of initial condition can include many, since it could be,  for instance, that of  bounded piecewise-continuous ones on [-tau, 0].

It is not given a reason  for the claim that the maximum number of real poles is three.

It is not given a clear explanation of the reason to conclude that the non-real poles of large amplitude depend, under some reasonable hypotheses,  only on “b” and the delay “tau” but not on the parameters “a” and “c”. Looking at the differential equation (1), this claim intuitively means that the complex poles of large amplitude of (1) are close to those complex of large amplitude of the differential equation dy/dt=b*dy(t-tau)/dt, but , why?. Furthermore, the whole solution of (1) can also depend “seriously” on the values of “a” and “c”  for certain initial conditions.

Eqn. (6): Which is its reason under the assumption of a real pole if it was said  previously that the maximum number of real poles is three but not how many of them really exist and under which conditions. Or,  under which parametrical conditions, there is a real pole?.

Author Response

Reviewer 1

 

We thank the reviewers for their helpful comments, suggestions, and for the time they spent analyzing this manuscript. We added some few phrases to address the questions raised by the reviewers. Main or relevant changes in the text are in blue, so the reviewers can see them easily. Here we respond in bold to all comments point by point.

 

Comments and Suggestions for Authors

The ideas explored in the paper are interesting in the field of delay  differential systems. However, it seems to us that the suggested approximations on the key differential equation have to be justified with more details and rigor.

Authors: Thank you for reading and analyzing this manuscript. We appreciate your nice comments about our paper and research.

Eqn. 2: Which is its analytical relation of the history function with the initial conditions of (1); and can it be displayed explicitly?.

Authors: In DDEs the history function in fact is the initial condition. Some researchers that use a constant history function just write them as initial conditions instead of giving the history. However, when the history function is not constant it is imperative to include (or define) a history function, and it plays crucial a role in the solution. The history appears in the residues which are part of the solution (see Eq. (5)). The history functions in the paper are taken as polynomials and trigonometric functions which are easy to deduce the plot.

Is it possible that zero-pole cancellations take place in (2) for potential relations between the initial conditions of (1), which would depend on the interval [-tau, 0],  and it parameters a, b, c?. Furthermore , the admissible class of functions of initial condition can include many, since it could be,  for instance, that of  bounded piecewise-continuous ones on [-tau, 0].

Authors: Yes, cancellations can occur but they are very unlikely due to the form of the denominator D(s) since it depends on many parameters (b,c,tau). Regardless of the cancellations the Laplace-Fourier method can find the solution by using Cauchy Residue theorem [1,2]. If there is a cancellation that means that the singularity is removable [1,3,4]. We added a brief comment to explain this aspect. Indeed, there are many admissible class of history functions and they play a role in the computation of the residues and the solution. We have added a comment to clarify this aspect. Interestingly we encountered this situation in our limit cycles paper, where it was shown that the relevant singularities were removable [4].

 

It is not given a reason for the claim that the maximum number of real poles is three.

Authors:  Thank you. We added a brief explanation in the revised paper about why there are at most 3 real poles. It can be shown that the real roots of the denominator D(s) are given by the intersection of the LambertW function and a line. Thus, we can have at most three real roots. In the case of no real poles the Laplace-Fourier solution would be given only in terms of the complex poles. As a particular case we have that when b=0 we can obtain 2 roots by using the Lambert function.  If we consider the case when τ=1, a=2 and c=-2, then we are guaranteed a real root at s=0. We obtain 2 additional real roots when b∈(-1,0), 1 additional real root when b∈[0,∞) and the single real root s=0 for b∈(-∞,-1].  For τ=1, a=-2 and c=-2, there are no real roots for b∈[0,10].

 

It is not given a clear explanation of the reason to conclude that the non-real poles of large amplitude depend, under some reasonable hypotheses,  only on “b” and the delay “tau” but not on the parameters “a” and “c”. Looking at the differential equation (1), this claim intuitively means that the complex poles of large amplitude of (1) are close to those complex of large amplitude of the differential equation dy/dt=b*dy(t-tau)/dt, but, why?. Furthermore, the whole solution of (1) can also depend “seriously” on the values of “a” and “c” for certain initial conditions.

Authors: Thank you for your interesting insight.  Eq. (4) is used as an approximation for the complex poles when |s| is large and as the reviewer mentioned does not depend on a and c.  However, as the reviewer mentions, the poles do also depend on a and c. Indeed, that is what we have in Eq. (31), which is a more accurate formula for the location of the poles.  In the Laplace-Fourier method, however, it is imperative that one uses the approximation from Eq. (4). We have shown that, in most cases, when using Eq. (4) the convergence is very fast. In the worst scenario when b is very small in comparison with a and c, we would need to use more terms in the Laplace-Fourier solution [1,5]. We added a brief explanation in the revised version related to this aspect.

Eqn. (6): Which is its reason under the assumption of a real pole if it was said previously that the maximum number of real poles is three but not how many of them really exist and under which conditions. Or, under which parametrical conditions, there is a real pole?

Authors: Thank you. We added a brief explanation in the revised paper about why there are at most 3 real poles. It can be shown that the real roots of the denominator D(s) are given by the intersection of the LambertW function and a line. Thus, we can have at most three intersections. Since we have 4 parameters (a,b,c,tau) a variety of situations can happen regarding the roots of D and finding those specific conditions in a four dimensional space can be challenging. However, from a practical viewpoint regardless of the values of the parameters we always can find the real roots and therefore the solution.

 

Authors: Thank you for reading and analyzing this manuscript.

 [1] Kerr, G., & González-Parra, G. (2022). Accuracy of the Laplace transform method for linear neutral delay differential equations. Mathematics and Computers in Simulation, 197, 308-326.

[2] Sherman, M., Kerr, G., & González-Parra, G. (2023). Analytic solutions of linear neutral and non-neutral delay differential equations using the Laplace transform method: featuring higher order poles and resonance. Journal of Engineering Mathematics, 140(1), 12.

[3] Sherman, M., Kerr, G., & González-Parra, G. (2022). Comparison of symbolic computations for solving linear delay differential equations using the Laplace transform method. Mathematical and Computational Applications, 27(5), 81.

[4] Kerr, G., Lopez, N., & González-Parra, G. (2024). Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit Cycles. Mathematical and Computational Applications, 29(1), 11.

[5] Kerr, G., González-Parra, G., & Sherman, M. (2022). A new method based on the Laplace transform and Fourier series for solving linear neutral delay differential equations. Applied Mathematics and Computation, 420, 126914.

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Authors, The work is good. It can be extended to second order also in future.

Comments for author File: Comments.pdf

Author Response

Reviewer 2

We thank the reviewers for their helpful comments, suggestions, and for the time they spent analyzing this manuscript. We added a few phrases to address the questions raised by the reviewers. Main or relevant changes in the text are in blue, so the reviewers can see them easily. Here we respond in bold to all comments point by point.

 

Paper Tittle: A new higher order convergence Laplace-Fourier method for linear neutral delay differential equations

Manuscript ID: mca-3510345

 

The manuscript presents a well-structured and thoroughly researched study on “A new higher order convergence Laplace-Fourier method for linear neutral delay differential equations”.

The authors have effectively addressed the problem, provided a clear theoretical framework, and employed appropriate mathematical techniques to derive meaningful results. The logical flow of the arguments is sound, and the use of examples and proofs to support the main claims is commendable. The results are both significant and relevant to the field, and the conclusions drawn are well-supported by the findings.

The overall quality of the writing is high, and the authors have done an excellent job in explaining complex concepts in a clear and accessible manner, making the paper suitable for a broad audience within the mathematical community.

After reviewing the manuscript thoroughly, I believe the work is suitable for publication and recommend its acceptance without the need for further revisions.

Authors: Thank you for reading and analyzing this manuscript. We appreciate your nice comments about our paper and research.

 

Reviewer 3 Report

Comments and Suggestions for Authors For the differential equations with constant delays no interest left, in fact such equations have been studied extensively qualitatively as well as quantitatively. This includes applications of Laplace transforms. Starting from 1972 various numerical schemes have been proposed and successfully applied to more completed equations (nonlinear, delay, advance, neutral, and the list continues). Thus, the present study is rather weak and does not add anything essentially new in the exiting literature. In conclusion my recommendation is against the publication of this paper.

 

Author Response

Reviewer 3

We thank the reviewers for their helpful comments, suggestions, and for the time they spent analyzing this manuscript. We added a few phrases to address the questions raised by the reviewers. Main or relevant changes in the text are in blue, so the reviewers can see them easily. Here we respond in bold to all comments point by point.

 

Comments and Suggestions for Authors

 

For the differential equations with constant delays no interest left, in fact such equations have been studied extensively qualitatively as well as quantitatively. This includes applications of Laplace transforms. Starting from 1972 various numerical schemes have been proposed and successfully applied to more completed equations (nonlinear, delay, advance, neutral, and the list continues). Thus, the present study is rather weak and does not add anything essentially new in the exiting literature. In conclusion my recommendation is against the publication of this paper.

 

Authors: Thank you for reading and analyzing this manuscript.

The higher order convergence Laplace-Fourier method proposed in this paper is novel. There is no similar method close to the one developed in this paper (besides the regular Laplace-Fourier method developed by some of the authors of this work). The higher order convergence Laplace-Fourier method produces very accurate analytical solutions that have several advantages. One main advantage is that the analytical solutions are very accurate and their accuracy can be further improved by adding more terms and by using higher degree polynomials. In [1] we have shown that even the regular Laplace-Fourier method produces more accurate solutions than the ones generated by the classic dde23 solver of Matlab which is a numerical solution that many researchers use.  Since the higher order convergence Laplace-Fourier method produces even more accurate solutions we have obtained a large improvement regarding accuracy, which is the main contribution of this research. Moreover, the Laplace-Fourier solution can be evaluated at any time, even for very large values of the variable t. On the other hand, the numerical solutions would require a long computation time for very large values of t.

Another main advantage of the higher order convergence Laplace-Fourier solution is that sensitivity analysis can be performed for different times.

Finally, we would like to point out that previous analytical solutions of linear DDEs examples have been used by numerous authors for testing, verifying and validating their numerical methods. Thus, the examples that we designed in this work can be used by other authors to test their numerical methods. Moreover, our work is not about existence or uniqueness of the solutions of DDE with constant delay.

We have added several sentences in the revised manuscript to emphasize these advantages of the higher order convergence Laplace-Fourier method.

 

[1] Kerr, G., González-Parra, G., & Sherman, M. (2022). A new method based on the Laplace transform and Fourier series for solving linear neutral delay differential equations. Applied Mathematics and Computation, 420, 126914.

 

Reviewer 4 Report

Comments and Suggestions for Authors

The review report is attached for authors to make necessary corrections in the manuscript.

Comments for author File: Comments.pdf

Author Response

Reviewer 4

We thank the reviewers for their helpful comments, suggestions, and for the time they spent analyzing this manuscript. We added a few phrases to address the questions raised by the reviewers. Main or relevant changes in the text are in blue, so the reviewers can see them easily. Here we respond in bold to all comments point by point.

Comments and Suggestions for Authors

 

 Suggestions for Authors

This paper proposes a new higher order convergence Laplace-Fourier method to obtain the solutions of linear neutral delay differential equations. The proposed method provides more accurate solutions than the ones provided by the pure Laplace method and the original Laplace-Fourier method. The validity of the new technique is corroborated by means of illustrative examples. Comparisons of the solutions of the new method with those generated by the pure Laplace method and the unmodified Laplace-Fourier approach are presented. The paper is nicely written with subtle analysis.

 

Authors: Thank you for reading and analyzing this manuscript. We appreciate your nice comments about our paper and research.

Please find my suggestions to improve the paper

(1) Discuss some practical applications of NDDEs for benefit of readers and solve one application-based NDDEs using the presented method in the paper.

 

Authors: Thank you.   We added a few sentences in the revised manuscript related to the applications of NDDEs to real world problems so the readers can see some of these applications. In [1,2] we used the pure Laplace method (simpler but less accurate than the Laplace-Fourier method) to solve DDEs including systems of DDEs.  Some examples included physically relevant linear system of NDDEs that are related to the current in a PEEC circuit. We have added a few more real world applications in the new version of the manuscript.

 

(2) The details can be discussed about the computational efficiency (e.g. runtime comparisons with other methods) and difficulty faced while solving these problems.

 

Authors: Thank you. We added a few sentences in the new version of the manuscript related to computational efficiency of the higher order convergence Laplace-Fourier method.  In [2] we have shown that even the regular Laplace-Fourier method produces more accurate solutions than the ones generated by the classic dde23 solver of Matlab which is a numerical solution. Moreover, the Laplace-Fourier solution can be evaluated very fast at any time and even for very large values of t. On the other hand, the numerical solutions would require a long computation time for very large values of t.  The tradeoff in the Laplace-Fourier method is that some theoretical/computational work needs to be done to obtain the analytical solution. In particular, the poles and residues must be found. We added a sentence in the revised manuscript related to the difficulty of solving the NDDEs.

 

(3) Authors can include comparisons with other available methods to show effectiveness and robustness of the proposed method.

 

Authors: Thank you.   In [2] we have shown that even the regular Laplace-Fourier method produces more accurate solutions than the ones generated by the classic dde23 solver of Matlab. We have added a paragraph in the revised paper, explaining that the higher order convergence Laplace-Fourier method solutions are more accurate than the simple Laplace-Fourier ones. Thus, the higher order convergence Laplace-Fourier method solutions are even more accurate than the numerical ones generated by the classic dde23 solver of Matlab.

 

(4) The conclusion could be expanded to suggest possible avenues for future research. For example, could this method be extended to higher-dimensional problems or other types of delay differential equations?

 

Authors: Thank you, that is an excellent point. Indeed, we have thought about future avenues of research related to the higher order convergence Laplace-Fourier method. In [1] we used the pure Laplace method (simpler but less accurate than Laplace-Fourier) to solve systems of DDEs.  Thus, one future project will involve solving DDE systems by the higher order convergence Laplace-Fourier method. We added a few sentences related to this important point. We can also use the new Laplace-Fourier method to solve NDDEs with delta function inputs, to generate analytic solutions which are indeed piecewise continuous. In our previous (2023) paper where we used the pure Laplace method, the solutions exhibited the Gibb’s phenomenon at the discontinuities.      

 

After these minor corrections paper can be accepted.

Authors: Thank you for reading and analyzing this manuscript. We appreciate your nice comments about our paper and research. The reviewer grasped the main ideas of the Laplace-Fourier method.

 

[1] Kerr, G., & González-Parra, G. (2022). Accuracy of the Laplace transform method for linear neutral delay differential equations. Mathematics and Computers in Simulation, 197, 308-326.

 

[2] Kerr, G., Lopez, N., & González-Parra, G. (2024). Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit Cycles. Mathematical and Computational Applications, 29(1), 11.

[3] Kerr, G., González-Parra, G., & Sherman, M. (2022). A new method based on the Laplace transform and Fourier series for solving linear neutral delay differential equations. Applied Mathematics and Computation, 420, 126914.

 

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The paper has been improved according to the suggestions. We have no further comments.

Reviewer 3 Report

Comments and Suggestions for Authors

To some extend I agree the method developed in the paper is new, but the big question is `whether the equation studied itself is interesting and has any future'.  Such equations have been extensively studied from a very long time, for example,  Bellman and Cooke's book Differential-Difference Equations, written in 1963. I have not seen any new numerical work on such equations from the last 20 years.