Comparison of Symbolic Computations for Solving Linear Delay Differential Equations Using the Laplace Transform Method

Sherman, Michelle; Kerr, Gilbert; González-Parra, Gilberto

doi:10.3390/mca27050081

Cited by 5 publications

(5 citation statements)

References 56 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the other hand, on the right-hand side of Figure 9, it can be seen that the error of the numerical solution produced by the dde23 built-in Matlab function increases and that it is much larger than the one generated by the analytical LT solution. Similar results have been found for other types of linear DDEs [32,59]. It is important to remark that the truncated LT solution was produced with only 15 terms, and can be evaluated at any time t. Example 5.…”

Section: Linear Rddessupporting

confidence: 72%

Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit Cycles

Kerr,

Lopez,

González-Parra

2024

MCA

Self Cite

View full text Add to dashboard Cite

In this paper we develop an approach for obtaining the solutions to systems of linear retarded and neutral delay differential equations. Our analytical approach is based on the Laplace transform, inverse Laplace transform and the Cauchy residue theorem. The obtained solutions have the form of infinite non-harmonic Fourier series. The main advantage of the proposed approach is the closed-form of the solutions, which are capable of accurately evaluating the solution at any time. Moreover, it allows one to study the asymptotic behavior of the solutions. A remarkable discovery, which to the best of our knowledge has never been presented in the literature, is that there are some particular linear systems of both retarded and neutral delay differential equations for which the solution asymptotically approaches a limit cycle. The well-known method of steps in many cases is unable to obtain the asymptotic behavior of the solution and would most likely fail to detect such cycles. Examples illustrating the Laplace transform method for linear systems of DDEs are presented and discussed. These examples are designed to facilitate a discussion on how the spectral properties of the matrices determine the manner in which one proceeds and how they impact the behavior of the solution. Comparisons with the exact solution provided by the method of steps are presented. Finally, we should mention that the solutions generated by the Laplace transform are, in most instances, extremely accurate even when the truncated series is limited to only a handful of terms and in many cases become more accurate as the independent variable increases.

show abstract

Section: Linear Rddessupporting

confidence: 72%

Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit Cycles

Kerr,

Lopez,

González-Parra

2024

MCA

Self Cite

View full text Add to dashboard Cite

show abstract

“…Furthermore, throughout this paper, we will assume that w = 0 in order to deal with the discontinuity on the input function. The case where w = 0 has already been studied in , , Sherman et al (2022).…”

Section: Framework For Linear Ddes and Laplace Transformmentioning

confidence: 99%

“…The linear NDDE (1) can be solved using the Laplace transform method, which is well suited for handling discontinuous functions (Bellman and Roth 1984;Debnath 2016;Spiegel 1965). Determining the solution requires one to find the infinite sequence of poles (Jamilla et al 2020a, b;Sherman et al 2022), then use Cauchy's residue theorem to obtain the solution in the original t-space (Conway 2012). For DDEs, the form of solution is a non-harmonic Fourier series Sedletskii 2000;Young 2001).…”

Section: Framework For Linear Ddes and Laplace Transformmentioning

confidence: 99%

Analytical solutions of linear delay-differential equations with Dirac delta function inputs using the Laplace transform

2023

View full text Add to dashboard Cite

In this paper, we propose a methodology for computing the analytic solutions of linear retarded delay-differential equations and neutral delay-differential equations that include Dirac delta function inputs. In numerous applications, the delta function serves as a convenient and effective surrogate for modeling high voltages, sudden shocks, large forces, impulse vaccinations, etc., applied over a short period of time. The solutions are obtained using the Laplace transform method, in conjunction with the Cauchy residue theorem. The accuracy of these solutions are assessed by comparing them with the ones provided by the method of steps. Numerical examples illustrating the methodology are presented and discussed. These examples show that the Laplace transform solution is very reliable for linear retarded delay-differential equations, because the analytic solution, for a single delta function input, is continuous. However, for linear neutral delay-differential equations with a delta function input the analytic solution is discontinuous. Consequently, the well-known Gibbs phenomenon is observed in the vicinity of the discontinuities. However, for neutral delay differential equations, we show that in some cases, the magnitude of the jumps at the discontinuities decrease, as time increases. Therefore, the Gibbs phenomenon of the Laplace solution dissipates.

show abstract

“…The authors of [18], developed a novel method coupling the Laplace transform (LT) with the Fourier series for the solution DDEs. Sherman et al [30] compared the performance of MAPLE and MATLAB for computing the method of steps and LT solutions for neutral and retarded linear DDEs. Akhmet et al [2] considered a general linear impulsive system of DEs with distributed delay.…”

Section: Introductionmentioning

confidence: 99%

On the numerical solution of second order delay differential equations via a novel approach

Aljawi,

Kamran,

Hussain

et al. 2024

J. Math. Computer Sci.

View full text Add to dashboard Cite

Comparison of Symbolic Computations for Solving Linear Delay Differential Equations Using the Laplace Transform Method

Cited by 5 publications

References 56 publications

Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit Cycles

Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit Cycles

Analytical solutions of linear delay-differential equations with Dirac delta function inputs using the Laplace transform

On the numerical solution of second order delay differential equations via a novel approach

Contact Info

Product

Resources

About