Analytic solutions of linear neutral and non-neutral delay differential equations using the Laplace transform method: featuring higher order poles and resonance

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

doi:10.1007/s10665-023-10276-5

Cited by 4 publications

(8 citation statements)

References 34 publications

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“…For additional details, we refer the interested reader to previous works [36,54]. Observe that the second equation (above) is essentially an exact match with the eigenvalue equation for C. Assuming that the matrix C is non-singular, with distinct eigenvalues one gets n (infinite) sequences of poles, with approximate locations given by…”

Section: Computation Of the Complex Polesmentioning

confidence: 99%

“…And there are four real poles, at r = 1/20, r = ln(29/30) and r = 0 (with multiplicity 2). Dealing with an infinite sequence of complex poles of order 2 typically requires one to incorporate the appropriate modifications, when applying the Cauchy residue theorem to compute the ILT [36]. Fortunately, it is not necessary to do so for this particular system.…”

Section: Linear Ndde Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Recently, a variety of linear DDEs have been solved using the Laplace transform (LT) [16,32,35,36]. For instance, in [36], the authors computed solutions that featured resonance for linear DDEs. Furthermore, in [29], the Laplace transform method (LTM) was employed by the authors to look at the stability of linear DDEs.…”

Section: Introductionmentioning

confidence: 99%

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

Kerr,

Lopez,

González-Parra

2024

MCA

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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.

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Section: Computation Of the Complex Polesmentioning

confidence: 99%

Section: Linear Ndde Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Kerr,

Lopez,

González-Parra

2024

MCA

Self Cite

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show abstract

“…Recently, a variety of linear DDEs have been solved using the Laplace transform (LT) [13,25,28,29]. For instance, in [29] the authors computed solutions that featured resonance for linear DDEs.…”

Section: Introductionmentioning

confidence: 99%

Analytical solutions of systems of linear retarded and neutral delay differential equations by the Laplace transform: featuring limit cycles

Kerr,

Lopez,

Gonzalez-Parra

2023

Preprint

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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 approaches asymptotically to 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 becomes more accurate as the independent variable increases.

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

2023

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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.

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Analytic solutions of linear neutral and non-neutral delay differential equations using the Laplace transform method: featuring higher order poles and resonance

Cited by 4 publications

References 34 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 systems of linear retarded and neutral 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

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