A new method based on the Laplace transform and Fourier series for solving linear neutral delay differential equations

Gilbert Kerr, Gilberto González-Parra and Michele Sherman

Applied Mathematics and Computation, 2022, vol. 420, issue C

Abstract: In this paper, we present a new method for solving linear neutral delay differential equations. We derive and illustrate the main features of this novel approach, that combines the Laplace transform method with (harmonic) Fourier series theory. Linear neutral delay differential equations are generally more difficult to solve because the time delay appears in the derivative of the state variable. We rely on computer algebra and numerical methods to implement the method. In addition, we derive an approximate formula for the location of the complex poles, which are required for computing the inverse Laplace transform. The form of the resulting solution, when only the Laplace method is used is a non-harmonic Fourier series. The accuracy of this solution can be improved by including more terms in the associated truncated series, but the convergence to the correct solution is slow. The main goal of this paper is to present a modified method which enables us to account for the terms which are excluded from these truncated Laplace series. That is, the terms in the tail of the infinite series. We include several examples where we compare the solutions generated by the standard Laplace method and the proposed Laplace-Fourier approach. Both solutions require using Cauchy’s residue theorem and finding the real and complex poles. It is shown that the Laplace-Fourier solution provides more accurate solutions than the conventional Laplace transform solution. Finally, since the Laplace-Fourier method generates a solution which is valid for all times, it allows us to accurately approximate the solution at any point with a single calculation.

Keywords: Neutral delay differential equations; Laplace transform; Fourier series; Cauchy’s residue theorem (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0096300321009978
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:420:y:2022:i:c:s0096300321009978

DOI: 10.1016/j.amc.2021.126914

Access Statistics for this article

Applied Mathematics and Computation is currently edited by Theodore Simos

More articles in Applied Mathematics and Computation from Elsevier
Bibliographic data for series maintained by Catherine Liu ().