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On the Construction of Exact Numerical Schemes for Linear Delay Models

Authors :
Carlos Julio Mayorga
María Ángeles Castro
Antonio Sirvent
Francisco Rodríguez
Source :
Mathematics, Vol 11, Iss 8, p 1836 (2023)
Publication Year :
2023
Publisher :
MDPI AG, 2023.

Abstract

Exact numerical schemes have previously been obtained for some linear retarded delay differential equations and systems. Those schemes were derived from explicit expressions of the exact solutions, and were expressed in the form of perturbed difference systems, involving the values at previous delay intervals. In this work, we propose to directly obtain expressions of the same type for the fundamental solutions of linear delay differential equations, by considering vector equations with vector components corresponding to delay-lagged values at previous intervals. From these expressions for the fundamental solutions, exact numerical schemes for arbitrary initial functions can be proposed, and they may also facilitate obtaining explicit exact solutions. We apply this approach to obtain an exact numerical scheme for the first order linear neutral equation x′(t)−γx′(t−τ)=αx(t)+βx(t−τ), with the general initial condition x(t)=φ(t) for −τ≤t≤0. The resulting expression reduces to those previously published for the corresponding retarded equations when γ=0.

Details

Language :
English
ISSN :
22277390
Volume :
11
Issue :
8
Database :
Directory of Open Access Journals
Journal :
Mathematics
Publication Type :
Academic Journal
Accession number :
edsdoj.b8ea588c2b4047c0838c09fc6060f7ab
Document Type :
article
Full Text :
https://doi.org/10.3390/math11081836