Barton, David A. W.; Krauskopf, Bernd; Wilson, R. Eddie Collocation schemes for periodic solutions of neutral delay differential equations. (English) Zbl 1108.65089 J. Difference Equ. Appl. 12, No. 11, 1087-1101 (2006). Summary: We introduce two collocation schemes for the computation of periodic solutions of neutral delay differential equations (NDDEs): one based on a direct discretisation of the underlying NDDE, and one based on a discretisation of a related delay differential difference equation (i.e. a delay differential equation (DDE) coupled with a difference equation). Numerical examples are used to demonstrate these schemes and their respective orders of convergence. Both collocation schemes are implemented in DDE-BIFTOOL, a numerical continuation tool for delay equations. Their use in a continuation setting is shown with one- and two-parameter bifurcation studies of a transmission line model. Cited in 18 Documents MSC: 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 34K28 Numerical approximation of solutions of functional-differential equations (MSC2010) 34K18 Bifurcation theory of functional-differential equations 34K13 Periodic solutions to functional-differential equations Keywords:continuation; bifurcation analysis; collocation; periodic solutions; delay differential difference equation; numerical examples; convergence Software:DDE-BIFTOOL; RADAR5 PDFBibTeX XMLCite \textit{D. A. W. Barton} et al., J. Difference Equ. Appl. 12, No. 11, 1087--1101 (2006; Zbl 1108.65089) Full Text: DOI Link References: [1] DOI: 10.1115/1.1569950 · doi:10.1115/1.1569950 [2] DOI: 10.1007/s002850050133 · Zbl 0908.92026 · doi:10.1007/s002850050133 [3] DOI: 10.1002/0470856211.ch5 · doi:10.1002/0470856211.ch5 [4] Kuang Y., Mathematics in Science and Engineering (1993) [5] Brayton R., Quarterly of Applied Mathematics 24 pp 215– (1966) · Zbl 0143.30701 · doi:10.1090/qam/204800 [6] Brayton R., Quarterly of Applied Mathematics 24 pp 289– (1967) · Zbl 0166.35102 · doi:10.1090/qam/99914 [7] DOI: 10.1063/1.1804092 · doi:10.1063/1.1804092 [8] DOI: 10.1147/rd.51.0002 · Zbl 0148.08405 · doi:10.1147/rd.51.0002 [9] DOI: 10.1006/jdeq.1996.0009 · Zbl 0840.34080 · doi:10.1006/jdeq.1996.0009 [10] Hale, J. and Verduyn Lunel, S. 1993. ”Introduction to functional differential equations. Applied Mathematical Sciences,”. New York: Springer. 99 · Zbl 0787.34002 [11] Diekmann O., Applied Mathematical Sciences 110 (1995) [12] Stépán G., Retarded Dynamical Systems (1989) · Zbl 0686.34044 [13] Thompson, S. and Shampine, L., 2004, A friendly Fortran DDE solver. In Proceedings of the 3rd international conference on the numerical solution of Volterra and delay equations. · Zbl 1089.65062 [14] DOI: 10.1093/imamci/19.1_and_2.5 · Zbl 1005.93026 · doi:10.1093/imamci/19.1_and_2.5 [15] Ascher U., Proceedings of A Working Conference on Codes for Boundary-value Problems in Ordinary Differential Equations pp 164– (1979) · doi:10.1007/3-540-09554-3_12 [16] DOI: 10.1137/S1064827599363381 · Zbl 0981.65082 · doi:10.1137/S1064827599363381 [17] DOI: 10.1007/s002110100313 · Zbl 1002.65089 · doi:10.1007/s002110100313 [18] DOI: 10.1080/10236190108808312 · Zbl 0998.65074 · doi:10.1080/10236190108808312 [19] DOI: 10.1142/S0218127498001595 · Zbl 0941.34070 · doi:10.1142/S0218127498001595 [20] DOI: 10.1142/S0218127491000555 · Zbl 0876.65060 · doi:10.1142/S0218127491000555 [21] Engelborghs, K., 2000, Numerical bifurcation analysis of delay differential equations, PhD. thesis, Department of Computer Science, Katholieke Universiteit Leuven, Leuven, Belgium. · Zbl 0970.65090 [22] DOI: 10.1016/0377-0427(84)90039-6 · Zbl 0538.65047 · doi:10.1016/0377-0427(84)90039-6 [23] DOI: 10.1137/0710052 · Zbl 0232.65065 · doi:10.1137/0710052 [24] Hairer E., Springer Series In Computational Mathematics, 2. ed. (1993) [25] DOI: 10.1007/BF01994847 · Zbl 0765.65069 · doi:10.1007/BF01994847 [26] DOI: 10.1023/A:1021994523890 · Zbl 0988.65061 · doi:10.1023/A:1021994523890 [27] DOI: 10.1137/0718072 · Zbl 0511.65053 · doi:10.1137/0718072 [28] Engelborghs K., Dynamics and Stability of Systems 14 pp 255– (1999) · Zbl 0936.34057 · doi:10.1080/026811199281994 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.