{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T07:38:01Z","timestamp":1787384281515,"version":"build-2736575974"},"reference-count":33,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2001,1]]},"abstract":"<jats:p>In this paper we investigate collocation methods for the computation of periodic solutions of autonomous delay differential equations (DDEs). Periodic solutions are found by solving a periodic two-point boundary value problem, which is an infinite-dimensional problem for DDEs, in contrast to the case of ordinary differential equations. We investigate three collocation methods based on piecewise polynomials. We discuss computational issues and show numerical orders of convergence using an extensive number of tests. We compare our numerical results with known theoretical convergence results for initial value problems for DDEs. In particular, we show how superconvergence at the mesh points can be lost or recovered depending on the DDE model under consideration and on the choice of collocation discretization. We end with a brief discussion of adaptive mesh selection.<\/jats:p>","DOI":"10.1137\/s1064827599363381","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"1593-1609","source":"Crossref","is-referenced-by-count":100,"title":["Collocation Methods for the Computation of Periodic Solutions of Delay Differential Equations"],"prefix":"10.1137","volume":"22","author":[{"given":"K.","family":"Engelborghs","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"T.","family":"Luzyanina","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"K. J. In 't","family":"Hout","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"D.","family":"Roose","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1145\/355945.355950"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971231"},{"key":"R3","unstructured":"N. Azbelev, V. Maksimov, L. Rakhmatullina, Vvedenie v teoriyu funktsionalno\u2010differentsialnykh uravnenii, \u201cNauka\u201d, 1991, 280\u20130, With an English summary92j:34123"},{"key":"R4","unstructured":"G. Bader,\n                      Numerische Behandlung von Randwertproblemen f\u00fcr Funktionaldifferentialgleichungen\n                      , Technical Report 227, Universit\u00e4t Heidelberg, Heidelberg, Germany, 1983."},{"key":"R5","doi-asserted-by":"crossref","unstructured":"G. Bader, Solving boundary value problems for functional\u2010differential equations by collocation, Progr. Sci. Comput., Vol. 5, Birkh\u00e4user Boston, Boston, MA, 1985, 227\u201324387e:65047","DOI":"10.1007\/978-1-4612-5160-6_13"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/0908047"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(84)90039-6"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"A. Bellen, A Runge\u2010Kutta\u2010Nystr\u00f6m method for delay differential equations, Progr. Sci. Comput., Vol. 5, Birkh\u00e4user Boston, Boston, MA, 1985, 271\u201328387i:65113","DOI":"10.1007\/978-1-4612-5160-6_16"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/BF02243775"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/0147039"},{"key":"R11","unstructured":"Eusebius Doedel, Patrick Leung, A numerical technique for bifurcation problems in delay differential equations, Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Man., 1981), Vol. 34, 1982, 225\u201323784f:65065"},{"key":"R12","unstructured":"E. J. Doedel, X. J. Wang, and T. F. Fairgrieve,\n                      AUTO94: Software for continuation and bifurcation problems in ordinary differential equations\n                      , Technical Report CRPC\u201095\u20102, Center for Research on Parallel Computing, California Institute of Technology, Pasadena, CA, 1995."},{"key":"R13","unstructured":"K. Engelborghs, V. Lemaire, J. B\u00e9lair, and D. Roose,\n                      Numerical bifurcation analysis of delay differential equations arising from physiology modeling\n                      , J. Math. Biol., to appear."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127498001595"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1126\/science.267326"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1007\/BF01403681"},{"key":"R17","volume-title":"Solving ordinary differential equations. I","author":"Hairer E.","year":"1993"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"J. K. Hale,\n                      Theory of Functional Differential Equations\n                      , Appl. Math. Sci., Springer\u2010Verlag, New York, 1977.","DOI":"10.1007\/978-1-4612-9892-2"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(88)90164-7"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1007\/BF01994847"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(97)00118-0"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1007\/BF02510918"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(74)90162-0"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-017-1965-0"},{"key":"R25","first-page":"357","volume":"3","author":"Liu Xinzhi","year":"1994","journal-title":"Dynam. Systems Appl."},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827594277673"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127497001709"},{"key":"R28","doi-asserted-by":"crossref","unstructured":"Roger Nussbaum, Periodic solutions of nonlinear autonomous functional differential equations, Lecture Notes in Math., Vol. 730, Springer, Berlin, 1979, 283\u201332581g:34090","DOI":"10.1007\/BFb0064325"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1137\/0140012"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(74)90281-9"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1137\/0715004"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(78)90049-9"},{"key":"R33","unstructured":"A. Zapp,\n                      Verzweigungseigenschaften bei nichtlinearen Differential gleichungen mit einer Zeitverz\u00f6gerung\n                      , Diplomarbeit, University of Cologne, Cologne, Germany, 1997."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1064827599363381","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:51:44Z","timestamp":1787334704000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1064827599363381"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,1]]},"references-count":33,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2001,1]]}},"alternative-id":["10.1137\/S1064827599363381"],"URL":"https:\/\/doi.org\/10.1137\/s1064827599363381","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2001,1]]}}}