{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:18:46Z","timestamp":1776842326632,"version":"3.51.2"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2005,5,31]],"date-time":"2005-05-31T00:00:00Z","timestamp":1117497600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2005,7]]},"DOI":"10.1007\/s00211-005-0617-2","type":"journal-article","created":{"date-parts":[[2005,5,31]],"date-time":"2005-05-31T16:01:14Z","timestamp":1117555274000},"page":"143-164","source":"Crossref","is-referenced-by-count":27,"title":["Asymptotically correct error estimation for collocation methods applied to singular boundary value problems"],"prefix":"10.1007","volume":"101","author":[{"given":"Othmar","family":"Koch","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2005,5,31]]},"reference":[{"key":"617_CR1","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1023\/A:1025531130904","volume":"33","author":"Auzinger","year":"2003","unstructured":"Auzinger, W., Kneisl, G., Koch, O., Weinm\u00fcller, E.: A collocation code for boundary value problems in ordinary differential equations. Numer. Algorithms 33, 27\u201339 (2003)","journal-title":"Numer. Algorithms"},{"key":"617_CR2","unstructured":"Auzinger, W., Koch, O., Weinm\u00fcller, E.: Analysis of a new error estimate for collocation methods applied to singular boundary value problems. To appear in SIAM J. Numer. Anal. Also available at http:\/\/www.math.tuwien.ac.at\/~inst115\/preprints.htm\/."},{"key":"617_CR3","unstructured":"Auzinger, W., Koch, O., Weinm\u00fcller, E.: Efficient mesh selection for collocation methods applied to singular BVPs. To appear in J. Comput. Appl. Math. Also available at http:\/\/www.math.tuwien.ac.at\/~inst115\/preprints.htm\/."},{"key":"617_CR4","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1023\/A:1021151821275","volume":"31","author":"Auzinger","year":"2002","unstructured":"Auzinger, W., Koch, O., Weinm\u00fcller, E.: Efficient collocation schemes for singular boundary value problems. Numer. Algorithms 31, 5\u201325 (2002)","journal-title":"Numer. Algorithms"},{"key":"617_CR5","doi-asserted-by":"crossref","first-page":"1067","DOI":"10.1103\/PhysRevB.37.1067","volume":"37","author":"Badralexe","year":"3","unstructured":"Badralexe, E., Freeman, A.J.: Eigenvalue equation for a general periodic potential and its multipole expansion solution. Phys. Rev. B 37(3), 1067\u20131084 (1988)","journal-title":"Phys. Rev. B"},{"key":"617_CR6","doi-asserted-by":"crossref","first-page":"582","DOI":"10.1137\/0710052","volume":"10","author":"Boor","year":"1973","unstructured":"de Boor, C., Swartz, B.: Collocation at Gaussian points. SIAM J. Numer. Anal. 10, 582\u2013606 (1973)","journal-title":"SIAM J. Numer. Anal."},{"key":"617_CR7","doi-asserted-by":"crossref","first-page":"4219","DOI":"10.1103\/PhysRevA.50.4219","volume":"50","author":"Carr","year":"1994","unstructured":"Carr, T.W., Erneux, T.: Understanding the bifurcation to traveling waves in a class-b laser using a degenerate Ginzburg-Landau equation. Phys. Rev. A 50, 4219\u20134227 (1994)","journal-title":"Phys. Rev. A"},{"key":"617_CR8","unstructured":"Coddington, E., Levison, N.: Theory of ordinary differential equations. McGraw-Hill, New York, 1955"},{"key":"617_CR9","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1103\/PhysRevA.42.149","volume":"42","author":"Fernandez","year":"1990","unstructured":"Fernandez, F.M., Ogilvie, J.F.: Approximate solutions to the Thomas-Fermi equation. Phys. Rev. A 42, 149\u2013154 (1990)","journal-title":"Phys. Rev. A"},{"key":"617_CR10","first-page":"803","volume":"36","author":"Gr\u00e4ff","year":"1985","unstructured":"Gr\u00e4ff, M., Scheidl, R., Troger, H., Weinm\u00fcller, E.: An investigation of the complete post-buckling behavior of axisymmetric spherical shells. ZAMP 36, 803\u2013821 (1985)","journal-title":"ZAMP"},{"key":"617_CR11","unstructured":"Hildebrand, F.B.: Introduction to numerical analysis. McGraw-Hill, New York, 2nd edition, 1974"},{"key":"617_CR12","doi-asserted-by":"crossref","first-page":"775","DOI":"10.1137\/0713063","volume":"13","author":"Hoog","year":"1976","unstructured":"de Hoog, F.R., Weiss, R.: Difference methods for boundary value problems with a singularity of the first kind. SIAM J. Numer. Anal. 13, 775\u2013813 (1976)","journal-title":"SIAM J. Numer. Anal."},{"key":"617_CR13","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1137\/0715013","volume":"15","author":"Hoog","year":"1978","unstructured":"de Hoog, F.R., Weiss, R.: Collocation methods for singular boundary value problems. SIAM J. Numer. Anal. 15, 198\u2013217 (1978)","journal-title":"SIAM J. Numer. Anal."},{"key":"617_CR14","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1090\/S0025-5718-1985-0771033-0","volume":"44","author":"Hoog","year":"1985","unstructured":"de Hoog, F.R., Weiss, R.: The application of Runge-Kutta schemes to singular initial value problems. Math. Comp. 44, 93\u2013103 (1985)","journal-title":"Math. Comp."},{"key":"617_CR15","doi-asserted-by":"crossref","first-page":"432","DOI":"10.1007\/BF00916648","volume":"47","author":"Jinqiao","year":"1996","unstructured":"Jinqiao, D., Ly, H.V., Titi, E.S.: The effect of nonlocal interactions on the dynamics of the Ginzburg-Landau equation. Z. Ang. Math. Phys. 47, 432\u2013455 (1996)","journal-title":"Z. Ang. Math. Phys."},{"key":"617_CR16","doi-asserted-by":"crossref","first-page":"674","DOI":"10.1137\/0113045","volume":"13","author":"Keller","year":"1965","unstructured":"Keller, H., Wolfe, A.: On the nonunique equilibrium states and buckling mechanism of spherical shells. J. Soc. Indust. Applied Math. 13, 674\u2013705 (1965)","journal-title":"J. Soc. Indust. Applied Math."},{"key":"617_CR17","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1524\/anly.2001.21.4.373","volume":"21","author":"Koch","year":"2001","unstructured":"Koch, O., Kofler, P., Weinm\u00fcller, E.: Initial value problems for systems of ordinary first and second order differential equations with a singularity of the first kind. Analysis 21, 373\u2013389 (2001)","journal-title":"Analysis"},{"key":"617_CR18","doi-asserted-by":"crossref","first-page":"1784","DOI":"10.1137\/S0036142900368095","volume":"38","author":"Koch","year":"6","unstructured":"Koch, O., Weinm\u00fcller, E.: Iterated Defect Correction for the solution of singular initial value problems. SIAM J. Numer. Anal. 38(6), 1784\u20131799 (2001)","journal-title":"SIAM J. Numer. Anal."},{"key":"617_CR19","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1016\/S0096-3003(02)00919-0","volume":"148","author":"Koch","year":"2","unstructured":"Koch, O., Weinm\u00fcller, E.: Analytical and numerical treatment of a singular initial value problem in avalanche modeling. Appl. Math. Comput. 148(2), 561\u2013570 (2003)","journal-title":"Appl. Math. Comput."},{"key":"617_CR20","doi-asserted-by":"crossref","unstructured":"Stetter, H.J.: Analysis of discretization methods for ordinary differential equations. Springer-Verlag, Berlin-Heidelberg-New York, 1973","DOI":"10.1007\/978-3-642-65471-8"},{"key":"617_CR21","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1007\/BF01432879","volume":"29","author":"Stetter","year":"1978","unstructured":"Stetter, H.J.: The defect correction principle and discretization methods. Numer. Math. 29, 425\u2013443 (1978)","journal-title":"Numer. Math."},{"key":"617_CR22","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1103\/PhysRev.169.413","volume":"169","author":"Tholfsen","year":"1968","unstructured":"Tholfsen, P., Meissner, H.: Cylindrically symmetric solutions of the Ginzburg-Landau equations. Phys. Rev. 169, 413\u2013416 (1968)","journal-title":"Phys. Rev."},{"key":"617_CR23","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1137\/0515023","volume":"15","author":"Weinm\u00fcller","year":"1984","unstructured":"Weinm\u00fcller, E.: On the boundary value problems for systems of ordinary second order differential equations with a singularity of the first kind. SIAM J. Math. Anal. 15, 287\u2013307 (1984)","journal-title":"SIAM J. Math. Anal."},{"key":"617_CR24","doi-asserted-by":"crossref","first-page":"1062","DOI":"10.1137\/0723074","volume":"23","author":"Weinm\u00fcller","year":"1986","unstructured":"Weinm\u00fcller, E.: Collocation for singular boundary value problems of second order. SIAM J. Numer. Anal. 23, 1062\u20131095 (1986)","journal-title":"SIAM J. Numer. Anal."},{"key":"617_CR25","doi-asserted-by":"crossref","first-page":"10976","DOI":"10.1103\/PhysRevB.42.10976","volume":"42","author":"Yeh","year":"17","unstructured":"Chin-Yu Yeh, A.-B. Chen, Nicholson, D.M., Butler, W.H.: Full-potential Korringa-Kohn-Rostoker band theory applied to the Mathieu potential. Phys. Rev. B 42(17), 10976\u201310982 (1990)","journal-title":"Phys. Rev. B"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-005-0617-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-005-0617-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-005-0617-2","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,3]],"date-time":"2023-05-03T09:03:43Z","timestamp":1683104623000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-005-0617-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,5,31]]},"references-count":25,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2005,7]]}},"alternative-id":["617"],"URL":"https:\/\/doi.org\/10.1007\/s00211-005-0617-2","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,5,31]]}}}