{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T06:39:25Z","timestamp":1778740765702,"version":"3.51.4"},"reference-count":20,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2018,11,13]],"date-time":"2018-11-13T00:00:00Z","timestamp":1542067200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2019,4]]},"DOI":"10.1007\/s10444-018-9643-3","type":"journal-article","created":{"date-parts":[[2018,11,12]],"date-time":"2018-11-12T19:28:25Z","timestamp":1542050905000},"page":"981-1004","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Theoretical and computational analysis of a nonlinear Schr\u00f6dinger problem with moving boundary"],"prefix":"10.1007","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9987-4634","authenticated-orcid":false,"given":"Daniele C. R.","family":"Gomes","sequence":"first","affiliation":[]},{"given":"Mauro A.","family":"Rincon","sequence":"additional","affiliation":[]},{"given":"Maria Darci G. da","family":"Silva","sequence":"additional","affiliation":[]},{"given":"Gladson O.","family":"Antunes","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,11,13]]},"reference":[{"key":"9643_CR1","doi-asserted-by":"publisher","first-page":"226","DOI":"10.1016\/j.apnum.2018.01.007","volume":"127","author":"RMP Almeida","year":"2018","unstructured":"Almeida, R.M.P., Duque, J.C.M., Ferreira, J., Robalo, R.J.: Finite element schemes for a class of nonlocal parabolic systems with moving boundaries. J. Appl. Numer. Math. 127, 226\u2013248 (2018)","journal-title":"J. Appl. Numer. Math."},{"issue":"1","key":"9643_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0022-0396(90)90086-5","volume":"85","author":"P Cannarsa","year":"1990","unstructured":"Cannarsa, P., Da Prato, G., Zolesio, J.P.: The damped wave equation in a moving domain. J. Differ. Equ. 85(1), 1\u201316 (1990)","journal-title":"J. Differ. Equ."},{"key":"9643_CR3","doi-asserted-by":"crossref","DOI":"10.1090\/cln\/010","volume-title":"Semilinear Schr\u00f6dinger Equations, vol. 10","author":"T Cazenave","year":"2003","unstructured":"Cazenave, T.: Semilinear Schr\u00f6dinger Equations, vol. 10. AMS, New York (2003)"},{"key":"9643_CR4","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719208","volume-title":"The Finite Element Method for Elliptic Problems, vol. 40","author":"FG Ciarlet","year":"2002","unstructured":"Ciarlet, F.G.: The Finite Element Method for Elliptic Problems, vol. 40. SIAM, Paris (2002)"},{"issue":"2","key":"9643_CR5","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/S0168-9274(03)00066-7","volume":"47","author":"HR Clark","year":"2003","unstructured":"Clark, H.R., Rincon, M.A., Rodrigues, R.D.: Beam equation with weak-internal damping in domain with moving boundary. J. Appl. Numer. Math. 47(2), 139\u2013157 (2003)","journal-title":"J. Appl. Numer. Math."},{"issue":"3","key":"9643_CR6","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1103\/RevModPhys.71.463","volume":"71","author":"F Dalfovo","year":"1999","unstructured":"Dalfovo, F., Giorgini, S.: Theory of bose-einstein condensation in trapped gases. Rev. Mod. Phys. 71(3), 463\u2013512 (1999)","journal-title":"Rev. Mod. Phys."},{"key":"9643_CR7","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1017\/S0305004100011919","volume":"24","author":"D Hartree","year":"1968","unstructured":"Hartree, D.: The wave mechanics of an atom with a non-coulomb central field. Part i. Theory and methods. Proc. Camb. Philos. Soc. 24, 89\u2013132 (1968)","journal-title":"Proc. Camb. Philos. Soc."},{"issue":"5","key":"9643_CR8","doi-asserted-by":"publisher","first-page":"479","DOI":"10.1016\/j.matpur.2012.01.001","volume":"98","author":"LI Ignat","year":"2012","unstructured":"Ignat, L.I., Zuazua, E.: Convergence rates for dispersive approximation schemes to nonlinear schrodinger equations. Journal de Math\u00e9,matiques Pures et Appliqu\u00e9es 98(5), 479\u2013517 (2012)","journal-title":"Journal de Math\u00e9,matiques Pures et Appliqu\u00e9es"},{"issue":"3","key":"9643_CR9","doi-asserted-by":"publisher","first-page":"777","DOI":"10.1016\/0022-247X(74)90273-X","volume":"46","author":"A Inoue","year":"1974","unstructured":"Inoue, A.: Sur \u25a1 u + u 3 = f $\\square {u} + u^{3} = f$ dans un domaine noncylindrique. J. Math. Anal. Appl. 46(3), 777\u2013819 (1974)","journal-title":"J. Math. Anal. Appl."},{"key":"9643_CR10","doi-asserted-by":"publisher","first-page":"240","DOI":"10.1016\/j.cam.2007.08.006","volume":"220","author":"J Jina","year":"2008","unstructured":"Jina, J., Wu, X.: Analysis of finite element method for one-dimensional time-dependent schrodinger equation on unbounded domain. J. Comput. Appl. Math. 220, 240\u2013256 (2008)","journal-title":"J. Comput. Appl. Math."},{"key":"9643_CR11","doi-asserted-by":"publisher","first-page":"1005","DOI":"10.1103\/PhysRevLett.15.1005","volume":"15","author":"PL Kelley","year":"1965","unstructured":"Kelley, P.L.: Self-focusing of optical beams. Phys. Rev. Lett. 15, 1005\u20131008 (1965)","journal-title":"Phys. Rev. Lett."},{"key":"9643_CR12","volume-title":"Quelques m\u00e9thodes de r\u00e9solutions des probl\u00e9mes aux limites non lin\u00e9aires","author":"JL Lions","year":"1969","unstructured":"Lions, J.L.: Quelques m\u00e9thodes de r\u00e9solutions des probl\u00e9mes aux limites non lin\u00e9aires. Dunod, Paris (1969)"},{"key":"9643_CR13","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1016\/S0168-9274(03)00063-1","volume":"47","author":"IS Liu","year":"2003","unstructured":"Liu, I.S., Rincon, M.A.: Effect of moving boundaries on the vibrating elastic string. Appl. Numer. Math. 47, 159\u2013172 (2003)","journal-title":"Appl. Numer. Math."},{"key":"9643_CR14","first-page":"685","volume":"319","author":"LA Medeiros","year":"1994","unstructured":"Medeiros, L.A., Miranda, M.M.: Contr\u00f4llabilit\u00e9 exacte de l\u2019\u00e9quation de schrodinger dans des domaines non cylindriques. CR Acad. Sci. 319, 685\u2013689 (1994)","journal-title":"CR Acad. Sci."},{"key":"9643_CR15","unstructured":"Medeiros, L.A., Miranda, M.M.: Exact controllability for schr\u00f6dinger equations in non cylindrical domains: 41o semin\u00e1rio brasileiro de an\u00e1lise. Rev Mat Apl (1995)"},{"key":"9643_CR16","doi-asserted-by":"publisher","first-page":"2200","DOI":"10.1016\/j.matcom.2010.04.013","volume":"80","author":"MA Rincon","year":"2010","unstructured":"Rincon, M.A., Lacerda, M.O.: Error analysis of thermal equation with moving ends. J. Math. Comput. Simul. 80, 2200\u20132215 (2010)","journal-title":"J. Math. Comput. Simul."},{"issue":"73","key":"9643_CR17","first-page":"1","volume":"8","author":"KE Strecker","year":"2003","unstructured":"Strecker, K.E., Partridge, G.B., Truscott, A.G., Hulet, R.G.: Bright matter wave solitons in bose-einstein condensates. New J. Phys. 8(73), 1\u201373 (2003)","journal-title":"New J. Phys."},{"key":"9643_CR18","unstructured":"Thom\u00e9e, V: Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational mathematics. 2nd edn., vol. 25. Springer, Rio de Janeiro (2006)"},{"key":"9643_CR19","first-page":"286","volume":"422","author":"T Wang","year":"2015","unstructured":"Wang, T.: Uniform point-wise error estimates of semi-implicit compact finite difference methods for the nonlinear schrodinger equation perturbed by wave operator. J. Comput. Appl. Math. 422, 286\u2013308 (2015)","journal-title":"J. Comput. Appl. Math."},{"issue":"4","key":"9643_CR20","doi-asserted-by":"publisher","first-page":"723","DOI":"10.1137\/0710062","volume":"10","author":"MF Wheeler","year":"1973","unstructured":"Wheeler, M.F.: A priori L2 error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 10(4), 723\u2013759 (1973)","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-018-9643-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10444-018-9643-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-018-9643-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,4]],"date-time":"2026-04-04T01:29:42Z","timestamp":1775266182000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10444-018-9643-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,13]]},"references-count":20,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,4]]}},"alternative-id":["9643"],"URL":"https:\/\/doi.org\/10.1007\/s10444-018-9643-3","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,11,13]]},"assertion":[{"value":"29 September 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 October 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 November 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}