{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T11:44:20Z","timestamp":1759837460687,"version":"3.37.3"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2022,5,21]],"date-time":"2022-05-21T00:00:00Z","timestamp":1653091200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,5,21]],"date-time":"2022-05-21T00:00:00Z","timestamp":1653091200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100004281","name":"narodowe centrum nauki","doi-asserted-by":"publisher","award":["2017\/25\/B\/ST1\/00945"],"award-info":[{"award-number":["2017\/25\/B\/ST1\/00945"]}],"id":[{"id":"10.13039\/501100004281","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2022,12]]},"DOI":"10.1007\/s11075-022-01324-9","type":"journal-article","created":{"date-parts":[[2022,5,21]],"date-time":"2022-05-21T09:02:33Z","timestamp":1653123753000},"page":"1829-1854","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Approximation of solutions of DDEs under nonstandard assumptions via Euler scheme"],"prefix":"10.1007","volume":"91","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7445-5305","authenticated-orcid":false,"given":"Natalia","family":"Czy\u017cewska","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pawe\u0142 M.","family":"Morkisz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pawe\u0142","family":"Przyby\u0142owicz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,5,21]]},"reference":[{"key":"1324_CR1","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1016\/S0377-0427(00)00476-3","volume":"125","author":"CTH Baker","year":"2000","unstructured":"Baker, C.T.H.: Retarded differential equations. J. Comput. Appl. Math. 125, 309\u2013335 (2000)","journal-title":"J. Comput. Appl. Math."},{"key":"1324_CR2","volume-title":"A Bibliography on the Numerical Solution of Delay Differential Equations","author":"CTH Baker","year":"1995","unstructured":"Baker, C.T.H., Paul, C.A.H., Will\u00e9, D.R.: A Bibliography on the Numerical Solution of Delay Differential Equations. Numerical Analysis Report 269, Mathematics Department, University of Manchester, U.K. (1995)"},{"key":"1324_CR3","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1007\/BF03028370","volume":"3","author":"CTH Baker","year":"1995","unstructured":"Baker, C.T.H., Paul, C.A.H., Will\u00e9, D.R.: Issues in the numerical solution of evolutionary delay differential equations. Adv. Comput. Math. 3, 171\u2013196 (1995)","journal-title":"Adv. Comput. Math."},{"volume-title":"Delay Differential Equations Directions New Recent Advanced","year":"2009","key":"1324_CR4","unstructured":"Balachandran, B., Kalm\u00e1r-Nagy, T., Gilsinn, D.E. (eds.): Delay Differential Equations Directions New Recent Advanced. Springer, New York (2009)"},{"key":"1324_CR5","doi-asserted-by":"crossref","unstructured":"Baleanu, D., Agheli, B., Darzi, R.: An optimal method for approximating the delay differential equations of noninteger order advances in difference equations 2018(284) (2018)","DOI":"10.1186\/s13662-018-1717-5"},{"issue":"3","key":"1324_CR6","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1007\/s002110050001","volume":"84","author":"A Bellen","year":"2000","unstructured":"Bellen, A., Maset, S.: Numerical solution of constant coefficient linear delay differential equations as abstract Cauchy problems. Numer. Math. 84 (3), 351\u2013374 (2000)","journal-title":"Numer. Math."},{"key":"1324_CR7","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198506546.001.0001","volume-title":"Numerical Methods For Delay Differential Equations","author":"A Bellen","year":"2003","unstructured":"Bellen, A., Zennaro, M.: Numerical Methods For Delay Differential Equations. Oxford, New York (2003)"},{"key":"1324_CR8","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4939-2107-2","volume-title":"Stability of Linear Delay Differential Equations. A Numerical Approach with MATLAB","author":"D Breda","year":"2015","unstructured":"Breda, D., Maset, S., Vermiglio, R.: Stability of Linear Delay Differential Equations. A Numerical Approach with MATLAB. Springer, New York (2015)"},{"key":"1324_CR9","doi-asserted-by":"publisher","DOI":"10.1002\/9781119121534","volume-title":"Numerical methods for ordinary differential equations, 3rd edn.","author":"JC Butcher","year":"2016","unstructured":"Butcher, J.C.: Numerical methods for ordinary differential equations, 3rd edn. Wiley, Chichester (2016)"},{"issue":"1\u20133","key":"1324_CR10","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1016\/0898-1221(94)00095-6","volume":"28","author":"KL Cooke","year":"1994","unstructured":"Cooke, K.L., Gy\u00f6ri, I.: Numerical approximation of the solutions of delay differential equations on an infinite interval using piecewise constant arguments. Comput. Math. Appl. 28(1\u20133), 81\u201392 (1994)","journal-title":"Comput. Math. Appl."},{"key":"1324_CR11","unstructured":"Czy\u017cewska, N., Morkisz, J., Kusiak. P., Oprocha, P., Pietrzyk, M., Przyby\u0142owicz, P., Rauch, \u0141., Szeliga, D.: On mathematical aspects of evolution of dislocation density in metallic materials. arXiv:2011.08504 (2020)"},{"key":"1324_CR12","volume-title":"Delay Equations. Functional-Complex-, and Nonlinear Analysis","author":"O Diekmann","year":"1995","unstructured":"Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Walther, H.O.: Delay Equations. Functional-Complex-, and Nonlinear Analysis. Springer, New York (1995)"},{"key":"1324_CR13","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9467-9","volume-title":"Ordinary and Delay Differential Equations","author":"RD Driver","year":"1977","unstructured":"Driver, R.D.: Ordinary and Delay Differential Equations. Springer, New York (1977)"},{"key":"1324_CR14","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1016\/0001-6160(84)90202-5","volume":"29","author":"Y Estrin","year":"1984","unstructured":"Estrin, Y., Mecking, H.: A unified phenomenological description of work hardening and creep based on one parameter models. Acta Metall. 29, 57\u201370 (1984)","journal-title":"Acta Metall."},{"key":"1324_CR15","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-47940-8","volume-title":"H\u00f6lder and locally h\u00f6lder Continuous Functions, and Open Sets of Class ck,Ck,\u03bb. Birkh\u00e4user","author":"R Fiorenza","year":"2016","unstructured":"Fiorenza, R.: H\u00f6lder and locally h\u00f6lder Continuous Functions, and Open Sets of Class ck, Ck, \u03bb. Birkh\u00e4user. Cham, Switzerland (2016)"},{"key":"1324_CR16","volume-title":"Mathematical Analysis for Physicists (in Polish)","author":"L G\u00f3rniewicz","year":"2012","unstructured":"G\u00f3rniewicz, L., Ingarden, R.S.: Mathematical Analysis for Physicists (in Polish). Wydawnictwo Naukowe UMK, Toru\u0144 (2012)"},{"key":"1324_CR17","volume-title":"Solving ordinary differential equations I. Nonstiff problems, 2nd revised edn.","author":"E Hairer","year":"2008","unstructured":"Hairer, E., N\u00f8rsett, S.P., Wanner, G.: Solving ordinary differential equations I. Nonstiff problems, 2nd revised edn. Springer, New York (2008)"},{"key":"1324_CR18","doi-asserted-by":"publisher","first-page":"719","DOI":"10.1007\/s11075-017-0457-z","volume":"79","author":"X Hu","year":"2018","unstructured":"Hu, X., Cong, Y., Hu, G.: Delay-dependent stability of linear multistep methods for DAEs with multiple delays. Numer. Algo. 79, 719\u2013739 (2018)","journal-title":"Numer. Algo."},{"key":"1324_CR19","unstructured":"Hale, J.K.: History of Delay Equations. In: Arino, O., Hbid, M., Dads E.A. (eds.) Delay Differential Equations and Applications. NATO Science Series (II.Mathematics, Physics and Chemistry), vol. 205. Springer Dordrecht (2006)"},{"issue":"1\u20132","key":"1324_CR20","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1016\/j.camwa.2004.01.003","volume":"48","author":"T Koto","year":"2004","unstructured":"Koto, T.: Method of lines approximation of delay differential equations. Computers and Mathematics with Applications 48(1\u20132), 45\u201359 (2004)","journal-title":"Computers and Mathematics with Applications"},{"key":"1324_CR21","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1126\/science.267326","volume":"197","author":"MC Mackey","year":"1977","unstructured":"Mackey, M.C., Glass, L.: Oscillation and chaos in physiological control systems. Science 197, 287\u2013289 (1977)","journal-title":"Science"},{"issue":"1","key":"1324_CR22","doi-asserted-by":"publisher","first-page":"18","DOI":"10.3390\/axioms10010018","volume":"10","author":"M Mahrouf","year":"2021","unstructured":"Mahrouf, M., Boukhouima, A., Houssine, Z., Lofti, M., Torres, D.F.M., Yousfi, N.: Modeling and forecasting of COVID-19 spreading by delayed stochastic differential equations. Axioms 10(1), 18 (2021)","journal-title":"Axioms"},{"key":"1324_CR23","doi-asserted-by":"publisher","first-page":"1865","DOI":"10.1016\/0001-6160(81)90112-7","volume":"29","author":"H Mecking","year":"1981","unstructured":"Mecking, H., Kocks, U.F.: Kinetics of flow and strain-hardening. Acta Metall. 29, 1865\u20131875 (1981)","journal-title":"Acta Metall."},{"issue":"3","key":"1324_CR24","doi-asserted-by":"publisher","first-page":"361","DOI":"10.1615\/IntJMultCompEng.2019030591","volume":"17","author":"P Morkisz","year":"2019","unstructured":"Morkisz, P., Oprocha, P., Przyby\u0142owicz, P., Czy\u017cewska, N., Kusiak, J., Szeliga, D., Rauch, \u0141., Pietrzyk, M.: Prediction of distribution of microstructural parameters in metallic materials described by differential equations with recrystallization term. Int. J. Multi. Comput. Eng. 17(3), 361\u2013371 (2019)","journal-title":"Int. J. Multi. Comput. Eng."},{"key":"1324_CR25","volume-title":"Computational Materials Engineering: Achieving High Accuracy and Efficiency in Metals Processing Simulations","author":"M Pietrzyk","year":"2015","unstructured":"Pietrzyk, M., Madej, \u0141., Rauch, \u0141., Szeliga, D.: Computational Materials Engineering: Achieving High Accuracy and Efficiency in Metals Processing Simulations. Butterworth-Heinemann, Elsevier, Amsterdam (2015)"},{"key":"1324_CR26","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-7646-8","volume-title":"An Introduction to Delay Differential Equations with Applications to the Life Sciences. Texts in Applied Mathematics 57","author":"H Smith","year":"2011","unstructured":"Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Texts in Applied Mathematics 57. Springer, New York (2011)"},{"issue":"317","key":"1324_CR27","first-page":"1","volume":"118","author":"D Szeliga","year":"2021","unstructured":"Szeliga, D., Czy\u017cewska, N., Klimczak, K., Kusiak, J., Morkisz, P., Oprocha, P., Pietrzyk, M., Przyby\u0142, owicz, P.: Sensivity analysis, identification and validation of the dislocation density-based model for metallic materials. Metall. Res. Technol. 118(317), 1\u201312 (2021)","journal-title":"Metall. Res. Technol."},{"issue":"1","key":"1324_CR28","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1016\/0377-0427(89)90071-X","volume":"25","author":"L Torelli","year":"1989","unstructured":"Torelli, L.: Stability of numerical methods for delay differential equations. J. Comput. Appl. Math. 25(1), 15\u201326 (1989)","journal-title":"J. Comput. Appl. Math."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-022-01324-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-022-01324-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-022-01324-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,5]],"date-time":"2023-02-05T21:07:12Z","timestamp":1675631232000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-022-01324-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,21]]},"references-count":28,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2022,12]]}},"alternative-id":["1324"],"URL":"https:\/\/doi.org\/10.1007\/s11075-022-01324-9","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"type":"print","value":"1017-1398"},{"type":"electronic","value":"1572-9265"}],"subject":[],"published":{"date-parts":[[2022,5,21]]},"assertion":[{"value":"28 June 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 April 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 May 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Conflict of interest"}}]}}