{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T11:51:41Z","timestamp":1778673101598,"version":"3.51.4"},"reference-count":53,"publisher":"American Mathematical Society (AMS)","issue":"338","license":[{"start":{"date-parts":[[2023,7,29]],"date-time":"2023-07-29T00:00:00Z","timestamp":1690588800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We derive optimal order a posteriori error estimates for fully discrete approximations of linear parabolic delay differential equations (PDDEs), in the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Superscript normal infinity Baseline left-parenthesis upper L squared right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">L^\\infty (L^2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -norm. For the discretization in time we use Backward Euler and Crank-Nicolson methods, while for the space discretization we use standard conforming finite element methods. A novel space-time reconstruction operator is introduced, which is a generalization of the elliptic reconstruction operator, and we call it as delay-dependent elliptic reconstruction operator. The related a posteriori error estimates for the delay-dependent elliptic reconstruction play key roles in deriving optimal order a posteriori error estimates in the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Superscript normal infinity Baseline left-parenthesis upper L squared right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">L^\\infty (L^2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -norm. Numerical experiments verify and complement our theoretical results.\n                  <\/p>","DOI":"10.1090\/mcom\/3761","type":"journal-article","created":{"date-parts":[[2022,6,1]],"date-time":"2022-06-01T09:21:22Z","timestamp":1654075282000},"page":"2609-2643","source":"Crossref","is-referenced-by-count":3,"title":["Delay-dependent elliptic reconstruction and optimal \ud835\udc3f^{\u221e}(\ud835\udc3f\u00b2) a posteriori error estimates for fully discrete delay parabolic problems"],"prefix":"10.1090","volume":"91","author":[{"given":"Wansheng","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lijun","family":"Yi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2022,7,29]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics (New York)","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032824","volume-title":"A posteriori error estimation in finite element analysis","author":"Ainsworth, Mark","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/047129411X"},{"issue":"254","key":"2","doi-asserted-by":"publisher","first-page":"511","DOI":"10.1090\/S0025-5718-05-01800-4","article-title":"A posteriori error estimates for the Crank-Nicolson method for parabolic equations","volume":"75","author":"Akrivis, Georgios","year":"2006","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"3","unstructured":"C. T. H. Baker, G. A. Bocharov, and F. A. Rihan, A report on the use of delay differential equations in numerical modelling in the biosciences, MCCM Technical Report, vol. 343, Manchester, 1999, ISSN 1360-1725."},{"issue":"3-4","key":"4","doi-asserted-by":"publisher","first-page":"284","DOI":"10.1016\/j.apnum.2005.04.009","article-title":"Discontinuous solutions of neutral delay differential equations","volume":"56","author":"Baker, Christopher T. H.","year":"2006","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"6","key":"5","doi-asserted-by":"publisher","first-page":"2845","DOI":"10.1137\/110839424","article-title":"A posteriori error control for fully discrete Crank-Nicolson schemes","volume":"50","author":"B\u00e4nsch, E.","year":"2012","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"6","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1016\/j.apnum.2011.08.008","article-title":"The effect of mesh modification in time on the error control of fully discrete approximations for parabolic equations","volume":"67","author":"B\u00e4nsch, E.","year":"2013","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"274","key":"7","doi-asserted-by":"publisher","first-page":"761","DOI":"10.1090\/S0025-5718-2010-02444-5","article-title":"Quasi-optimal and robust a posteriori error estimates in \ud835\udc3f^{\u221e}(\ud835\udc3f\u00b2) for the approximation of Allen-Cahn equations past singularities","volume":"80","author":"Bartels, S\u00f6ren","year":"2011","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"8","series-title":"Numerical Mathematics and Scientific Computation","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198506546.001.0001","volume-title":"Numerical methods for delay differential equations","author":"Bellen, Alfredo","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0198506546"},{"key":"9","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1016\/j.amc.2013.12.188","article-title":"Convergence and stability analysis of the \ud835\udf03-method for delayed diffusion mathematical models","volume":"231","author":"Blanco-Cocom, Luis","year":"2014","journal-title":"Appl. Math. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0096-3003","issn-type":"print"},{"key":"10","series-title":"Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-75934-0","volume-title":"The mathematical theory of finite element methods","volume":"15","author":"Brenner, Susanne C.","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9780387759333","edition":"3"},{"key":"11","series-title":"Cambridge Monographs on Applied and Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511543234","volume-title":"Collocation methods for Volterra integral and related functional differential equations","volume":"15","author":"Brunner, Hermann","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0521806151"},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1515\/jiip-2020-0010","article-title":"A time delay dynamical model for outbreak of 2019-nCoV and the parameter identification","volume":"28","author":"Chen, Yu","year":"2020","journal-title":"J. Inverse Ill-Posed Probl.","ISSN":"https:\/\/id.crossref.org\/issn\/0928-0219","issn-type":"print"},{"key":"13","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1051\/m2an\/197509R200771","article-title":"Approximation by finite element functions using local regularization","volume":"9","author":"Cl\u00e9ment, Ph.","year":"1975","journal-title":"Rev. Fran\\c{c}aise Automat. Informat. Recherche Op\\'{e}rationnelle S\\'{e}r. Rouge Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0397-9342","issn-type":"print"},{"issue":"5","key":"14","doi-asserted-by":"publisher","first-page":"531","DOI":"10.1016\/j.apnum.2010.01.003","article-title":"Non-Fickian delay reaction-diffusion equations: theoretical and numerical study","volume":"60","author":"Branco, J. R.","year":"2010","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"3-4","key":"15","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1023\/A:1019107718128","article-title":"A delay differential equation solver based on a continuous Runge-Kutta method with defect control","volume":"16","author":"Enright, W. H.","year":"1997","journal-title":"Numer. Algorithms","ISSN":"https:\/\/id.crossref.org\/issn\/1017-1398","issn-type":"print"},{"issue":"5-6","key":"16","doi-asserted-by":"publisher","first-page":"860","DOI":"10.1016\/j.mcm.2009.05.015","article-title":"Numerical solutions of diffusion mathematical models with delay","volume":"50","author":"Garc\u00eda, P.","year":"2009","journal-title":"Math. Comput. Modelling","ISSN":"https:\/\/id.crossref.org\/issn\/0895-7177","issn-type":"print"},{"issue":"7-8","key":"17","doi-asserted-by":"publisher","first-page":"1279","DOI":"10.1016\/j.mcm.2010.02.016","article-title":"Convergence of two implicit numerical schemes for diffusion mathematical models with delay","volume":"52","author":"Garc\u00eda, P.","year":"2010","journal-title":"Math. Comput. Modelling","ISSN":"https:\/\/id.crossref.org\/issn\/0895-7177","issn-type":"print"},{"key":"18","isbn-type":"print","first-page":"19","article-title":"Diffusion and hereditary effects in a class of population models","author":"Green, David, Jr.","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/0121483606"},{"issue":"2","key":"19","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1016\/j.cam.2005.03.010","article-title":"Open issues in devising software for the numerical solution of implicit delay differential equations","volume":"185","author":"Guglielmi, Nicola","year":"2006","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"3","key":"20","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1007\/s10444-007-9044-5","article-title":"Computing breaking points in implicit delay differential equations","volume":"29","author":"Guglielmi, Nicola","year":"2008","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"issue":"4","key":"21","doi-asserted-by":"publisher","first-page":"2428","DOI":"10.1137\/100801238","article-title":"Asymptotic expansions for regularized state-dependent neutral delay equations","volume":"44","author":"Guglielmi, Nicola","year":"2012","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"3","key":"22","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1007\/s10543-014-0480-6","article-title":"Implicit Euler and Lie splitting discretizations of nonlinear parabolic equations with delay","volume":"54","author":"Hansen, Eskil","year":"2014","journal-title":"BIT","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3835","issn-type":"print"},{"issue":"1-3","key":"23","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1016\/0168-9274(95)00051-U","article-title":"Existence and stability of fixed points for a discretised nonlinear reaction-diffusion equation with delay","volume":"18","author":"Higham, Desmond J.","year":"1995","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"1","key":"24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1093\/imanum\/6.1.1","article-title":"On the stability of predictor-corrector methods for parabolic equations with delay","volume":"6","author":"van der Houwen, P. J.","year":"1986","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"3-4","key":"25","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1016\/j.apnum.2005.04.021","article-title":"Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations","volume":"56","author":"Jackiewicz, Z.","year":"2006","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"key":"26","series-title":"Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-017-1965-0","volume-title":"Introduction to the theory and applications of functional-differential equations","volume":"463","author":"Kolmanovskii, V.","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/0792355040"},{"key":"27","unstructured":"J. X. Kuang and Y. H. Cong, Stability of Numerical Methods for Delay Differential Equations, Science Press, Beijing, 2005."},{"issue":"256","key":"28","doi-asserted-by":"publisher","first-page":"1627","DOI":"10.1090\/S0025-5718-06-01858-8","article-title":"Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems","volume":"75","author":"Lakkis, Omar","year":"2006","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"294","key":"29","doi-asserted-by":"publisher","first-page":"1537","DOI":"10.1090\/S0025-5718-2014-02912-8","article-title":"A comparison of duality and energy a posteriori estimates for L_{\u221e}(0,T;L\u2082(\u03a9)) in parabolic problems","volume":"84","author":"Lakkis, Omar","year":"2015","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"30","doi-asserted-by":"publisher","first-page":"1401","DOI":"10.1016\/j.nonrwa.2011.11.005","article-title":"Long time behavior of non-Fickian delay reaction-diffusion equations","volume":"13","author":"Li, Dongfang","year":"2012","journal-title":"Nonlinear Anal. Real World Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1468-1218","issn-type":"print"},{"issue":"6","key":"31","doi-asserted-by":"publisher","first-page":"4290","DOI":"10.1137\/080741148","article-title":"High order contractive Runge-Kutta methods for Volterra functional differential equations","volume":"47","author":"Li, Shoufu","year":"2010","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"6","key":"32","doi-asserted-by":"publisher","first-page":"2570","DOI":"10.1137\/130944837","article-title":"\ud835\udc35-convergence theory of Runge-Kutta methods for stiff Volterra functional differential equations with infinite integration interval","volume":"53","author":"Li, Shoufu","year":"2015","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"33","unstructured":"S. F. Li, Numerical Analysis for Stiff Ordinary and Functional Differential Equations, Xiangtan University Press, Xiangtan, 2010."},{"key":"34","doi-asserted-by":"publisher","first-page":"160","DOI":"10.1016\/j.amc.2015.04.104","article-title":"Convergence and asymptotic stability of Galerkin methods for linear parabolic equations with delays","volume":"264","author":"Liang, Hui","year":"2015","journal-title":"Appl. Math. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0096-3003","issn-type":"print"},{"issue":"4","key":"35","doi-asserted-by":"publisher","first-page":"2757","DOI":"10.1137\/080715135","article-title":"An anisotropic error estimator for the Crank-Nicolson method: application to a parabolic problem","volume":"31","author":"Lozinski, Alexei","year":"2009","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"4","key":"36","doi-asserted-by":"publisher","first-page":"1843","DOI":"10.1137\/130908543","article-title":"Good behavior with respect to the stiffness in the numerical integration of retarded functional differential equations","volume":"52","author":"Maset, S.","year":"2014","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"37","doi-asserted-by":"publisher","first-page":"1585","DOI":"10.1137\/S0036142902406314","article-title":"Elliptic reconstruction and a posteriori error estimates for parabolic problems","volume":"41","author":"Makridakis, Charalambos","year":"2003","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"38","doi-asserted-by":"publisher","first-page":"743","DOI":"10.1016\/j.camwa.2008.02.011","article-title":"Analytic-numerical solutions of diffusion mathematical models with delays","volume":"56","author":"Reyes, Elia","year":"2008","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"issue":"190","key":"39","doi-asserted-by":"publisher","first-page":"483","DOI":"10.2307\/2008497","article-title":"Finite element interpolation of nonsmooth functions satisfying boundary conditions","volume":"54","author":"Scott, L. Ridgway","year":"1990","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"40","doi-asserted-by":"publisher","first-page":"498","DOI":"10.1093\/imanum\/dry078","article-title":"Long-time \ud835\udc3f^{\u221e}(\ud835\udc3f\u00b2) a posteriori error estimates for fully discrete parabolic problems","volume":"40","author":"Sutton, Oliver J.","year":"2020","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"key":"41","series-title":"Numerical Mathematics and Scientific Computation","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199679423.001.0001","volume-title":"A posteriori error estimation techniques for finite element methods","author":"Verf\u00fcrth, R\u00fcdiger","year":"2013","ISBN":"https:\/\/id.crossref.org\/isbn\/9780199679423"},{"issue":"4","key":"42","doi-asserted-by":"publisher","first-page":"2181","DOI":"10.1137\/060654116","article-title":"Stability analysis of \u0398-methods for nonlinear neutral functional differential equations","volume":"30","author":"Wang, Wansheng","year":"2008","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"2","key":"43","doi-asserted-by":"publisher","first-page":"122","DOI":"10.1016\/j.apnum.2006.11.002","article-title":"Nonlinear stability of one-leg methods for delay differential equations of neutral type","volume":"58","author":"Wang, Wan-Sheng","year":"2008","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"2","key":"44","doi-asserted-by":"publisher","first-page":"592","DOI":"10.1016\/j.cam.2008.05.050","article-title":"Nonlinear stability of general linear methods for neutral delay differential equations","volume":"224","author":"Wang, Wan-Sheng","year":"2009","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"8","key":"45","doi-asserted-by":"publisher","first-page":"3319","DOI":"10.1016\/j.apm.2008.10.038","article-title":"Stability of continuous Runge-Kutta-type methods for nonlinear neutral delay-differential equations","volume":"33","author":"Wang, Wansheng","year":"2009","journal-title":"Appl. Math. Model.","ISSN":"https:\/\/id.crossref.org\/issn\/0307-904X","issn-type":"print"},{"issue":"3","key":"46","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1007\/s00211-009-0281-z","article-title":"Preserving stability implicit Euler method for nonlinear Volterra and neutral functional differential equations in Banach space","volume":"115","author":"Wang, Wansheng","year":"2010","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"2","key":"47","doi-asserted-by":"publisher","first-page":"A1095--A1120","DOI":"10.1137\/17M1143514","article-title":"A posteriori error analysis for Crank-Nicolson-Galerkin type methods for reaction-diffusion equations with delay","volume":"40","author":"Wang, Wansheng","year":"2018","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"1","key":"48","doi-asserted-by":"publisher","first-page":"Paper No. 13, 27","DOI":"10.1007\/s10915-020-01262-5","article-title":"A posteriori error estimates for fully discrete finite element method for generalized diffusion equation with delay","volume":"84","author":"Wang, Wansheng","year":"2020","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"issue":"3-5","key":"49","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1016\/0168-9274(92)90016-7","article-title":"The tracking of derivative discontinuities in systems of delay-differential equations","volume":"9","author":"Will\u00e9, David R.","year":"1992","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"key":"50","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4050-1","volume-title":"Theory and applications of partial functional-differential equations","volume":"119","author":"Wu, Jianhong","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/038794771X"},{"key":"51","doi-asserted-by":"crossref","unstructured":"Y. Yan, Y. Chen, K. J. Liu, X. Y. Luo, B. X. Xu, Y. Jiang, and J. Cheng, Modeling and prediction for the trend of outbreak of NCP based on a time-delay dynamic system (Chinese), Sci. Sin. Math. 50 (2020), 385\u2013392.","DOI":"10.1360\/SSM-2020-0026"},{"key":"52","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1016\/j.cam.2016.04.016","article-title":"The compact and Crank-Nicolson ADI schemes for two-dimensional semilinear multidelay parabolic equations","volume":"306","author":"Zhang, Qifeng","year":"2016","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"1","key":"53","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1023\/A:1021930104326","article-title":"Stability in the numerical solution of linear parabolic equations with a delay term","volume":"41","author":"Zubik-Kowal, Barbara","year":"2001","journal-title":"BIT","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3835","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2022-91-338\/S0025-5718-2022-03761-3\/S0025-5718-2022-03761-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:41:47Z","timestamp":1776832907000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2022-91-338\/S0025-5718-2022-03761-3\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,29]]},"references-count":53,"journal-issue":{"issue":"338","published-print":{"date-parts":[[2022,11]]}},"alternative-id":["S0025-5718-2022-03761-3"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3761","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2022,7,29]]}}}