{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,27]],"date-time":"2026-07-27T15:01:59Z","timestamp":1785164519262,"version":"3.55.0"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,7,27]],"date-time":"2026-07-27T00:00:00Z","timestamp":1785110400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,7,27]],"date-time":"2026-07-27T00:00:00Z","timestamp":1785110400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100004731","name":"Natural Science Foundation of Zhejiang Province","doi-asserted-by":"publisher","award":["LQ24A010002"],"award-info":[{"award-number":["LQ24A010002"]}],"id":[{"id":"10.13039\/501100004731","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2026,9]]},"DOI":"10.1007\/s10915-026-03418-1","type":"journal-article","created":{"date-parts":[[2026,7,27]],"date-time":"2026-07-27T14:21:49Z","timestamp":1785162109000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Time Fractional Hemivariational Inequality Arising from a Dynamic Contact Problem: Analysis and Numerical Simulation"],"prefix":"10.1007","volume":"108","author":[{"given":"Hailing","family":"Xuan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0810-6921","authenticated-orcid":false,"given":"Lele","family":"Yuan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaoliang","family":"Cheng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,7,27]]},"reference":[{"key":"3418_CR1","doi-asserted-by":"publisher","first-page":"660","DOI":"10.1134\/S0012266110050058","volume":"46","author":"AA Alikhanov","year":"2010","unstructured":"Alikhanov, A.A.: A priori estimates for solutions of boundary value problems for fractional-order equations. Differ. Equ. 46, 660\u2013666 (2010)","journal-title":"Differ. Equ."},{"key":"3418_CR2","volume-title":"Theoretical Numerical Analysis: A Functional Analysis Framework","author":"KE Atkinson","year":"2009","unstructured":"Atkinson, K.E., Han, W.: Theoretical Numerical Analysis: A Functional Analysis Framework, 3rd edn. Springer-Verlag, New York (2009)","edition":"3"},{"key":"3418_CR3","doi-asserted-by":"publisher","first-page":"108026","DOI":"10.1016\/j.cnsns.2024.108026","volume":"134","author":"Y Bai","year":"2024","unstructured":"Bai, Y., Costea, N., Zeng, S.: Existence results for variational-hemivariational inequality systems with nonlinear couplings. Commun. Nonlinear Sci. Numer. Simul. 134, 108026 (2024)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"3418_CR4","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1016\/S0020-7683(99)00075-X","volume":"37","author":"JR Barber","year":"2000","unstructured":"Barber, J.R., Ciavarella, M.: Contact mechanics. Int. J. Solids Struct. 37, 29\u201343 (2000)","journal-title":"Int. J. Solids Struct."},{"key":"3418_CR5","doi-asserted-by":"publisher","first-page":"841","DOI":"10.1016\/j.jmaa.2014.09.078","volume":"423","author":"K Bartosz","year":"2015","unstructured":"Bartosz, K., Cheng, X., Kalita, P., et al.: Rothe method for parabolic variational-hemivariational inequalities. J. Math. Anal. Appl. 423, 841\u2013862 (2015)","journal-title":"J. Math. Anal. Appl."},{"key":"3418_CR6","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-75934-0","volume-title":"The Mathematical Theory of Finite Elements Methods","author":"SC Brenner","year":"2008","unstructured":"Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Elements Methods, 3rd edn. Springer-Verlag, New York (2008)","edition":"3"},{"key":"3418_CR7","doi-asserted-by":"publisher","first-page":"2943","DOI":"10.1016\/j.camwa.2018.08.058","volume":"77","author":"X Cheng","year":"2019","unstructured":"Cheng, X., Xiao, Q., Mig\u00f3rski, S., Ochal, A.: Error estimate for quasistatic history-dependent contact model. Comput. Math. Appl. 77, 2943\u20132952 (2019)","journal-title":"Comput. Math. Appl."},{"key":"3418_CR8","doi-asserted-by":"publisher","first-page":"108058","DOI":"10.1016\/j.cnsns.2024.108058","volume":"135","author":"X Chu","year":"2024","unstructured":"Chu, X., Chen, T., Huang, N., Li, X.: A new class of general fractional differential quasivariational and quasivariational-hemivariational inequalities with variable constraint sets. Commun. Nonlinear Sci. Numer. Simul. 135, 108058 (2024)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"3418_CR9","volume-title":"The Finite Element Method for Elliptic Problems","author":"PG Ciarlet","year":"1978","unstructured":"Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)"},{"key":"3418_CR10","doi-asserted-by":"publisher","first-page":"861","DOI":"10.1016\/S0168-874X(01)00072-5","volume":"37","author":"A Czekanski","year":"2001","unstructured":"Czekanski, A., Meguid, S.A.: Analysis of dynamic frictional contact problems using variational inequalities. Finite Elem. Anal. Des. 37, 861\u2013879 (2001)","journal-title":"Finite Elem. Anal. Des."},{"key":"3418_CR11","doi-asserted-by":"publisher","first-page":"1323","DOI":"10.1080\/02331934.2016.1262863","volume":"66","author":"J Gwinner","year":"2016","unstructured":"Gwinner, J.: Lagrange Multipliers and mixed formulations for some inequality constrained variational inequalities and some nonsmooth unilateral problems. Optimization 66, 1323\u20131336 (2016)","journal-title":"Optimization"},{"key":"3418_CR12","doi-asserted-by":"publisher","first-page":"3891","DOI":"10.1137\/140963248","volume":"46","author":"W Han","year":"2014","unstructured":"Han, W., Mig\u00f3rski, S., Sofonea, M.: A class of variational-hemivariational inequalities with applications to frictional contact problems. SIAM J. Math. Anal. 46, 3891\u20133912 (2014)","journal-title":"SIAM J. Math. Anal."},{"key":"3418_CR13","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1017\/S0962492919000023","volume":"28","author":"W Han","year":"2019","unstructured":"Han, W., Sofonea, M.: Numerical analysis of hemivariational inequalities in contact mechanics. Acta Numer 28, 175\u2013286 (2019)","journal-title":"Acta Numer"},{"key":"3418_CR14","doi-asserted-by":"publisher","first-page":"640","DOI":"10.1137\/16M1072085","volume":"55","author":"W Han","year":"2017","unstructured":"Han, W., Sofonea, M., Barboteu, M.: Numerical analysis of elliptic hemivariational inequalities. SIAM J. Numer. Anal. 55, 640\u2013663 (2017)","journal-title":"SIAM J. Numer. Anal."},{"key":"3418_CR15","doi-asserted-by":"publisher","first-page":"3147","DOI":"10.1016\/j.cma.2004.08.006","volume":"194","author":"S H\u00fceber","year":"2005","unstructured":"H\u00fceber, S., Wohlmuth, B.I.: A primal-dual active set strategy for non-linear multibody contact problems. Comput. Methods Appl. Mech. Eng. 194, 3147\u20133166 (2005)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"3418_CR16","doi-asserted-by":"publisher","first-page":"106718","DOI":"10.1016\/j.cnsns.2022.106718","volume":"115","author":"J Han","year":"2022","unstructured":"Han, J., Li, C., Zeng, S.: Applications of generalized fractional hemivariational inequalities in solid viscoelastic contact mechanics. Commun. Nonlinear Sci. Numer. Simul. 115, 106718 (2022)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"3418_CR17","doi-asserted-by":"publisher","first-page":"108098","DOI":"10.1016\/j.cnsns.2024.108098","volume":"136","author":"J Hao","year":"2024","unstructured":"Hao, J., Li, M.: A new class of fractional Navier-Stokes system coupled with multivalued boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 136, 108098 (2024)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"3418_CR18","doi-asserted-by":"publisher","first-page":"1032","DOI":"10.1177\/10812865211054185","volume":"27","author":"J Hao","year":"2022","unstructured":"Hao, J., Wang, J., Han, J.: History-dependent fractional hemivariational inequality with time-delay system for a class of new frictionless quasistatic contact problems. Math. Mech. Solids 27, 1032\u20131052 (2022)","journal-title":"Math. Mech. Solids"},{"key":"3418_CR19","doi-asserted-by":"publisher","first-page":"969","DOI":"10.1080\/02331934.2022.2133544","volume":"73","author":"J Hao","year":"2024","unstructured":"Hao, J., Wang, J., Lu, L.: Coupled system of fractional hemivariational inequalities with applications. Optimization 73, 969\u2013994 (2024)","journal-title":"Optimization"},{"key":"3418_CR20","doi-asserted-by":"publisher","first-page":"113735","DOI":"10.1016\/j.jde.2025.113735","volume":"450","author":"J Hao","year":"2026","unstructured":"Hao, J., Caraballo, T.: Solvability and convergence results for the temperature and concentration field in incompressible Navier-Stokes equations with boundary conditions. J. Differ. Equ. 450, 113735 (2026)","journal-title":"J. Differ. Equ."},{"key":"3418_CR21","volume-title":"Theory and applications of fractional differential equations","author":"A Kilbas","year":"2006","unstructured":"Kilbas, A., Srivastava, H., Trujillo, J.: Theory and applications of fractional differential equations. Elsevier, Amsterdam (2006)"},{"key":"3418_CR22","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970845","volume-title":"Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Method","author":"N Kikuchi","year":"1988","unstructured":"Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Method. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1988)"},{"key":"3418_CR23","doi-asserted-by":"publisher","first-page":"4729","DOI":"10.1016\/j.na.2012.03.023","volume":"75","author":"A Kulig","year":"2012","unstructured":"Kulig, A., Mig\u00f3rski, S.: Solvability and continuous dependence results for second order nonlinear evolution inclusions with a Volterra-type operator. Nonlinear Anal. 75, 4729\u20134746 (2012)","journal-title":"Nonlinear Anal."},{"key":"3418_CR24","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1080\/01630563.2012.706673","volume":"34","author":"C Li","year":"2013","unstructured":"Li, C., Zeng, F.: The finite difference methods for fractional ordinary differential equations. Numer. Funct. Anal. Optim. 34, 149\u2013179 (2013)","journal-title":"Numer. Funct. Anal. Optim."},{"key":"3418_CR25","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-4232-5","volume-title":"Nonlinear Inclusions and Hemivariational Inequalities","author":"S Mig\u00f3rski","year":"2013","unstructured":"Mig\u00f3rski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities. Springer, New York (2013)"},{"key":"3418_CR26","doi-asserted-by":"publisher","first-page":"e201800172","DOI":"10.1002\/zamm.201800172","volume":"99","author":"Z Peng","year":"2019","unstructured":"Peng, Z.: Existence of a class of variational inequalities modelling quasi-static viscoelastic contact problems. Z. Angew. Math. Mech. 99, e201800172 (2019)","journal-title":"Z. Angew. Math. Mech."},{"key":"3418_CR27","volume-title":"Variational-Hemivariational Inequalities with Applications","author":"M Sofonea","year":"2018","unstructured":"Sofonea, M., Mig\u00f3rski, S.: Variational-Hemivariational Inequalities with Applications. Chapman and Hall\/CRC Press, New York (2018)"},{"key":"3418_CR28","doi-asserted-by":"publisher","first-page":"1945","DOI":"10.1002\/nme.1620361110","volume":"36","author":"GE Stavroulakis","year":"1993","unstructured":"Stavroulakis, G.E., Panagiotopoulos, P.D.: Convex multilevel decomposition algorithms for nonmonotone problems. Int. J. Numer. Meth. Eng. 36, 1945\u20131966 (1993)","journal-title":"Int. J. Numer. Meth. Eng."},{"key":"3418_CR29","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1016\/0045-7825(93)90020-X","volume":"110","author":"MA Tzaferopoulos","year":"1993","unstructured":"Tzaferopoulos, M.A., Panagiotopoulos, P.D.: Delamination of composites as a substationarity problem: Numerical approximation and algorithms. Comput. Methods Appl. Mech. Eng. 110, 63\u201385 (1993)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"3418_CR30","doi-asserted-by":"publisher","first-page":"2207","DOI":"10.1007\/s11425-019-1672-4","volume":"63","author":"S Wang","year":"2020","unstructured":"Wang, S., Xu, W., Han, W., Chen, W.: Numerical analysis of history-dependent variational-hemivariational inequalities. Sci China Math 63, 2207\u20132232 (2020)","journal-title":"Sci China Math"},{"key":"3418_CR31","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1007\/s10915-023-02212-7","volume":"95","author":"F Wang","year":"2023","unstructured":"Wang, F., Shah, S., Wu, B.: Discontinuous Galerkin methods for hemivariational inequalities in contact mechanics. J. Sci. Comput. 95, 87 (2023)","journal-title":"J. Sci. Comput."},{"key":"3418_CR32","doi-asserted-by":"publisher","first-page":"108165","DOI":"10.1016\/j.cnsns.2024.108165","volume":"137","author":"X Sun","year":"2024","unstructured":"Sun, X., Cheng, X., Xuan, H.: Error estimates and numerical simulations of a thermoviscoelastic contact problem with damage and long memory. Commun. Nonlinear Sci. Numer. Simul. 137, 108165 (2024)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"3418_CR33","doi-asserted-by":"publisher","first-page":"82","DOI":"10.1007\/s10915-023-02310-6","volume":"96","author":"W Xiao","year":"2023","unstructured":"Xiao, W., Ling, M.: Virtual element method for a history-dependent variational-hemivariational inequality in contact problems. J. Sci. Comput. 96, 82 (2023)","journal-title":"J. Sci. Comput."},{"key":"3418_CR34","doi-asserted-by":"publisher","first-page":"569","DOI":"10.4208\/nmtma.OA-2019-0130","volume":"13","author":"H Xuan","year":"2020","unstructured":"Xuan, H., Cheng, X., Han, W., Xiao, Q.: Numerical analysis of a dynamic contact problem with history-dependent operators. Num. Math. Theory Methods Appl. 13, 569\u2013594 (2020)","journal-title":"Num. Math. Theory Methods Appl."},{"key":"3418_CR35","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1007\/s10915-025-02815-2","volume":"103","author":"H Xuan","year":"2025","unstructured":"Xuan, H., Cheng, X., Yuan, L.: Numerical studies of a class of thermoviscoelastic frictional contact problem described by fractional differential hemivariational inequalities. J. Sci. Comput. 103, 4 (2025)","journal-title":"J. Sci. Comput."},{"key":"3418_CR36","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1016\/j.cnsns.2017.07.016","volume":"56","author":"S Zeng","year":"2018","unstructured":"Zeng, S., Mig\u00f3rski, S.: A class of time-fractional hemivariational inequalities with application to frictional contact problem. Commun. Nonlinear Sci. Numer. Simul. 56, 34\u201348 (2018)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03418-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-026-03418-1","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03418-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,27]],"date-time":"2026-07-27T14:21:50Z","timestamp":1785162110000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-026-03418-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,27]]},"references-count":36,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,9]]}},"alternative-id":["3418"],"URL":"https:\/\/doi.org\/10.1007\/s10915-026-03418-1","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,27]]},"assertion":[{"value":"28 April 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 June 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 July 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 July 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.","order":1,"name":"Ethics","label":"Conflict of interest","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"91"}}