{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T17:30:28Z","timestamp":1778088628538,"version":"3.51.4"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2018,3,21]],"date-time":"2018-03-21T00:00:00Z","timestamp":1521590400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"name":"National Natural Science Fund","award":["11661058"],"award-info":[{"award-number":["11661058"]}]},{"name":"National Natural Science Fund","award":["11361035"],"award-info":[{"award-number":["11361035"]}]},{"name":"Natural Science Fund of Inner Mongolia Autonomous Region","award":["2016M- S0102"],"award-info":[{"award-number":["2016M- S0102"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2019,2]]},"DOI":"10.1007\/s11075-018-0496-0","type":"journal-article","created":{"date-parts":[[2018,3,21]],"date-time":"2018-03-21T10:15:25Z","timestamp":1521627325000},"page":"533-555","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":80,"title":["Some second-order \ud835\udf03 schemes combined with finite element method for nonlinear fractional cable equation"],"prefix":"10.1007","volume":"80","author":[{"given":"Yang","family":"Liu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yanwei","family":"Du","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hong","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1034-2349","authenticated-orcid":false,"given":"Fawang","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yajun","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2018,3,21]]},"reference":[{"key":"496_CR1","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1016\/j.aml.2016.03.005","volume":"59","author":"Y Zhao","year":"2016","unstructured":"Zhao, Y., Zhang, Y., Shi, D., Liu, F., Turner, I.: Superconvergence analysis of nonconforming finite element method for two-dimensional time fractional diffusion equations. Appl. Math. Lett. 59, 38\u201347 (2016)","journal-title":"Appl. Math. Lett."},{"issue":"15","key":"496_CR2","doi-asserted-by":"publisher","first-page":"3871","DOI":"10.1016\/j.apm.2013.10.007","volume":"38","author":"F Liu","year":"2014","unstructured":"Liu, F., Zhuang, P., Turner, I., Burrage, K., Anh, V.: A new fractional finite volume method for solving the fractional diffusion equation. Appl. Math. Model. 38(15), 3871\u20133878 (2014)","journal-title":"Appl. Math. Model."},{"key":"496_CR3","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1016\/j.cma.2015.02.027","volume":"290","author":"H Wang","year":"2015","unstructured":"Wang, H., Yang, D., Zhu, S.F.: A Petrov-Galerkin finite element method for variable-coefficient fractional diffusion equations. Comput. Methods Appl. Mech. Eng. 290, 45\u201356 (2015)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"12","key":"496_CR4","doi-asserted-by":"publisher","first-page":"2594","DOI":"10.1016\/j.cpc.2012.07.011","volume":"183","author":"SB Yuste","year":"2012","unstructured":"Yuste, S.B., Quintana-Murillo, J.: A finite difference method with non-uniform time steps for fractional diffusion equations. Comput. Phys. Comm. 183(12), 2594\u20132600 (2012)","journal-title":"Comput. Phys. Comm."},{"issue":"1","key":"496_CR5","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1016\/j.cam.2004.01.033","volume":"172","author":"MM Meerschaert","year":"2004","unstructured":"Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172(1), 65\u201377 (2004)","journal-title":"J. Comput. Appl. Math."},{"key":"496_CR6","doi-asserted-by":"publisher","first-page":"3153","DOI":"10.1137\/110827624","volume":"33","author":"JC Li","year":"2011","unstructured":"Li, J.C., Huang, Y.Q., Lin, Y.P.: Developing finite element methods for Maxwell\u2019s equations in a cole-cole dispersive medium. SIAM J. Sci. Comput. 33, 3153\u20133174 (2011)","journal-title":"SIAM J. Sci. Comput."},{"key":"496_CR7","doi-asserted-by":"crossref","first-page":"2534","DOI":"10.1016\/j.amc.2010.07.066","volume":"217","author":"H Zhang","year":"2010","unstructured":"Zhang, H., Liu, F., Anh, V.: Galerkin finite element approximation of symmetric space-fractional partial differential equations. Appl. Math. Comput. 217, 2534\u20132545 (2010)","journal-title":"Appl. Math. Comput."},{"key":"496_CR8","doi-asserted-by":"publisher","first-page":"1287","DOI":"10.1007\/s11425-013-4584-2","volume":"56","author":"YJ Jiang","year":"2013","unstructured":"Jiang, Y.J., Ma, J.T.: Moving finite element methods for time fractional partial differential equations. Sci. China Math. 56, 1287\u20131300 (2013)","journal-title":"Sci. China Math."},{"key":"496_CR9","doi-asserted-by":"publisher","first-page":"855","DOI":"10.1016\/j.camwa.2011.02.045","volume":"62","author":"CP Li","year":"2011","unstructured":"Li, C.P., Zhao, Z.G., Chen, Y.Q.: Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Math. Appl. 62, 855\u2013875 (2011)","journal-title":"Comput. Math. Appl."},{"key":"496_CR10","unstructured":"Ding, H.F., Li, C.P.: A novel second-order midpoint approximation formula for Riemann-Liouville derivative: algorithm and application. arXiv: 1605.02177 (2016)"},{"issue":"4","key":"496_CR11","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1016\/j.camwa.2015.05.015","volume":"70","author":"Y Liu","year":"2015","unstructured":"Liu, Y., Du, Y.W., Li, H., He, S., Gao, W.: Finite difference\/finite element method for a nonlinear time-fractional fourth-order reaction-diffusion problem. Comput. Math. Appl. 70(4), 573\u2013591 (2015)","journal-title":"Comput. Math. Appl."},{"issue":"10","key":"496_CR12","doi-asserted-by":"publisher","first-page":"2474","DOI":"10.1016\/j.camwa.2015.09.012","volume":"70","author":"Y Liu","year":"2015","unstructured":"Liu, Y., Du, Y.W., Li, H., Li, J.C., He, S.: A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative. Comput. Math. Appl. 70(10), 2474\u20132492 (2015)","journal-title":"Comput. Math. Appl."},{"issue":"3","key":"496_CR13","doi-asserted-by":"publisher","first-page":"749","DOI":"10.1007\/s11075-015-0065-8","volume":"72","author":"LB Feng","year":"2016","unstructured":"Feng, L.B., Zhuang, P., Liu, F., Turner, I., Gu, Y.T.: Finite element method for space-time fractional diffusion equation. Numer. Algor. 72(3), 749\u2013767 (2016)","journal-title":"Numer. Algor."},{"key":"496_CR14","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1016\/j.jcp.2014.10.051","volume":"281","author":"B Jin","year":"2015","unstructured":"Jin, B., Lazarov, R., Liu, Y.K., Zhou, Z.: The Galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys. 281, 825\u2013843 (2015)","journal-title":"J. Comput. Phys."},{"key":"496_CR15","doi-asserted-by":"publisher","first-page":"1703","DOI":"10.1090\/S0025-5718-2015-02917-2","volume":"84","author":"WY Tian","year":"2015","unstructured":"Tian, W.Y., Zhou, H., Deng, W.H.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84, 1703\u20131727 (2015)","journal-title":"Math. Comput."},{"key":"496_CR16","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jcp.2014.08.012","volume":"277","author":"ZB Wang","year":"2014","unstructured":"Wang, Z.B., Vong, S.W.: Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation. J. Comput. Phys. 277, 1\u201315 (2014)","journal-title":"J. Comput. Phys."},{"key":"496_CR17","doi-asserted-by":"publisher","first-page":"264","DOI":"10.1016\/j.jcp.2014.06.031","volume":"293","author":"WP Bu","year":"2015","unstructured":"Bu, W.P., Tang, Y.F., Wu, Y.C., Yang, J.Y.: Finite difference\/finite element method for two-dimensional space and time fractional Bloch-Torrey equations. J. Comput. Phys. 293, 264\u2013279 (2015)","journal-title":"J. Comput. Phys."},{"issue":"6","key":"496_CR18","doi-asserted-by":"publisher","first-page":"A2976","DOI":"10.1137\/130910865","volume":"35","author":"F Zeng","year":"2013","unstructured":"Zeng, F., Li, C., Liu, F., Turner, I.: The use of finite difference\/element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35(6), A2976\u2013A3000 (2013)","journal-title":"SIAM J. Sci. Comput."},{"key":"496_CR19","doi-asserted-by":"publisher","first-page":"1369","DOI":"10.1090\/S0025-5718-2010-02438-X","volume":"80","author":"YM Lin","year":"2011","unstructured":"Lin, Y.M., Li, X.J., Xu, C.J.: Finite difference\/spectral approximations for the fractional Cable equation. Math. Comput. 80, 1369\u20131396 (2011)","journal-title":"Math. Comput."},{"issue":"6","key":"496_CR20","doi-asserted-by":"publisher","first-page":"761","DOI":"10.1007\/s00285-009-0251-1","volume":"59","author":"TAM Langlands","year":"2009","unstructured":"Langlands, T.A.M., Henry, B., Wearne, S.: Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions. J. Math. Biol. 59(6), 761\u2013808 (2009)","journal-title":"J. Math. Biol."},{"key":"496_CR21","doi-asserted-by":"publisher","first-page":"2535","DOI":"10.1007\/s11071-016-2843-9","volume":"85","author":"Y Liu","year":"2016","unstructured":"Liu, Y., Du, Y.W., Li, H., Wang, J.F.: A two-grid finite element approximation for a nonlinear time-fractional Cable equation. Nonlinear Dyn. 85, 2535\u20132548 (2016)","journal-title":"Nonlinear Dyn."},{"issue":"1-2","key":"496_CR22","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1007\/s11071-014-1854-7","volume":"80","author":"AH Bhrawy","year":"2015","unstructured":"Bhrawy, A.H., Zaky, M.A.: Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation. Nonlinear Dyn. 80(1-2), 101\u2013116 (2015)","journal-title":"Nonlinear Dyn."},{"issue":"2","key":"496_CR23","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1007\/s11075-015-0055-x","volume":"72","author":"P Zhuang","year":"2016","unstructured":"Zhuang, P., Liu, F., Turner, I., Anh, V.: Galerkin finite element method and error analysis for the fractional cable equation. Numer. Algor. 72(2), 447\u2013466 (2016)","journal-title":"Numer. Algor."},{"issue":"1-2","key":"496_CR24","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1007\/s12190-015-0944-0","volume":"52","author":"JC Liu","year":"2016","unstructured":"Liu, J.C., Li, H., Liu, Y.: A new fully discrete finite difference\/element approximation for fractional Cable equation. J. Appl. Math. Comput. 52(1-2), 345\u2013361 (2016)","journal-title":"J. Appl. Math. Comput."},{"issue":"1","key":"496_CR25","doi-asserted-by":"publisher","first-page":"252","DOI":"10.1007\/s10915-015-0136-y","volume":"68","author":"B Yu","year":"2016","unstructured":"Yu, B., Jiang, X.Y.: Numerical identification of the fractional derivatives in the two-dimensional fractional Cable equation. J. Sci. Comput. 68(1), 252\u2013272 (2016)","journal-title":"J. Sci. Comput."},{"key":"496_CR26","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1140\/epjp\/i2016-16061-3","volume":"131","author":"YJ Wang","year":"2016","unstructured":"Wang, Y.J., Liu, Y., Li, H., Wang, J.F.: Finite element method combined with second-order time discrete scheme for nonlinear fractional Cable equation. Eur. Phys. J. Plus. 131, 61 (2016). https:\/\/doi.org\/10.1140\/epjp\/i2016-16061-3","journal-title":"Eur. Phys. J. Plus."},{"key":"496_CR27","doi-asserted-by":"publisher","first-page":"108","DOI":"10.1016\/j.jcp.2017.04.078","volume":"344","author":"YW Du","year":"2017","unstructured":"Du, Y.W., Liu, Y., Li, H., Fang, Z.C., He, S.: Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation. J. Comput. Phys. 344, 108\u2013126 (2017)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"496_CR28","doi-asserted-by":"publisher","first-page":"970","DOI":"10.1002\/num.22038","volume":"32","author":"H Sun","year":"2016","unstructured":"Sun, H., Sun, Z.Z., Gao, G.H.: Some temporal second order difference schemes for fractional wave equations. Numer. Methods Partial Differential Eq. 32 (3), 970\u20131001 (2016)","journal-title":"Numer. Methods Partial Differential Eq."},{"key":"496_CR29","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1016\/j.jcp.2014.09.031","volume":"280","author":"AA Alikhanov","year":"2015","unstructured":"Alikhanov, A.A.: A new difference scheme for the fractional diffusion equation. J. Comput. Phys. 280, 424\u2013438 (2015)","journal-title":"J. Comput. Phys."},{"key":"496_CR30","doi-asserted-by":"publisher","first-page":"510","DOI":"10.1016\/j.jcp.2014.09.033","volume":"280","author":"GH Gao","year":"2015","unstructured":"Gao, G.H., Sun, H.W., Sun, Z.Z.: Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence. J. Comput. Phys. 280, 510\u2013528 (2015)","journal-title":"J. Comput. Phys."},{"issue":"6","key":"496_CR31","doi-asserted-by":"publisher","first-page":"1298","DOI":"10.1016\/j.camwa.2016.08.015","volume":"73","author":"Y Liu","year":"2017","unstructured":"Liu, Y., Zhang, M., Li, H., Li, J.C.: High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional subdiffusion equation. Comput. Math. Appl. 73(6), 1298\u20131314 (2017)","journal-title":"Comput. Math. Appl."},{"issue":"3","key":"496_CR32","doi-asserted-by":"publisher","first-page":"959","DOI":"10.1007\/s10915-014-9956-4","volume":"64","author":"CC Ji","year":"2015","unstructured":"Ji, C.C., Sun, Z.Z.: A high-order compact finite difference scheme for the fractional sub-diffusion equation. J. Sci. Comput. 64(3), 959\u2013985 (2015)","journal-title":"J. Sci. Comput."},{"issue":"2","key":"496_CR33","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1007\/s11075-016-0160-5","volume":"74","author":"M Li","year":"2017","unstructured":"Li, M., Huang, C.M., Wang, P.D.: Galerkin finite element method for nonlinear fractional Schr\u00f6dinger equations. Numer. Algor. 74(2), 499\u2013525 (2017)","journal-title":"Numer. Algor."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-018-0496-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-018-0496-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-018-0496-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,10,29]],"date-time":"2020-10-29T21:22:29Z","timestamp":1604006549000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-018-0496-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,3,21]]},"references-count":33,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,2]]}},"alternative-id":["496"],"URL":"https:\/\/doi.org\/10.1007\/s11075-018-0496-0","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,3,21]]},"assertion":[{"value":"29 April 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 February 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 March 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}