{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T08:55:38Z","timestamp":1773046538159,"version":"3.50.1"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T00:00:00Z","timestamp":1584576000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T00:00:00Z","timestamp":1584576000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["11671131"],"award-info":[{"award-number":["11671131"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2020,5]]},"DOI":"10.1007\/s40314-020-1134-8","type":"journal-article","created":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T16:04:00Z","timestamp":1584633840000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Weak Galerkin finite-element method for time-fractional nonlinear integro-differential equations"],"prefix":"10.1007","volume":"39","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0659-8771","authenticated-orcid":false,"given":"Haifeng","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Da","family":"Xu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jing","family":"Guo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,3,19]]},"reference":[{"key":"1134_CR1","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1080\/096031000331815","volume":"10","author":"JT Barkoulas","year":"2000","unstructured":"Barkoulas JT, Baum CF, Travlos N (2000) Long memory in the Greek stock market. Appl. Financ. Econ. 10:177\u2013184","journal-title":"Appl. Financ. Econ."},{"key":"1134_CR2","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1016\/j.cam.2018.08.044","volume":"348","author":"Y Chen","year":"2019","unstructured":"Chen Y, Zhang T (2019) A weak Galerkin finite element method for Burgers\u2019 equation. J. Comput. Appl. Math. 348:103\u2013119","journal-title":"J. Comput. Appl. Math."},{"key":"1134_CR3","volume-title":"Theory of viscoelasticity: an introduction","author":"R Christensen","year":"2012","unstructured":"Christensen R (2012) Theory of viscoelasticity: an introduction. Elsevier, Oxford"},{"key":"1134_CR4","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1007\/BF00281373","volume":"31","author":"M Gurtin","year":"1968","unstructured":"Gurtin M, Pipkin A (1968) A general theory of heat conduction with finite wave speeds. Arch. Ration. Mech. An. 31:113\u2013126","journal-title":"Arch. Ration. Mech. An."},{"key":"1134_CR5","first-page":"197","volume":"36","author":"B Jin","year":"2015","unstructured":"Jin B, Lazarov R, Zhou Z (2015) An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36:197\u2013221","journal-title":"IMA J. Numer. Anal."},{"key":"1134_CR6","doi-asserted-by":"crossref","first-page":"2004","DOI":"10.1002\/num.21786","volume":"29","author":"Q Li","year":"2013","unstructured":"Li Q, Wang J (2013) Weak Galerkin finite element methods for parabolic equations. Numer. Methods Part Differ. Equ. 29:2004\u20132024","journal-title":"Numer. Methods Part Differ. Equ."},{"key":"1134_CR7","first-page":"86","volume":"24","author":"D Li","year":"2018","unstructured":"Li D, Liao H, Sun W, Wang J, Zhang J (2018) Analysis of L1-Galerkin FEMs for time-fractional nonlinear parabolic problems. Commun. Comput. Phys. 24:86\u2013103","journal-title":"Commun. Comput. Phys."},{"key":"1134_CR8","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1137\/0727002","volume":"27","author":"J L\u00f3pez-Marcos","year":"1990","unstructured":"L\u00f3pez-Marcos J (1990) A difference scheme for a nonlinear partial integrodifferential equation. SIAM J. Numer. Anal. 27:20\u201331","journal-title":"SIAM J. Numer. Anal."},{"key":"1134_CR9","doi-asserted-by":"publisher","first-page":"704","DOI":"10.1137\/0517050","volume":"17","author":"C Lubich","year":"1986","unstructured":"Lubich C (1986) Discretized fractional calculus. SIAM J. Math. Anal. 17:704\u2013719","journal-title":"SIAM J. Math. Anal."},{"key":"1134_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1090\/S0025-5718-96-00677-1","volume":"65","author":"C Lubich","year":"1996","unstructured":"Lubich C, Sloan I, Thom\u00e9e V (1996) Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term. Math. Comput. 65:1\u201317","journal-title":"Math. Comput."},{"key":"1134_CR11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0370-1573(00)00070-3","volume":"339","author":"R Metzler","year":"2000","unstructured":"Metzler R, Klafter J (2000) The random walk\u2019s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339:1\u201377","journal-title":"Phys. Rep."},{"key":"1134_CR12","doi-asserted-by":"publisher","first-page":"R161","DOI":"10.1088\/0305-4470\/37\/31\/R01","volume":"37","author":"R Metzler","year":"2004","unstructured":"Metzler R, Klafter J (2004) The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A-Math. Theor. 37:R161","journal-title":"J. Phys. A-Math. Theor."},{"key":"1134_CR13","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/j.cam.2014.08.006","volume":"275","author":"L Mu","year":"2015","unstructured":"Mu L, Wang X, Ye X (2015) A modified weak Galerkin finite element method for the Stokes equations. J. Comput. Appl. Math. 275:79\u201390","journal-title":"J. Comput. Appl. Math."},{"key":"1134_CR14","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1016\/j.cam.2018.05.007","volume":"354","author":"L Qiao","year":"2019","unstructured":"Qiao L, Xu D, Wang Z (2019) An ADI difference scheme based on fractional trapezoidal rule for fractional integro-differential equation with a weakly singular kernel. Appl. Math. Comput. 354:103\u2013114","journal-title":"Appl. Math. Comput."},{"key":"1134_CR15","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719413","volume-title":"Mathematical analysis of viscoelastic flows","author":"M Renardy","year":"2000","unstructured":"Renardy M (2000) Mathematical analysis of viscoelastic flows. SIAM, Philadephia"},{"key":"1134_CR16","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1016\/j.apnum.2005.03.003","volume":"56","author":"Z Sun","year":"2006","unstructured":"Sun Z, Wu X (2006) A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56:193\u2013209","journal-title":"Appl. Numer. Math."},{"key":"1134_CR17","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1016\/j.cam.2012.10.003","volume":"241","author":"J Wang","year":"2013","unstructured":"Wang J, Ye X (2013) A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241:103\u2013115","journal-title":"J. Comput. Appl. Math."},{"key":"1134_CR18","doi-asserted-by":"publisher","first-page":"2101","DOI":"10.1090\/S0025-5718-2014-02852-4","volume":"83","author":"J Wang","year":"2014","unstructured":"Wang J, Ye X (2014) A weak Galerkin mixed finite element method for second order elliptic problems. Math. Comput. 83:2101\u20132126","journal-title":"Math. Comput."},{"key":"1134_CR19","first-page":"241","volume":"58","author":"D Xu","year":"1993","unstructured":"Xu D (1993) Finite element methods of the two nonlinear integro-differential equations. Appl. Math. Comput. 58:241\u2013273","journal-title":"Appl. Math. Comput."},{"key":"1134_CR20","doi-asserted-by":"publisher","first-page":"1303","DOI":"10.1016\/S0898-1221(02)00102-5","volume":"43","author":"D Xu","year":"2002","unstructured":"Xu D (2002) Uniform L1 error bounds for the semidiscrete solution of a Volterra equation with completely monotonic convolution kernel. Comput. Math. Appl. 43:1303\u20131318","journal-title":"Comput. Math. Appl."},{"key":"1134_CR21","doi-asserted-by":"publisher","first-page":"881","DOI":"10.1007\/s10444-013-9331-2","volume":"40","author":"D Xu","year":"2014","unstructured":"Xu D (2014) The long time error analysis in the second order difference type method of an evolutionary integral equation with completely monotonic kernel. Adv. Comput. Math. 40:881\u2013922","journal-title":"Adv. Comput. Math."},{"key":"1134_CR22","first-page":"51","volume":"20","author":"D Xu","year":"2020","unstructured":"Xu D (2020) Analytical and numerical solutions of a class of nonlinear integro-differential equations with L1 kernels. Nonlinear Anal. Real World Appl. 20:51","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"1134_CR23","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.cam.2016.03.025","volume":"280","author":"T Zhang","year":"2016","unstructured":"Zhang T, Tang L (2016) A weak finite element method for elliptic problems in one space dimension. Appl. Math. Comput. 280:1\u201310","journal-title":"Appl. Math. Comput."},{"key":"1134_CR24","doi-asserted-by":"publisher","first-page":"244","DOI":"10.1016\/j.camwa.2019.06.027","volume":"79","author":"J Zhou","year":"2020","unstructured":"Zhou J, Xu D (2020) Alternating direction implicit difference scheme for the multi-term time-fractional integro-differential equation with a weakly singular kernel. Comput. Math. Appl. 79:244\u2013255","journal-title":"Comput. Math. Appl."},{"key":"1134_CR25","doi-asserted-by":"publisher","first-page":"181","DOI":"10.4208\/eajam.260617.151117a","volume":"8","author":"J Zhou","year":"2018","unstructured":"Zhou J, Xu D, Chen H (2018) A weak Galerkin finite element method for multi-term time-fractional diffusion equations. East Asian J. Appl. Math. 8:181\u2013193","journal-title":"East Asian J. Appl. Math."}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-020-1134-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s40314-020-1134-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-020-1134-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,19]],"date-time":"2021-03-19T01:02:38Z","timestamp":1616115758000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s40314-020-1134-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,19]]},"references-count":25,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,5]]}},"alternative-id":["1134"],"URL":"https:\/\/doi.org\/10.1007\/s40314-020-1134-8","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,3,19]]},"assertion":[{"value":"12 June 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 February 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 March 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 March 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"109"}}