{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T12:21:36Z","timestamp":1777292496816,"version":"3.51.4"},"reference-count":22,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2022,9,12]],"date-time":"2022-09-12T00:00:00Z","timestamp":1662940800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,9,12]],"date-time":"2022-09-12T00:00:00Z","timestamp":1662940800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12101089"],"award-info":[{"award-number":["12101089"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100018542","name":"Natural Science Foundation of Sichuan Province","doi-asserted-by":"publisher","award":["2022NSFSC1844"],"award-info":[{"award-number":["2022NSFSC1844"]}],"id":[{"id":"10.13039\/501100018542","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2022,10]]},"DOI":"10.1007\/s40314-022-02022-x","type":"journal-article","created":{"date-parts":[[2022,9,12]],"date-time":"2022-09-12T12:02:36Z","timestamp":1662984156000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A high-order and extra-dof-free generalized finite element method for time-fractional diffusion equation"],"prefix":"10.1007","volume":"41","author":[{"given":"Yi","family":"Yang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2191-4739","authenticated-orcid":false,"given":"Jin","family":"Huang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hu","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,9,12]]},"reference":[{"key":"2022_CR1","volume-title":"Sobolev Spaces","author":"RA Adams","year":"1975","unstructured":"Adams RA (1975) Sobolev Spaces. Academic Press, New York-London"},{"key":"2022_CR2","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1016\/j.jcp.2014.09.031","volume":"280","author":"AA Alikhanov","year":"2015","unstructured":"Alikhanov AA (2015) A new difference scheme for the time fractional diffusion equation. J Comput Phys 280:424\u2013438","journal-title":"J Comput Phys"},{"key":"2022_CR3","doi-asserted-by":"publisher","first-page":"727","DOI":"10.1002\/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N","volume":"40","author":"I Babu\u0161ka","year":"1997","unstructured":"Babu\u0161ka I, Melenk JM (1997) The partition of unity method. Int J Numer Methods Eng 40:727\u2013758","journal-title":"Int J Numer Methods Eng"},{"key":"2022_CR4","doi-asserted-by":"publisher","first-page":"558","DOI":"10.1002\/num.20112","volume":"22","author":"VJ Ervin","year":"2006","unstructured":"Ervin VJ, Roop JP (2006) Variational formulation for the stationary fractional advection dispersion equation. Numer Meth Part D E 22:558\u2013576","journal-title":"Numer Meth Part D E"},{"key":"2022_CR5","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1016\/j.jcp.2013.11.017","volume":"259","author":"GH Gao","year":"2014","unstructured":"Gao GH, Sun ZZ, Zhang HW (2014) A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J Comput Phys 259:33\u201350","journal-title":"J Comput Phys"},{"key":"2022_CR6","doi-asserted-by":"publisher","first-page":"799","DOI":"10.1515\/fca-2015-0048","volume":"18","author":"R Gorenflo","year":"2015","unstructured":"Gorenflo R, Luchko Y, Yamamoto M (2015) Time-fractional diffusion equation in the fractional Sobolev spaces. Fract Calc Appl Anal 18:799\u2013820","journal-title":"Fract Calc Appl Anal"},{"key":"2022_CR7","volume-title":"Applications of fractional calculus in physics","year":"2000","unstructured":"Hilfer R (ed) (2000) Applications of fractional calculus in physics. World Scientific, Singapore"},{"key":"2022_CR8","doi-asserted-by":"crossref","unstructured":"Huang CB, An N, Yu XJ (2020) A local discontinuous Galerkin method for time-fractional diffusion equation with discontinuous coefficient. Appl Numer Math 151","DOI":"10.1016\/j.apnum.2019.11.003"},{"key":"2022_CR9","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1137\/120873984","volume":"51","author":"BT Jin","year":"2013","unstructured":"Jin BT, Lazarov R, Zhou Z (2013) Error estimates for a semidiscrete finite element method for fractional order parabolic equations. SIAM J Numer Anal 51:445\u2013466","journal-title":"SIAM J Numer Anal"},{"key":"2022_CR10","volume-title":"Theory and applications of fractional differential equations","author":"AA Kilbas","year":"2006","unstructured":"Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam"},{"key":"2022_CR11","doi-asserted-by":"publisher","first-page":"276","DOI":"10.1515\/fca-2018-0018","volume":"21","author":"A Kubica","year":"2018","unstructured":"Kubica A, Yamamoto M (2018) Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients. Fract Calc Appl Anal 21:276\u2013311","journal-title":"Fract Calc Appl Anal"},{"key":"2022_CR12","doi-asserted-by":"publisher","first-page":"1533","DOI":"10.1016\/j.jcp.2007.02.001","volume":"225","author":"Y Lin","year":"2007","unstructured":"Lin Y, Xu CJ (2007) Finite difference\/spectral approximations for the time-fractional diffusion equation. J Comput Phys 225:1533\u20131552","journal-title":"J Comput Phys"},{"key":"2022_CR13","doi-asserted-by":"publisher","first-page":"888","DOI":"10.1515\/fca-2016-0048","volume":"19","author":"Y Liu","year":"2016","unstructured":"Liu Y, Rundell W, Yammamoto M (2016) Strong maximum principle for fractional diffusion equations and application to an inverse source problem. Fract Calc Appl Anal 19:888\u2013906","journal-title":"Fract Calc Appl Anal"},{"key":"2022_CR14","doi-asserted-by":"publisher","first-page":"1766","DOI":"10.1016\/j.camwa.2009.08.015","volume":"59","author":"Y Luchko","year":"2010","unstructured":"Luchko Y (2010) Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation. Comput Math Appl 59:1766\u20131772","journal-title":"Comput Math Appl"},{"key":"2022_CR15","doi-asserted-by":"publisher","first-page":"141","DOI":"10.2478\/s13540-012-0010-7","volume":"15","author":"Y Luchko","year":"2012","unstructured":"Luchko Y (2012) Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation. Fract Calc Appl Anal 15:141\u2013160","journal-title":"Fract Calc Appl Anal"},{"key":"2022_CR16","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1016\/S0045-7825(96)01087-0","volume":"139","author":"JM Melenk","year":"1996","unstructured":"Melenk JM, Babu\u0161ka I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139:289\u2013314","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2022_CR17","volume-title":"Fractional differential equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny I (1999) Fractional differential equations. Academic Press, San Diego"},{"key":"2022_CR18","doi-asserted-by":"crossref","unstructured":"Tian R (2013) Extra-dof-free and linearly independent enrichments in gfem. Comput Methods Appl Mech 536 Eng 266:1\u201322","DOI":"10.1016\/j.cma.2013.07.005"},{"key":"2022_CR19","doi-asserted-by":"publisher","first-page":"426","DOI":"10.1016\/j.jmaa.2011.04.058","volume":"382","author":"K Sakamoto","year":"2011","unstructured":"Sakamoto K, Yamamoto M (2011) Initial value\/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J Math Anal Appl 382:426\u2013447","journal-title":"J Math Anal Appl"},{"key":"2022_CR20","doi-asserted-by":"crossref","unstructured":"Toprakseven S (2021) A weak Galerkin finite element method for time fractional reaction-diffusion-convection problems with variable coefficients. Appl Numer Math","DOI":"10.1016\/j.apnum.2021.05.021"},{"key":"2022_CR21","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/j.jcp.2014.02.008","volume":"265","author":"YN Zhang","year":"2014","unstructured":"Zhang YN, Sun ZZ, Liao HL (2014) Finite difference methods for the time fractional diffusion equation on non-uniform meshes. J Comput Phys 265:195\u2013210","journal-title":"J Comput Phys"},{"key":"2022_CR22","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1016\/j.jcp.2018.08.033","volume":"375","author":"T Zhang","year":"2018","unstructured":"Zhang T, Guo QX (2018) The finite difference\/finite volume method for solving the fractional diffusion equation. J Comput Phys 375:120\u2013134","journal-title":"J Comput Phys"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-022-02022-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-022-02022-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-022-02022-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,7]],"date-time":"2022-10-07T19:22:44Z","timestamp":1665170564000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-022-02022-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,12]]},"references-count":22,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2022,10]]}},"alternative-id":["2022"],"URL":"https:\/\/doi.org\/10.1007\/s40314-022-02022-x","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,12]]},"assertion":[{"value":"20 December 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 August 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 September 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 September 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"308"}}