{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T02:52:17Z","timestamp":1773370337700,"version":"3.50.1"},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2019,3,5]],"date-time":"2019-03-05T00:00:00Z","timestamp":1551744000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"name":"National Sciences Foundation of China","award":["11671131"],"award-info":[{"award-number":["11671131"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2019,6]]},"DOI":"10.1007\/s40314-019-0807-7","type":"journal-article","created":{"date-parts":[[2019,3,5]],"date-time":"2019-03-05T18:02:07Z","timestamp":1551808927000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Weak Galerkin finite element method for the parabolic integro-differential equation with weakly singular kernel"],"prefix":"10.1007","volume":"38","author":[{"given":"Jun","family":"Zhou","sequence":"first","affiliation":[]},{"given":"Da","family":"Xu","sequence":"additional","affiliation":[]},{"given":"Xiuxiu","family":"Dai","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,3,5]]},"reference":[{"key":"807_CR1","first-page":"717","volume":"20","author":"Y Aimin","year":"2016","unstructured":"Aimin Y, Yang H, Jie L et al (2016) On steady heat flow problem involving Yang-Srivastava-Machado fractional derivative without singular kernel. Therm Sci 20:717\u2013S721","journal-title":"Therm Sci"},{"key":"807_CR2","first-page":"141","volume":"28","author":"H Chen","year":"2006","unstructured":"Chen H, Xu D (2006) A second-order fully discrete difference scheme for a partial integro-differential equation (in Chinese). Math Numer Sin 28:141\u2013154","journal-title":"Math Numer Sin"},{"key":"807_CR3","doi-asserted-by":"publisher","first-page":"559","DOI":"10.4208\/nmtma.2012.m11032","volume":"5","author":"H Chen","year":"2012","unstructured":"Chen H, Xu D (2012) A compact difference scheme for an evolution equation with a weakly singular kernel. Numer Math Theory Methods Appl 5:559\u2013572","journal-title":"Numer Math Theory Methods Appl"},{"key":"807_CR4","doi-asserted-by":"publisher","first-page":"587","DOI":"10.1090\/S0025-5718-1992-1122059-2","volume":"58","author":"C Chen","year":"1992","unstructured":"Chen C, Thomee V, Wahlbin LB (1992) Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel. Math Comp 58:587\u2013602","journal-title":"Math Comp"},{"key":"807_CR5","unstructured":"Chen Y, Zhang T (2016) A weak Galerkin finite element method for Burgers\u2019 equation. arXiv:1607.05622"},{"key":"807_CR6","doi-asserted-by":"publisher","first-page":"195","DOI":"10.4208\/jcm.1401-m4385","volume":"32","author":"F Gao","year":"2014","unstructured":"Gao F, Mu L (2014) On $$L^2$$ L 2 error estimate for weak Galerkin finte element for parabolic problems. J Comp Math 32:195\u2013204","journal-title":"J Comp Math"},{"key":"807_CR7","doi-asserted-by":"publisher","first-page":"6130","DOI":"10.1002\/mma.5124","volume":"41","author":"H Li","year":"2018","unstructured":"Li H, Jiang W (2018) A space-time spectral collocation method for the two-dimensional nonlinear Riesz space fractional diffusion equations. Math Methods Appl Sci 41:6130\u20136144","journal-title":"Math Methods Appl Sci"},{"key":"807_CR8","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 PDEs 29:2004\u20132024","journal-title":"Numer Methods PDEs"},{"key":"807_CR9","doi-asserted-by":"publisher","first-page":"855","DOI":"10.1016\/j.camwa.2011.02.045","volume":"62","author":"C Li","year":"2011","unstructured":"Li C, Zhao Z, Chen Y (2011) Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comp Math Appl 62:855\u2013875","journal-title":"Comp Math Appl"},{"key":"807_CR10","doi-asserted-by":"publisher","first-page":"2108","DOI":"10.1137\/080718942","volume":"47","author":"Y Lin","year":"2009","unstructured":"Lin Y, Xu C (2009) A space-time spectral method for the time fractional diffusion equation. SIAM J Numer Anal 47:2108\u20132131","journal-title":"SIAM J Numer Anal"},{"key":"807_CR11","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1137\/0727002","volume":"27","author":"JC Lopez-Marcos","year":"1990","unstructured":"Lopez-Marcos JC (1990) A difference scheme for a nonlinear partial integro-differential equation. SIAM J Numer Anal 27:20\u201331","journal-title":"SIAM J Numer Anal"},{"key":"807_CR12","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1017\/S0334270000007268","volume":"35","author":"W Mclean","year":"1993","unstructured":"Mclean W, Thomee V (1993) Numerical solution of an evolution equation with a positive type memory term. J Aust Math Soc Ser B 35:23\u201370","journal-title":"J Aust Math Soc Ser B"},{"key":"807_CR13","volume-title":"Fractional differential equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny I (1999) Fractional differential equations. Academic Press, San Diego"},{"key":"807_CR14","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1016\/0168-9274(93)90012-G","volume":"11","author":"T Tang","year":"1993","unstructured":"Tang T (1993) A finite difference scheme for partial integro-differential equations with a weakly singular kernel. Appl Numer Math 11:309\u2013319","journal-title":"Appl Numer Math"},{"key":"807_CR15","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":"807_CR16","doi-asserted-by":"publisher","first-page":"723","DOI":"10.1137\/0710062","volume":"10","author":"MF Wheeler","year":"1973","unstructured":"Wheeler MF (1973) A priori $$L^2$$ L 2 error estimates for Galerkin approximations to parabolic partial differential equations. SIMA J Numer Anal 10:723\u2013759","journal-title":"SIMA J Numer Anal"},{"key":"807_CR17","doi-asserted-by":"publisher","first-page":"1161","DOI":"10.2298\/TSCI161216326Y","volume":"21","author":"X Yang","year":"2017","unstructured":"Yang X (2017) Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems. Therm Sci 21:1161\u20131171","journal-title":"Therm Sci"},{"key":"807_CR18","first-page":"45","volume":"19","author":"X Yang","year":"2018","unstructured":"Yang X (2018) New rheological problems involving general fractional derivatives with nonsingular power-law kernels. Proc Rom Acad Ser A Math Phys Tech Sci Inf Sci 19:45\u201352","journal-title":"Proc Rom Acad Ser A Math Phys Tech Sci Inf Sci"},{"key":"807_CR19","doi-asserted-by":"publisher","first-page":"276","DOI":"10.1016\/j.physa.2017.04.054","volume":"481","author":"X Yang","year":"2017","unstructured":"Yang X, Machado J (2017) A new fractional operator of variable order: application in the description of anomalous diffusion. Phys A Stat Mech Appl 481:276\u2013283","journal-title":"Phys A Stat Mech Appl"},{"key":"807_CR20","doi-asserted-by":"publisher","first-page":"824","DOI":"10.1016\/j.jcp.2013.09.016","volume":"256","author":"X Yang","year":"2014","unstructured":"Yang X, Zhang H, Xu D (2014) Orthogonal spline collocation method for the two-dimensional fractional sub-diffusion equation. J Comp Phys 256:824\u2013837","journal-title":"J Comp Phys"},{"key":"807_CR21","doi-asserted-by":"publisher","first-page":"753","DOI":"10.2298\/TSCI151224222Y","volume":"20","author":"X Yang","year":"2016","unstructured":"Yang X, Srivastava H, Tenreiro Machado J (2016) A new fractional derivative without singular kernel: application to the modelling of the steady heat flow. Therm Sci 20:753\u2013756","journal-title":"Therm Sci"},{"key":"807_CR22","doi-asserted-by":"publisher","first-page":"3567","DOI":"10.1140\/epjst\/e2018-00020-2","volume":"226","author":"X Yang","year":"2017","unstructured":"Yang X, Gao F, Machado J et al (2017) A new fractional derivative involving the normalized sinc function without singular kernel. Eur Phys J Spec Top 226:3567\u20133575","journal-title":"Eur Phys J Spec Top"},{"key":"807_CR23","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1016\/j.cam.2017.10.007","volume":"339","author":"X Yang","year":"2018","unstructured":"Yang X, Gao F, Srivastava H (2018) A new computational approach for solving nonlinear local fractional PDEs. J Comp Appl Math 339:285\u2013296","journal-title":"J Comp Appl Math"},{"key":"807_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.amc.2016.01.018","volume":"280","author":"T Zhang","year":"2016","unstructured":"Zhang T, Tang L (2016) A weak finite element method for elliptic problem in one space dimension. Appl Math Comp 280:1\u201310","journal-title":"Appl Math Comp"},{"key":"807_CR25","doi-asserted-by":"publisher","first-page":"2011","DOI":"10.1080\/00207160.2017.1347259","volume":"95","author":"H Zhang","year":"2018","unstructured":"Zhang H, Yang X (2018) The BDF orthogonal spline collocation method for the two dimensional evolution equation with memory. Int J Comp Math 95:2011\u20132025","journal-title":"Int J Comp Math"},{"key":"807_CR26","doi-asserted-by":"publisher","first-page":"1710","DOI":"10.1016\/j.camwa.2014.10.019","volume":"68","author":"H Zhang","year":"2014","unstructured":"Zhang H, Yang X, Han X (2014) Discrete-time orthogonal spline collocation method with application to two-dimensional fractional cable equation. Comp Math Appl 68:1710\u20131722","journal-title":"Comp Math Appl"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-019-0807-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s40314-019-0807-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-019-0807-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,11,29]],"date-time":"2020-11-29T06:59:57Z","timestamp":1606633197000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s40314-019-0807-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,5]]},"references-count":26,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,6]]}},"alternative-id":["807"],"URL":"https:\/\/doi.org\/10.1007\/s40314-019-0807-7","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,3,5]]},"assertion":[{"value":"23 October 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 December 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 January 2019","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 March 2019","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"38"}}