{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T01:42:43Z","timestamp":1768786963976,"version":"3.49.0"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,2,17]],"date-time":"2021-02-17T00:00:00Z","timestamp":1613520000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,2,17]],"date-time":"2021-02-17T00:00:00Z","timestamp":1613520000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2021,4]]},"DOI":"10.1007\/s10915-021-01412-3","type":"journal-article","created":{"date-parts":[[2021,2,19]],"date-time":"2021-02-19T07:08:45Z","timestamp":1613718525000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Finite Element Method for Fractional Parabolic Integro-Differential Equations with Smooth and Nonsmooth Initial Data"],"prefix":"10.1007","volume":"87","author":[{"given":"Shantiram","family":"Mahata","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5910-8943","authenticated-orcid":false,"given":"Rajen Kumar","family":"Sinha","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,2,17]]},"reference":[{"issue":"2","key":"1412_CR1","first-page":"161","volume":"54","author":"YA Amer","year":"2018","unstructured":"Amer, Y.A., Mahdy, A.M.S., Youssef, E.S.M.: Solving fractional integro-differential equations by using Sumudu transform method and Hermite spectral collocation method. CMC Comput. Mater. Continua 54(2), 161\u2013180 (2018)","journal-title":"CMC Comput. Mater. Continua"},{"issue":"2","key":"1412_CR2","doi-asserted-by":"publisher","first-page":"521","DOI":"10.1016\/j.chaos.2007.08.001","volume":"40","author":"A Arikoglu","year":"2009","unstructured":"Arikoglu, A., Ozkol, I.: Solution of fractional integro-differential equations by using fractional differential transform method. Chaos Solitons Fractals 40(2), 521\u2013529 (2009)","journal-title":"Chaos Solitons Fractals"},{"issue":"1","key":"1412_CR3","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1051\/m2an\/1998320100851","volume":"32","author":"NY Bakaev","year":"1998","unstructured":"Bakaev, N.Y., Larsson, S., Thom\u00e9e, V.: Euler, backward, type methods for parabolic integro-differential equations in Banach space. RAIRO Mod\u00e9l. Math. Anal. Num\u00e9r. 32(1), 85\u201399 (1998)","journal-title":"RAIRO Mod\u00e9l. Math. Anal. Num\u00e9r."},{"issue":"12","key":"1412_CR4","doi-asserted-by":"publisher","first-page":"4587","DOI":"10.1016\/j.na.2010.02.035","volume":"72","author":"K Balachandran","year":"2010","unstructured":"Balachandran, K., Trujillo, J.J.: The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. Nonlinear Anal. 72(12), 4587\u20134593 (2010)","journal-title":"Nonlinear Anal."},{"issue":"1","key":"1412_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00211-014-0685-2","volume":"131","author":"E Bazhlekova","year":"2015","unstructured":"Bazhlekova, E., Jin, B., Lazarov, R., Zhou, Z.: An analysis of the Rayleigh\u2013Stokes problem for a generalized second-grade fluid. Numer. Math. 131(1), 1\u201331 (2015)","journal-title":"Numer. Math."},{"issue":"3","key":"1412_CR6","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1007\/BF02575943","volume":"25","author":"JR Cannon","year":"1988","unstructured":"Cannon, J.R., Lin, Y.: Nonclassical $$H^1$$ projection and Galerkin methods for nonlinear parabolic integro-differential equations. Calcolo 25(3), 187\u2013201 (1988)","journal-title":"Calcolo"},{"key":"1412_CR7","doi-asserted-by":"publisher","DOI":"10.1142\/3594","volume-title":"Finite Element Methods for Integrodifferential Equations","author":"C Chen","year":"1998","unstructured":"Chen, C., Shih, T.: Finite Element Methods for Integrodifferential Equations, vol. 9. World Scientific, River Edge (1998)"},{"issue":"254","key":"1412_CR8","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1090\/S0025-5718-06-01788-1","volume":"75","author":"E Cuesta","year":"2006","unstructured":"Cuesta, E., Lubich, C., Palencia, C.: Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comput. 75(254), 673\u2013696 (2006)","journal-title":"Math. Comput."},{"issue":"1","key":"1412_CR9","doi-asserted-by":"publisher","first-page":"1460030","DOI":"10.1080\/25742558.2018.1460030","volume":"5","author":"MM El-Borai","year":"2018","unstructured":"El-Borai, M.M., El-Nadi, K.E.S., Ahmed, H.M., El-Owaidy, H.M., Ghanem, A.S., Sakthivel, R.: Existence and stability for fractional parabolic integro-partial differential equations with fractional Brownian motion and nonlocal condition. Cogent Math. Stat. 5(1), 1460030 (2018)","journal-title":"Cogent Math. Stat."},{"key":"1412_CR10","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4355-5","volume-title":"Theory and Practice of Finite Elements","author":"A Ern","year":"2004","unstructured":"Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements, vol. 159. Springer, New York (2004)"},{"key":"1412_CR11","doi-asserted-by":"crossref","unstructured":"Fujita, H., Suzuki, T.: Evolution problems. In: Handbook of Numerical Analysis, vol. II, Handb. Numer. Anal., II, pp. 789\u2013928. North-Holland, Amsterdam (1991)","DOI":"10.1016\/S1570-8659(05)80043-2"},{"issue":"3\u20134","key":"1412_CR12","first-page":"251","volume":"20","author":"F Hecht","year":"2012","unstructured":"Hecht, F.: New development in freefem++. J. Numer. Math. 20(3\u20134), 251\u2013265 (2012)","journal-title":"J. Numer. Math."},{"issue":"3","key":"1412_CR13","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1007\/s00233-009-9164-y","volume":"79","author":"L Hu","year":"2009","unstructured":"Hu, L., Ren, Y., Sakthivel, R.: Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays. Semigr. Forum 79(3), 507\u2013514 (2009)","journal-title":"Semigr. Forum"},{"issue":"3","key":"1412_CR14","doi-asserted-by":"publisher","first-page":"1127","DOI":"10.1016\/j.camwa.2011.03.037","volume":"62","author":"L Huang","year":"2011","unstructured":"Huang, L., Li, X.-F., Zhao, Y., Duan, X.-Y.: Approximate solution of fractional integro-differential equations by Taylor expansion method. Comput. Math. Appl. 62(3), 1127\u20131134 (2011)","journal-title":"Comput. Math. Appl."},{"issue":"1","key":"1412_CR15","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1137\/120873984","volume":"51","author":"B Jin","year":"2013","unstructured":"Jin, B., Lazarov, R., Zhou, Z.: Error estimates for a semidiscrete finite element method for fractional order parabolic equations. SIAM. J. Numer. Anal 51(1), 445\u2013466 (2013)","journal-title":"SIAM. J. Numer. Anal"},{"issue":"1","key":"1412_CR16","doi-asserted-by":"publisher","first-page":"A146","DOI":"10.1137\/140979563","volume":"38","author":"B Jin","year":"2016","unstructured":"Jin, B., Lazarov, R., Zhou, Z.: Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM. J. Sci. Comput. 38(1), A146\u2013A170 (2016)","journal-title":"SIAM. J. Sci. Comput."},{"issue":"1","key":"1412_CR17","doi-asserted-by":"publisher","first-page":"518","DOI":"10.1093\/imanum\/drx019","volume":"38","author":"B Jin","year":"2018","unstructured":"Jin, B., Li, B., Zhou, Z.: An analysis of the Crank\u2013Nicolson method for subdiffusion. IMA J. Numer. Anal. 38(1), 518\u2013541 (2018)","journal-title":"IMA J. Numer. Anal."},{"issue":"1","key":"1412_CR18","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1007\/s10915-017-0450-7","volume":"74","author":"S Karaa","year":"2018","unstructured":"Karaa, S., Mustapha, K., Pani, A.K.: Optimal error analysis of a FEM for fractional diffusion problems by energy arguments. J. Sci. Comput. 74(1), 519\u2013535 (2018)","journal-title":"J. Sci. Comput."},{"key":"1412_CR19","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-0208(06)80001-0","volume-title":"Theory and Applications of Fractional Differential Equations","author":"AA Kilbas","year":"2006","unstructured":"Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204. Elsevier, Amsterdam (2006)"},{"issue":"4","key":"1412_CR20","doi-asserted-by":"publisher","first-page":"1047","DOI":"10.1137\/0728056","volume":"28","author":"Y Lin","year":"1991","unstructured":"Lin, Y., Thom\u00e9e, V., Wahlbin, L.B.: Ritz\u2013Volterra projections to finite-element spaces and applications to integrodifferential and related equations. SIAM. J. Numer. Anal. 28(4), 1047\u20131070 (1991)","journal-title":"SIAM. J. Numer. Anal."},{"issue":"3","key":"1412_CR21","doi-asserted-by":"publisher","first-page":"704","DOI":"10.1137\/0517050","volume":"17","author":"C Lubich","year":"1986","unstructured":"Lubich, C.: Discretized fractional calculus. SIAM. J. Math. Anal. 17(3), 704\u2013719 (1986)","journal-title":"SIAM. J. Math. Anal."},{"issue":"2","key":"1412_CR22","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1007\/BF01398686","volume":"52","author":"C Lubich","year":"1988","unstructured":"Lubich, C.: Convolution quadrature and discretized operational calculus. I. Numer. Math. 52(2), 129\u2013145 (1988)","journal-title":"Numer. Math."},{"issue":"3","key":"1412_CR23","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1023\/B:BITN.0000046813.23911.2d","volume":"44","author":"C Lubich","year":"2004","unstructured":"Lubich, C.: Convolution quadrature revisited. BIT 44(3), 503\u2013514 (2004)","journal-title":"BIT"},{"issue":"213","key":"1412_CR24","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.H., Thom\u00e9e, V.: Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term. Math. Comput. 65(213), 1\u201317 (1996)","journal-title":"Math. Comput."},{"issue":"12","key":"1412_CR25","doi-asserted-by":"crossref","first-page":"6750","DOI":"10.1016\/j.amc.2012.12.072","volume":"219","author":"X Ma","year":"2013","unstructured":"Ma, X., Huang, C.: Numerical solution of fractional integro-differential equations by a hybrid collocation method. Appl. Math. Comput. 219(12), 6750\u20136760 (2013)","journal-title":"Appl. Math. Comput."},{"issue":"4","key":"1412_CR26","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1216\/jie.2020.32.457","volume":"32","author":"S Mahata","year":"2020","unstructured":"Mahata, S., Sinha, R.K.: On the existence, uniqueness and stability results for time-fractional parabolic integro-differential equations. J. Integral Equ. Appl. 32(4), 457\u2013477 (2020)","journal-title":"J. Integral Equ. Appl."},{"key":"1412_CR27","unstructured":"Maleknejad, K., Sahlan, M.N., Ostadi, A.: Numerical solution of fractional integro-differential equation by using cubic B-spline wavelets. In: Proceedings of the World Congress on Engineering, vol. 1 (2013)"},{"key":"1412_CR28","doi-asserted-by":"crossref","unstructured":"Mohammed, D.S.: Numerical solution of fractional integro-differential equations by least squares method and shifted Chebyshev polynomial. Math. Probl. Eng. Art. ID 431965, 5 (2014)","DOI":"10.1155\/2014\/431965"},{"issue":"1","key":"1412_CR29","doi-asserted-by":"crossref","first-page":"754","DOI":"10.1016\/j.amc.2006.04.041","volume":"182","author":"S Momani","year":"2006","unstructured":"Momani, S., Noor, M.A.: Numerical methods for fourth-order fractional integro-differential equations. Appl. Math. Comput. 182(1), 754\u2013760 (2006)","journal-title":"Appl. Math. Comput."},{"issue":"313","key":"1412_CR30","doi-asserted-by":"publisher","first-page":"2259","DOI":"10.1090\/mcom\/3304","volume":"87","author":"K Mustapha","year":"2018","unstructured":"Mustapha, K.: FEM for time-fractional diffusion equations, novel optimal error analyses. Math. Comput. 87(313), 2259\u20132272 (2018)","journal-title":"Math. Comput."},{"issue":"3","key":"1412_CR31","first-page":"1000142","volume":"3","author":"T Oyedepo","year":"2016","unstructured":"Oyedepo, T., Taiwo, O.A., Abubakar, J.U., Ogunwobi, Z.O.: Numerical studies for solving fractional integro-differential equations by using least squares method and Bernstein polynomials. Fluid Mech. Open Access 3(3), 1000142 (2016)","journal-title":"Fluid Mech. Open Access"},{"key":"1412_CR32","volume-title":"Fractional Differential Equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny, I.: Fractional Differential Equations, vol. 198. Academic Press, San Diego (1999)"},{"key":"1412_CR33","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1016\/j.apnum.2020.01.003","volume":"151","author":"L Qiao","year":"2020","unstructured":"Qiao, L., Wang, Z., Xu, D.: An alternating direction implicit orthogonal spline collocation method for the two dimensional multi-term time fractional integro-differential equation. Appl. Numer. Math. 151, 199\u2013212 (2020)","journal-title":"Appl. Numer. Math."},{"issue":"10","key":"1412_CR34","doi-asserted-by":"publisher","first-page":"2055","DOI":"10.1080\/00207160.2019.1677896","volume":"97","author":"W Qiu","year":"2020","unstructured":"Qiu, W., Xu, D., Chen, H.: A formally second-order BDF finite difference scheme for the integro-differential equations with the multi-term kernels. Int. J. Comput. Math. 97(10), 2055\u20132073 (2020)","journal-title":"Int. J. Comput. Math."},{"issue":"1","key":"1412_CR35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.amc.2005.09.059","volume":"176","author":"EA Rawashdeh","year":"2006","unstructured":"Rawashdeh, E.A.: Numerical solution of fractional integro-differential equations by collocation method. Appl. Math. Comput. 176(1), 1\u20136 (2006)","journal-title":"Appl. Math. Comput."},{"issue":"13","key":"1412_CR36","doi-asserted-by":"publisher","first-page":"2050","DOI":"10.1177\/1077546310395977","volume":"17","author":"A Saadatmandi","year":"2011","unstructured":"Saadatmandi, A., Dehghan, M.: A Legendre collocation method for fractional integro-differential equations. J. Vib. Control 17(13), 2050\u20132058 (2011)","journal-title":"J. Vib. Control"},{"issue":"2","key":"1412_CR37","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1137\/0725022","volume":"25","author":"JM Sanz-Serna","year":"1988","unstructured":"Sanz-Serna, J.M.: A numerical method for a partial integro-differential equation. SIAM J. Numer. Anal. 25(2), 319\u2013327 (1988)","journal-title":"SIAM J. Numer. Anal."},{"key":"1412_CR38","volume-title":"Galerkin Finite Element Methods for Parabolic Problems","author":"V Thom\u00e9e","year":"2006","unstructured":"Thom\u00e9e, V.: Galerkin Finite Element Methods for Parabolic Problems, vol. 25. Springer, Berlin (2006)"},{"issue":"187","key":"1412_CR39","doi-asserted-by":"publisher","first-page":"121","DOI":"10.2307\/2008352","volume":"53","author":"V Thom\u00e9e","year":"1989","unstructured":"Thom\u00e9e, V., Zhang, N.-Y.: Error estimates for semidiscrete finite element methods for parabolic integro-differential equations. Math. Comput. 53(187), 121\u2013139 (1989)","journal-title":"Math. Comput."},{"issue":"1","key":"1412_CR40","doi-asserted-by":"publisher","first-page":"73","DOI":"10.26637\/MJM0601\/0011","volume":"6","author":"SI Unhale","year":"2018","unstructured":"Unhale, S.I., Kendre, S.D.: Numerical solution of nonlinear fractional integro-differential equation by collocation method, 2018. Malaya J. Mat. 6(1), 73\u201379 (2018)","journal-title":"Malaya J. Mat."},{"issue":"04","key":"1412_CR41","doi-asserted-by":"publisher","first-page":"1750071","DOI":"10.1142\/S1793557117500711","volume":"10","author":"S Zaeri","year":"2017","unstructured":"Zaeri, S., Saeedi, H., Izadi, M.: Fractional integration operator for numerical solution of the integro-partial time fractional diffusion heat equation with weakly singular kernel. Asian-Eur. J. Math. 10(04), 1750071 (2017)","journal-title":"Asian-Eur. J. Math."},{"issue":"2","key":"1412_CR42","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.: Alternating direction implicit difference scheme for the multi-term time-fractional integro-differential equation with a weakly singular kernel. Comput. Math. Appl. 79(2), 244\u2013255 (2020)","journal-title":"Comput. Math. Appl."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01412-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-021-01412-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01412-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,30]],"date-time":"2023-10-30T13:31:59Z","timestamp":1698672719000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-021-01412-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,17]]},"references-count":42,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,4]]}},"alternative-id":["1412"],"URL":"https:\/\/doi.org\/10.1007\/s10915-021-01412-3","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,2,17]]},"assertion":[{"value":"3 September 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 December 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 January 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 February 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"7"}}