{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T02:07:00Z","timestamp":1778551620345,"version":"3.51.4"},"reference-count":24,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2021,4,14]],"date-time":"2021-04-14T00:00:00Z","timestamp":1618358400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,4,14]],"date-time":"2021-04-14T00:00:00Z","timestamp":1618358400000},"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":["12071403"],"award-info":[{"award-number":["12071403"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002767","name":"Hunan Provincial Science and Technology Department","doi-asserted-by":"publisher","award":["2018WK4006"],"award-info":[{"award-number":["2018WK4006"]}],"id":[{"id":"10.13039\/501100002767","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11601460"],"award-info":[{"award-number":["11601460"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2021,12]]},"DOI":"10.1007\/s11075-021-01102-z","type":"journal-article","created":{"date-parts":[[2021,4,14]],"date-time":"2021-04-14T01:02:41Z","timestamp":1618362161000},"page":"1965-1988","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Efficient difference method for time-space fractional diffusion equation with Robin fractional derivative boundary condition"],"prefix":"10.1007","volume":"88","author":[{"given":"Biao","family":"Zhang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Weiping","family":"Bu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2075-5040","authenticated-orcid":false,"given":"Aiguo","family":"Xiao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,4,14]]},"reference":[{"key":"1102_CR1","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1016\/0378-4371(92)90441-R","volume":"185","author":"M Giona","year":"1992","unstructured":"Giona, M., Roman, H. E.: Fractional diffusion equation for transport phenomenna in random media. Physica A. 185, 87\u201397 (1992)","journal-title":"Physica A."},{"key":"1102_CR2","doi-asserted-by":"publisher","first-page":"521","DOI":"10.1016\/S0301-0104(02)00714-0","volume":"284","author":"R Gorenflflo","year":"2002","unstructured":"Gorenflflo, R., Mainardi, F., Moretti, D., Pagnini, G., Paradisi, P.: Discrete random walk models for space-time fractional diffusion. Chem. Phys. 284, 521\u2013541 (2002)","journal-title":"Chem. Phys."},{"key":"1102_CR3","volume-title":"Fractional Differential Equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)"},{"key":"1102_CR4","doi-asserted-by":"publisher","first-page":"414","DOI":"10.1016\/j.cam.2018.03.007","volume":"339","author":"B Baeumer","year":"2018","unstructured":"Baeumer, B., Kov\u00e1cs, M., Meerschaert, M., Sankaranarayanan, H.: Reprint of: Boundary conditions for fractional diffusion. J. Comput. Appl. Math. 339, 414\u2013430 (2018)","journal-title":"J. Comput. Appl. Math."},{"key":"1102_CR5","doi-asserted-by":"publisher","first-page":"1089","DOI":"10.1016\/j.jcp.2018.10.010","volume":"376","author":"JF Kelly","year":"2019","unstructured":"Kelly, J.F., Sankaranarayanan, H., Meerschaert, M.: Boundary conditions for two-sided fractional diffusion. J. Comput. Phys. 376, 1089\u20131107 (2019)","journal-title":"J. Comput. Phys."},{"key":"1102_CR6","first-page":"399","volume":"15","author":"A Boutiara","year":"2020","unstructured":"Boutiara, A., Benbachir, M., Guerbati, K.: Caputo type fractional differential equation with nonlocal Erdelyi-Kober type integral boundary conditions in Banach spaces. Sur. Math. Appl. 15, 399\u2013418 (2020)","journal-title":"Sur. Math. Appl."},{"key":"1102_CR7","doi-asserted-by":"publisher","first-page":"8561","DOI":"10.1002\/2016WR019178","volume":"52","author":"Y Zhang","year":"2016","unstructured":"Zhang, Y., Christopher, T., Labolle, E., Neupauer, R., Sun, H.: Bounded fractional diffusion in geological media: Definition and Lagrangian approximation. Water Resour. Res. 52, 8561\u20138577 (2016)","journal-title":"Water Resour. Res."},{"key":"1102_CR8","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1007\/s10483-016-2036-6","volume":"37","author":"B Guo","year":"2016","unstructured":"Guo, B., Xu, Q., Yin, Z.: Implicit finite difference method for fractional percolation equation with Dirichlet and fractional boundary conditions. Appl. Math. Mech. 37, 403\u2013416 (2016)","journal-title":"Appl. Math. Mech."},{"key":"1102_CR9","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1016\/j.jcp.2014.08.021","volume":"293","author":"J Jia","year":"2015","unstructured":"Jia, J., Wang, H.: Fast finite difference methods for space-fractional diffusion equations with fractional derivative boundary conditions. J. Comput. Phys. 293, 359\u2013369 (2015)","journal-title":"J. Comput. Phys."},{"key":"1102_CR10","unstructured":"Liu, T., Hou, M.: Finite difference approximation for one-dimensional Riesz fractional diffusion equation with fractional boundary condition, vol. 55 (2018)"},{"issue":"4","key":"1102_CR11","doi-asserted-by":"publisher","first-page":"1383","DOI":"10.1002\/num.22355","volume":"35","author":"C Xie","year":"2019","unstructured":"Xie, C., Fang, S.: A second-order finite difference method for fractional diffusion equation with Dirichlet and fractional boundary conditions. Numer. Methods Partial Differ. Equ. 35(4), 1383\u20131395 (2019)","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"1102_CR12","doi-asserted-by":"publisher","first-page":"3473","DOI":"10.1002\/mma.6132","volume":"43","author":"C Xie","year":"2020","unstructured":"Xie, C., Fang, S.: Finite difference scheme for time-space fractional diffusion equation with fractional boundary conditions. Math. Methods Appl. Sci. 43, 3473\u20133487 (2020)","journal-title":"Math. Methods Appl. Sci."},{"key":"1102_CR13","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.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193\u2013209 (2006)","journal-title":"Appl. Numer. Math."},{"key":"1102_CR14","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, C.: Finite difference\/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533\u20131552 (2007)","journal-title":"J. Comput. Phys."},{"key":"1102_CR15","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1016\/j.jcp.2013.11.017","volume":"259","author":"G Gao","year":"2014","unstructured":"Gao, G, Sun, Z., Zhang, H.: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259, 33\u201350 (2014)","journal-title":"J. Comput. Phys."},{"key":"1102_CR16","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1016\/j.jcp.2014.09.031","volume":"280","author":"A Alikhanov","year":"2015","unstructured":"Alikhanov, A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424\u2013438 (2015)","journal-title":"J. Comput. Phys."},{"key":"1102_CR17","doi-asserted-by":"crossref","unstructured":"Stynes, M., $O^{\\prime }$Riordan, E., Gracia, J.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55(2), 1057\u20131079 (2017)","DOI":"10.1137\/16M1082329"},{"issue":"4","key":"1102_CR18","doi-asserted-by":"publisher","first-page":"815","DOI":"10.1515\/cmam-2019-0042","volume":"20","author":"C Huang","year":"2020","unstructured":"Huang, C., Liu, X., Meng, X., Stynes, M.: Error analysis of a finite difference method on graded meshes for a multiterm time-fractional initial-boundary value problem. Comput. Meth. Appl. Math. 20(4), 815\u2013825 (2020)","journal-title":"Comput. Meth. Appl. Math."},{"issue":"4","key":"1102_CR19","doi-asserted-by":"publisher","first-page":"834","DOI":"10.4208\/eajam.010418.020718","volume":"8","author":"J Shen","year":"2018","unstructured":"Shen, J., Sun, Z., Du, R.: Fast finite difference schemes for time-fractional diffusion equations with a weak singularity at initial time. East Asian J. Appl. Math. 8(4), 834\u2013858 (2018)","journal-title":"East Asian J. Appl. Math."},{"key":"1102_CR20","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/j.jcp.2014.02.008","volume":"265","author":"Y Zhang","year":"2014","unstructured":"Zhang, Y., Sun, Z., Liao, H.: Finite difference methods for the time fractional diffusion equation on non-uniform meshes. J. Comput. Phys. 265, 195\u2013210 (2014)","journal-title":"J. Comput. Phys."},{"issue":"2","key":"1102_CR21","doi-asserted-by":"publisher","first-page":"1112","DOI":"10.1137\/17M1131829","volume":"56","author":"H Liao","year":"2018","unstructured":"Liao, H., Li, D., Zhang, J., Zhao, Y.: Sharp error estimate of nonuniform L1 formula for time-fractional reaction-subdiffusion equations. SIAM J. Numer. Anal. 56(2), 1112\u20131133 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"1102_CR22","first-page":"614","volume":"316","author":"C Li","year":"2016","unstructured":"Li, C., Yi, Q., Chen, A.: Finite difference methods with non-uniform meshes for nonlinear fractional differential equations. Comput. Appl. Math. 316, 614\u2013631 (2016)","journal-title":"Comput. Appl. Math."},{"issue":"3","key":"1102_CR23","doi-asserted-by":"publisher","first-page":"650","DOI":"10.4208\/cicp.OA-2016-0136","volume":"21","author":"S Jiang","year":"2017","unstructured":"Jiang, S., Zhang, J., Zhang, Q., Zhang, Z.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations. Commun. Comput. Phys. 21(3), 650\u2013678 (2017)","journal-title":"Commun. Comput. Phys."},{"key":"1102_CR24","volume-title":"Fractional Integrals and Derivatives: Theory and Applications","author":"S Samko","year":"1993","unstructured":"Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon Breach, Yverdon (1993)"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-021-01102-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-021-01102-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-021-01102-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,16]],"date-time":"2021-11-16T09:23:17Z","timestamp":1637054597000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-021-01102-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,14]]},"references-count":24,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,12]]}},"alternative-id":["1102"],"URL":"https:\/\/doi.org\/10.1007\/s11075-021-01102-z","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,14]]},"assertion":[{"value":"7 September 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 March 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 April 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}