{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T10:27:08Z","timestamp":1779100028749,"version":"3.51.4"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2018,2,17]],"date-time":"2018-02-17T00:00:00Z","timestamp":1518825600000},"content-version":"tdm","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":[[2018,9]]},"DOI":"10.1007\/s10915-018-0672-3","type":"journal-article","created":{"date-parts":[[2018,2,17]],"date-time":"2018-02-17T04:48:31Z","timestamp":1518842911000},"page":"1502-1520","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":31,"title":["WSGD-OSC Scheme for Two-Dimensional Distributed Order Fractional Reaction\u2013Diffusion Equation"],"prefix":"10.1007","volume":"76","author":[{"given":"Xuehua","family":"Yang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haixiang","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Da","family":"Xu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2018,2,17]]},"reference":[{"key":"672_CR1","first-page":"409","volume":"12","author":"Y Luchko","year":"2009","unstructured":"Luchko, Y.: Boundary value problems for the generalized time fractional diffusion equation of distributed order. Fract. Calc. Appl. Anal. 12, 409\u2013422 (2009)","journal-title":"Fract. Calc. Appl. Anal."},{"key":"672_CR2","doi-asserted-by":"crossref","first-page":"110","DOI":"10.2478\/s13540-011-0008-6","volume":"14","author":"Y Luchko","year":"2011","unstructured":"Luchko, Y.: Maximum principle and its application for the time-fractional diffusion equations. Fract. Calc. Appl. Anal. 14, 110\u2013124 (2011)","journal-title":"Fract. Calc. Appl. Anal."},{"key":"672_CR3","doi-asserted-by":"publisher","first-page":"1114","DOI":"10.2478\/s13540-014-0217-x","volume":"17","author":"Z Li","year":"2014","unstructured":"Li, Z., Luchko, Y., Yamamoto, M.: Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations. Fract. Calc. Appl. Anal. 17, 1114\u20131136 (2014)","journal-title":"Fract. Calc. Appl. Anal."},{"key":"672_CR4","doi-asserted-by":"publisher","first-page":"605","DOI":"10.3934\/cpaa.2014.13.605","volume":"13","author":"J Jia","year":"2014","unstructured":"Jia, J., Peng, J., Li, K.: Well-posedness of abstract distributed-order fractional diffusion equations. Commun. Pure Appl. Anal. 13, 605\u2013621 (2014)","journal-title":"Commun. Pure Appl. Anal."},{"key":"672_CR5","doi-asserted-by":"publisher","first-page":"216","DOI":"10.1016\/j.jmaa.2010.12.056","volume":"379","author":"MM Meerschaert","year":"2011","unstructured":"Meerschaert, M.M., Nane, E., Vellaisamy, P.: Distributed-order fractional diffusions on bounded domains. J. Math. Anal. Appl. 379, 216\u2013228 (2011)","journal-title":"J. Math. Anal. Appl."},{"issue":"9\u201310","key":"672_CR6","doi-asserted-by":"publisher","first-page":"1267","DOI":"10.1177\/1077546307087452","volume":"14","author":"F Mainardi","year":"2008","unstructured":"Mainardi, F., Mura, A., Pagnini, G., Gorenflo, R.: Time-fractional diffusion of distributed order. J. Vib. Control 14(9\u201310), 1267\u20131290 (2008)","journal-title":"J. Vib. Control"},{"key":"672_CR7","doi-asserted-by":"publisher","first-page":"652","DOI":"10.1016\/j.jcp.2015.06.025","volume":"298","author":"H Ye","year":"2015","unstructured":"Ye, H., Liu, F., Anh, V.: Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains. J. Comput. Phys. 298, 652\u2013660 (2015)","journal-title":"J. Comput. Phys."},{"key":"672_CR8","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1093\/imamat\/hxu015","volume":"80","author":"H Ye","year":"2015","unstructured":"Ye, H., Liu, F., Anh, V., Turner, I.: Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains. IMA J. Appl. Math. 80, 825\u2013838 (2015)","journal-title":"IMA J. Appl. Math."},{"key":"672_CR9","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1016\/j.jcp.2013.11.013","volume":"259","author":"JT Katsikadelis","year":"2014","unstructured":"Katsikadelis, J.T.: Numerical solution of distributed order fractional differential equations. J. Comput. Phys. 259, 11\u201322 (2014)","journal-title":"J. Comput. Phys."},{"key":"672_CR10","doi-asserted-by":"publisher","first-page":"926","DOI":"10.1016\/j.camwa.2015.02.023","volume":"69","author":"GH Gao","year":"2015","unstructured":"Gao, G.H., Sun, Z.Z.: Two alternating direction implicit difference schemes with the extrapolation method for the two-dimensional distributed-order differential equations. Comput. Math. Appl. 69, 926\u2013948 (2015)","journal-title":"Comput. Math. Appl."},{"key":"672_CR11","doi-asserted-by":"publisher","first-page":"1281","DOI":"10.1007\/s10915-015-0064-x","volume":"66","author":"GH Gao","year":"2016","unstructured":"Gao, G.H., Sun, Z.Z.: Two alternating direction implicit difference schemes for two-dimensional distributed-order fractional diffusion equations. J. Sci. Comput. 66, 1281\u20131312 (2016)","journal-title":"J. Sci. Comput."},{"key":"672_CR12","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1016\/j.jcp.2015.05.047","volume":"298","author":"GH Gao","year":"2015","unstructured":"Gao, G.H., Sun, H.W., Sun, Z.Z.: Some high-order difference scheme for the distributed-order differential equations. J. Comput. Phys. 298, 337\u2013359 (2015)","journal-title":"J. Comput. Phys."},{"key":"672_CR13","doi-asserted-by":"publisher","first-page":"131","DOI":"10.4208\/eajam.020615.030216a","volume":"6","author":"R Du","year":"2016","unstructured":"Du, R., Hao, Z.P., Sun, Z.Z.: Lubich second-order methods for distributed-order time-fractional differential equations with smooth solutions. EAJAM 6, 131\u2013151 (2016)","journal-title":"EAJAM"},{"key":"672_CR14","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1515\/fca-2016-0005","volume":"19","author":"B Jin","year":"2016","unstructured":"Jin, B., Lazarov, R., Sheen, D., Zhou, Z.: Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data. Fract. Calc. Appl. Anal. 19, 69\u201393 (2016)","journal-title":"Fract. Calc. Appl. Anal."},{"key":"672_CR15","doi-asserted-by":"publisher","first-page":"84","DOI":"10.1016\/j.jcp.2016.03.044","volume":"315","author":"H Chen","year":"2016","unstructured":"Chen, H., L\u00fc, S.J., Chen, W.P.: Finite difference\/spectral approximations for the distributed order time fractional reaction\u2013diffusion equation on an unbounded domain. J. Comput. Phys. 315, 84\u201397 (2016)","journal-title":"J. Comput. Phys."},{"key":"672_CR16","doi-asserted-by":"publisher","first-page":"216","DOI":"10.1016\/j.cam.2014.07.029","volume":"275","author":"ML Morgado","year":"2015","unstructured":"Morgado, M.L., Rebelo, M.: Numerical approximation of distributed order reaction diffusion equations. J. Comput. Appl. Math. 275, 216\u2013227 (2015)","journal-title":"J. Comput. Appl. Math."},{"key":"672_CR17","doi-asserted-by":"publisher","first-page":"755","DOI":"10.1137\/0729047","volume":"29","author":"Y Yan","year":"1992","unstructured":"Yan, Y., Fairweather, G.: Orthogonal spline collocation methods for some partial integro-differential equations. SIAM J. Numer. Anal. 29, 755\u2013768 (1992)","journal-title":"SIAM J. Numer. Anal."},{"key":"672_CR18","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1016\/S0377-0427(00)00509-4","volume":"128","author":"B Bialecki","year":"2001","unstructured":"Bialecki, B., Fairweather, G.: Orthogonal spline collocation methods for partial differential equations. J. Comput. Appl. Math. 128, 55\u201382 (2001)","journal-title":"J. Comput. Appl. Math."},{"key":"672_CR19","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1007\/s10915-014-9853-x","volume":"62","author":"B Bialecki","year":"2015","unstructured":"Bialecki, B., Fairweather, G., L\u00f3pez-Marcos, J.C.: The extrapolated Crank\u2013Nicolson orthogonal spline collocation method for a quasilinear parabolic problem with nonlocal boundary conditions. J. Sci. Comput. 62, 265\u2013283 (2015)","journal-title":"J. Sci. Comput."},{"key":"672_CR20","doi-asserted-by":"publisher","first-page":"6248","DOI":"10.1016\/j.jcp.2012.04.001","volume":"231","author":"RI Fernandes","year":"2012","unstructured":"Fernandes, R.I., Fairweather, G.: An ADI extrapolated Crank\u2013Nicolson orthogonal spline collocation method for nonlinear reaction\u2013diffusion systems. J. Comput. Phys. 231, 6248\u20136267 (2012)","journal-title":"J. Comput. Phys."},{"key":"672_CR21","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1016\/j.jcp.2015.07.016","volume":"299","author":"RI Fernandes","year":"2015","unstructured":"Fernandes, R.I., Bialecki, B., Fairweather, G.: An ADI extrapolated Crank\u2013Nicolson orthogonal spline collocation method for nonlinear reaction\u2013diffusion systems on evolving domains. J. Comput. Phys. 299, 561\u2013580 (2015)","journal-title":"J. Comput. Phys."},{"key":"672_CR22","doi-asserted-by":"publisher","first-page":"1217","DOI":"10.1007\/s10915-015-0003-x","volume":"65","author":"G Fairweather","year":"2015","unstructured":"Fairweather, G., Yang, X.H., Xu, D., Zhang, H.Z.: An ADI Crank\u2013Nicolson orthogonal spline collocation method for the two-dimensional fractional diffusion-wave equation. J. Sci. Comput. 65, 1217\u20131239 (2015)","journal-title":"J. Sci. Comput."},{"key":"672_CR23","doi-asserted-by":"publisher","first-page":"1534","DOI":"10.1002\/num.21958","volume":"31","author":"G Fairweather","year":"2015","unstructured":"Fairweather, G., Zhang, H.Z., Yang, X.H., Xu, D.: A backward Euler orthogonal spline collocation method for the time-fractional Fokker\u2013Plank equation. Numer. Methods Partial Differ. Equ. 31, 1534\u20131550 (2015)","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"672_CR24","doi-asserted-by":"publisher","first-page":"824","DOI":"10.1016\/j.jcp.2013.09.016","volume":"256","author":"XH Yang","year":"2014","unstructured":"Yang, X.H., Zhang, H.X., Xu, D.: Orthogonal spline collocation method for the two-dimensional fractional sub-diffusion equation. J. Comput. Phys. 256, 824\u2013837 (2014)","journal-title":"J. Comput. Phys."},{"key":"672_CR25","doi-asserted-by":"publisher","first-page":"1710","DOI":"10.1016\/j.camwa.2014.10.019","volume":"68","author":"HX Zhang","year":"2014","unstructured":"Zhang, H.X., Yang, X.H., Han, X.L.: Discrete-time orthogonal spline collocation method with application to two-dimensional fractional Cable equation. Comput. Math. Appl. 68, 1710\u20131722 (2014)","journal-title":"Comput. Math. Appl."},{"key":"672_CR26","doi-asserted-by":"publisher","first-page":"1703","DOI":"10.1090\/S0025-5718-2015-02917-2","volume":"84","author":"WY Tian","year":"2015","unstructured":"Tian, W.Y., Zhou, H., Deng, W.H.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84, 1703\u20131727 (2015)","journal-title":"Math. Comput."},{"key":"672_CR27","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1007\/s10915-012-9661-0","volume":"56","author":"H Zhou","year":"2013","unstructured":"Zhou, H., Tian, W.Y., Deng, W.H.: Quasi-compact finite difference schemes for space fractional diffusion equations. J. Sci. Comput. 56, 45\u201366 (2013)","journal-title":"J. Sci. Comput."},{"key":"672_CR28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jcp.2014.08.012","volume":"277","author":"Z Wang","year":"2014","unstructured":"Wang, Z., Vong, S.: Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation. J. Comput. Phys. 277, 1\u201315 (2014)","journal-title":"J. Comput. Phys."},{"key":"672_CR29","doi-asserted-by":"publisher","first-page":"605","DOI":"10.1137\/0717050","volume":"17","author":"P Percell","year":"1980","unstructured":"Percell, P., Wheeler, M.F.: A $$C^1$$ C 1 finite element collocation method for elliptic equations. SIAM J. Numer. Anal. 17, 605\u2013622 (1980)","journal-title":"SIAM J. Numer. Anal."},{"key":"672_CR30","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1002\/num.1690090207","volume":"9","author":"RI Fernandes","year":"1993","unstructured":"Fernandes, R.I., Fairweather, G.: Analysis of alternating direction collocation methods for parabolic and hyperbolic problems in two space variables. Numer. Methods Partial Differ. Equ. 9, 191\u2013211 (1993)","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"672_CR31","doi-asserted-by":"publisher","first-page":"282","DOI":"10.1137\/0723020","volume":"23","author":"CE Greenwell-Yanik","year":"1986","unstructured":"Greenwell-Yanik, C.E., Fairweather, G.: Analyses of spline collocation methods for parabolic and hyperbolic problems in two space variables. SIAM J. Numer. Anal. 23, 282\u2013296 (1986)","journal-title":"SIAM J. Numer. Anal."},{"key":"672_CR32","doi-asserted-by":"publisher","first-page":"1047","DOI":"10.1137\/0728056","volume":"28","author":"YP Lin","year":"1991","unstructured":"Lin, Y.P., 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, 1047\u20131070 (1991)","journal-title":"SIAM J. Numer. Anal."},{"key":"672_CR33","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1016\/j.jcp.2014.10.051","volume":"281","author":"B Jin","year":"2015","unstructured":"Jin, B., Lazarov, R., Liu, Y., Zhou, Z.: The Galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys. 281, 825\u2013843 (2015)","journal-title":"J. Comput. Phys."},{"key":"672_CR34","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1016\/j.jcp.2014.08.050","volume":"293","author":"W McLean","year":"2015","unstructured":"McLean, W., Mustapha, K.: Time-stepping error bounds for fractional diffusion problems with non-smooth initial data. J. Comput. Phys. 293, 201\u2013217 (2015)","journal-title":"J. Comput. Phys."},{"key":"672_CR35","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1093\/imanum\/dru018","volume":"35","author":"B Jin","year":"2015","unstructured":"Jin, B., Lazarov, R., Pasciak, J., Zhou, Z.: Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion. IMA J. Numer. Anal. 35, 561\u2013582 (2015)","journal-title":"IMA J. Numer. Anal."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-018-0672-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-018-0672-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-018-0672-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,14]],"date-time":"2022-08-14T13:31:03Z","timestamp":1660483863000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-018-0672-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,2,17]]},"references-count":35,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2018,9]]}},"alternative-id":["672"],"URL":"https:\/\/doi.org\/10.1007\/s10915-018-0672-3","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,2,17]]},"assertion":[{"value":"31 October 2016","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 February 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 February 2018","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 February 2018","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}