{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T13:48:05Z","timestamp":1774360085107,"version":"3.50.1"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2017,1,25]],"date-time":"2017-01-25T00:00:00Z","timestamp":1485302400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Basic Research Program of China","doi-asserted-by":"crossref","award":["2015CB251601"],"award-info":[{"award-number":["2015CB251601"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100012166","name":"National Basic Research Program of China","doi-asserted-by":"crossref","award":["2013CB227900"],"award-info":[{"award-number":["2013CB227900"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["Nos. 51322401, 51421003, U1261201"],"award-info":[{"award-number":["Nos. 51322401, 51421003, U1261201"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Fundamental Research Funds for the Central Universities(China University of Mining and Technology)","award":["Nos. 2014YC09, 2014ZDPY08"],"award-info":[{"award-number":["Nos. 2014YC09, 2014ZDPY08"]}]},{"name":"the 111 Project","award":["No. B07028"],"award-info":[{"award-number":["No. B07028"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2017,10]]},"DOI":"10.1007\/s11075-017-0271-7","type":"journal-article","created":{"date-parts":[[2017,1,25]],"date-time":"2017-01-25T03:54:43Z","timestamp":1485316483000},"page":"573-598","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":51,"title":["A second-order compact difference scheme for the fourth-order fractional sub-diffusion equation"],"prefix":"10.1007","volume":"76","author":[{"given":"Pu","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Hai","family":"Pu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2017,1,25]]},"reference":[{"key":"271_CR1","unstructured":"Podlubny, I.: Fractional Differential Equations. Academic (1999)"},{"key":"271_CR2","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)","DOI":"10.1142\/3779"},{"key":"271_CR3","volume-title":"Theory and Applications of Fractional Differential Equation","author":"AA Kilbas","year":"2006","unstructured":"Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equation. Elsevier, Amsterdam (2006)"},{"key":"271_CR4","doi-asserted-by":"crossref","first-page":"3975","DOI":"10.1103\/PhysRevLett.71.3975","volume":"71","author":"TH Solomon","year":"1993","unstructured":"Solomon, T.H., Weeks, E.R., Swinney, H.L.: Observations of anomalous diffusion and Lvy flights in a 2-dimensional rotating flow. Phys. Rev. Lett. 71, 3975\u20133979 (1993)","journal-title":"Phys. Rev. Lett."},{"key":"271_CR5","doi-asserted-by":"crossref","DOI":"10.1002\/9783527622979","volume-title":"Anomalous Transport: Foundations and Applications","author":"R Klages","year":"2008","unstructured":"Klages, R., Radons, G., Sokolov, I.M.: Anomalous Transport: Foundations and Applications. Wiley, Weinheim (2008)"},{"key":"271_CR6","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1016\/0378-4371(85)90028-7","volume":"132","author":"V Balakrishnan","year":"1985","unstructured":"Balakrishnan, V.: Anomalous diffusion in one dimension. Phys. A 132, 569\u2013580 (1985)","journal-title":"Phys. A"},{"key":"271_CR7","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1063\/1.528578","volume":"30","author":"WR Schneider","year":"1989","unstructured":"Schneider, W.R., Wyss, W.: Fractional diffusion and wave equations. J. Math. Phys. 30, 134\u2013144 (1989)","journal-title":"J. Math. Phys."},{"key":"271_CR8","unstructured":"Mainardi, F.: Fractional diffusive waves in viscoelastic solids. In: Wegner, J.I., Norwood, F.R (eds.) Nonlinear Waves in Solids. ASME\/AMR, pp. 93\u201397. Fairfield, NJ (1995)"},{"key":"271_CR9","doi-asserted-by":"crossref","unstructured":"Mainardi, F.: Some basic problems in continuum and statistical mechanics. In: rpinteri, A., Mainardi, F (eds.) Fractals and Fractional Calculus in Continuum Mechanics. Springer, Wien (1997)","DOI":"10.1007\/978-3-7091-2664-6_7"},{"key":"271_CR10","doi-asserted-by":"crossref","unstructured":"Das, S.: Functional Fractional Calculus. Springer, Berlin (2011)","DOI":"10.1007\/978-3-642-20545-3"},{"key":"271_CR11","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1007\/s11071-012-0710-x","volume":"71","author":"\u017e Tomovski","year":"2013","unstructured":"Tomovski, \u017e., Sandev, T.: Exact solutions for fractional diffusion equation in a bounded domain with different boundary conditions. Nonlinear Dynam. 71, 671\u2013683 (2013)","journal-title":"Nonlinear Dynam."},{"key":"271_CR12","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1016\/j.jcp.2004.11.025","volume":"205","author":"TAM Langlands","year":"2005","unstructured":"Langlands, T.A.M., Henry, B.I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205, 719\u2013736 (2005)","journal-title":"J. Comput. Phys."},{"key":"271_CR13","doi-asserted-by":"crossref","first-page":"1862","DOI":"10.1137\/030602666","volume":"42","author":"SB Yuste","year":"2005","unstructured":"Yuste, S.B., Acedo, L.: An explicit finite difference method and a new Von-Neumann type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42, 1862\u20131874 (2005)","journal-title":"SIAM J. Numer. Anal."},{"key":"271_CR14","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1016\/j.jcp.2005.12.006","volume":"216","author":"S Yuste","year":"2006","unstructured":"Yuste, S.: Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216, 264\u2013274 (2006)","journal-title":"J. Comput. Phys."},{"key":"271_CR15","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/j.jcp.2005.08.008","volume":"213","author":"C Tadjeran","year":"2006","unstructured":"Tadjeran, C., Meerschaert, M.M., Scheffler, H.P.: A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213, 205\u2013213 (2006)","journal-title":"J. Comput. Phys."},{"key":"271_CR16","doi-asserted-by":"crossref","first-page":"1533","DOI":"10.1016\/j.jcp.2007.02.001","volume":"225","author":"X Lin","year":"2007","unstructured":"Lin, X., 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":"271_CR17","doi-asserted-by":"crossref","first-page":"886","DOI":"10.1016\/j.jcp.2007.05.012","volume":"227","author":"CM Chen","year":"2007","unstructured":"Chen, C.M., Liu, F., Turner, I., Anh, V.: A Fourier method for the fractional diffusion equation describing subdiffusion. J. Comput. Phys. 227, 886\u2013897 (2007)","journal-title":"J. Comput. Phys."},{"key":"271_CR18","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1137\/060673114","volume":"46","author":"P Zhuang","year":"2008","unstructured":"Zhuang, P., Liu, F., Anh, V., Turner, I.: New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation. SIAM J. Numer. Anal. 46, 1079\u20131095 (2008)","journal-title":"SIAM J. Numer. Anal."},{"key":"271_CR19","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1016\/j.cam.2009.02.013","volume":"231","author":"F Liu","year":"2009","unstructured":"Liu, F., Yang, C., Burrage, K.: Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term. J. Comput. Appl. Math. 231, 160\u2013176 (2009)","journal-title":"J. Comput. Appl. Math."},{"key":"271_CR20","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.apnum.2005.03.003","volume":"56","author":"ZZ Sun","year":"2006","unstructured":"Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193\u2013209 (2006)","journal-title":"Appl. Numer. Math."},{"key":"271_CR21","doi-asserted-by":"crossref","first-page":"7792","DOI":"10.1016\/j.jcp.2009.07.021","volume":"228","author":"M Cui","year":"2009","unstructured":"Cui, M.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228, 7792\u20137804 (2009)","journal-title":"J. Comput. Phys."},{"key":"271_CR22","doi-asserted-by":"crossref","first-page":"1740","DOI":"10.1137\/090771715","volume":"32","author":"CM Chen","year":"2010","unstructured":"Chen, C.M., Liu, F., Anh, V., Turner, I.: Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equations. SIAM J. Sci. Comput 32, 1740\u20131760 (2010)","journal-title":"SIAM J. Sci. Comput"},{"key":"271_CR23","doi-asserted-by":"crossref","first-page":"586","DOI":"10.1016\/j.jcp.2010.10.007","volume":"230","author":"GH Gao","year":"2011","unstructured":"Gao, G.H., Sun, Z.Z.: A compact difference scheme for the fractional subdiffusion equations. J. Comput. Phys. 230, 586\u2013595 (2011)","journal-title":"J. Comput. Phys."},{"key":"271_CR24","doi-asserted-by":"crossref","first-page":"2302","DOI":"10.1137\/100812707","volume":"49","author":"YN Zhang","year":"2011","unstructured":"Zhang, Y.N., Sun, Z.Z., Wu, H.W.: Error estimates of Crank-Nicolson type difference schemes for the subdiffusion equation. SIAM J. Numer. Anal. 49, 2302\u20132322 (2011)","journal-title":"SIAM J. Numer. Anal."},{"issue":"15-16","key":"271_CR25","doi-asserted-by":"crossref","first-page":"3848","DOI":"10.1016\/j.apm.2013.10.037","volume":"38","author":"X Zhao","year":"2014","unstructured":"Zhao, X., Xu, Q.: Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient. Appl. Math. Model 38(15-16), 3848\u20133859 (2014)","journal-title":"Appl. Math. Model"},{"issue":"2","key":"271_CR26","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1007\/s10915-015-0040-5","volume":"66","author":"S Vong","year":"2015","unstructured":"Vong, S., Lyu, P., Wang, Z.: A compact difference scheme for fractional sub-diffusion equations with the spatially variable coefficient under Neumann boundary conditions. J. Sci. Comput. 66(2), 725\u2013739 (2015)","journal-title":"J. Sci. Comput."},{"key":"271_CR27","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/j.jcp.2014.08.015","volume":"293","author":"X Zhao","year":"2015","unstructured":"Zhao, X., Sun, Z.Z., Karniadakis, G.E.: Second-order approximations for variable order fractional derivatives: algorithms and applications. J. Comput. Phys. 293, 184\u2013200 (2015)","journal-title":"J. Comput. Phys."},{"key":"271_CR28","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1016\/j.jcp.2014.09.033","volume":"280","author":"GH Gao","year":"2015","unstructured":"Gao, G.H., Sun, H.W., Sun, Z.Z.: Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence. J. Comput. Phys. 280, 510\u2013528 (2015)","journal-title":"J. Comput. Phys."},{"key":"271_CR29","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1016\/j.jcp.2014.09.031","volume":"280","author":"AA Alikhanov","year":"2015","unstructured":"Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424\u2013438 (2015)","journal-title":"J. Comput. Phys."},{"key":"271_CR30","volume-title":"Fourier Transforms","author":"IN Sneddon","year":"1951","unstructured":"Sneddon, I.N.: Fourier Transforms. McGraw Hill, New York (1951)"},{"key":"271_CR31","volume-title":"The Fractional Calculus","author":"KB Oldhan","year":"1974","unstructured":"Oldhan, K.B., Spainer, J.: The Fractional Calculus. Academic Press, New York (1974)"},{"issue":"2","key":"271_CR32","doi-asserted-by":"crossref","first-page":"1336","DOI":"10.1103\/PhysRevE.53.R1336","volume":"53","author":"VI Karpman","year":"1996","unstructured":"Karpman, V.I.: Stabilization of soliton instabilities by higher-order dispersion: fourth order nonlinear Schr\u00f6dinger-type equations. Phys. Rev. E 53(2), 1336\u20131339 (1996)","journal-title":"Phys. Rev. E"},{"key":"271_CR33","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/S0167-2789(00)00078-6","volume":"144","author":"VI Karpman","year":"2000","unstructured":"Karpman, V.I., Shagalov, A.G.: Stability of soliton described by nonlinear Schr\u00f6dinger-type equations with higher order dispersion. Phys. D 144, 194\u2013210 (2000)","journal-title":"Phys. D"},{"key":"271_CR34","doi-asserted-by":"crossref","first-page":"1497","DOI":"10.1016\/S0045-7949(01)00026-8","volume":"79","author":"OmP Agrawal","year":"2001","unstructured":"Agrawal, Om P.: A general solution for a fourth-order fractional diffusion-wave equation in a bounded domain. Comput. Struct. 79, 1497\u20131501 (2001)","journal-title":"Comput. Struct."},{"key":"271_CR35","volume-title":"Solving a fourth-order fractional diffusion-wave equation in a bounded domain by decomposition method","author":"H Jafari","year":"2007","unstructured":"Jafari, H., Dehghan, M., Sayevand, K.: Solving a fourth-order fractional diffusion-wave equation in a bounded domain by decomposition method. Wiley InterScience, New York (2007)"},{"key":"271_CR36","doi-asserted-by":"crossref","first-page":"2227","DOI":"10.1016\/j.camwa.2010.09.022","volume":"61","author":"A Golbabai","year":"2011","unstructured":"Golbabai, A., Sayevand, K.: Fractional calculus\u2014a new approach to the analysis of generalized fourthorder diffusion-wave equations. Appl. Math. Comput. 61, 2227\u20132231 (2011)","journal-title":"Appl. Math. Comput."},{"key":"271_CR37","first-page":"703","volume":"243","author":"Y Liu","year":"2014","unstructured":"Liu, Y., Fang, Z.C., Li, H., He, S.: A mixed finite element method for a time-fractional fourth-order partial differential equation. Appl. Math. Comput. 243, 703\u2013717 (2014)","journal-title":"Appl. Math. Comput."},{"issue":"4","key":"271_CR38","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1016\/j.camwa.2015.05.015","volume":"70","author":"Y Liu","year":"2015","unstructured":"Liu, Y., Du, Y., Li, H., He, S., Gao, W.: Finite difference\/finite element method for a nonlinear time-fractional fourth-order reaction-diffusion problem. Comput. Math. Appl. 70(4), 573\u2013591 (2015)","journal-title":"Comput. Math. Appl."},{"key":"271_CR39","first-page":"5019","volume":"218","author":"XL Hu","year":"2012","unstructured":"Hu, X.L., Zhang, L.M.: On finite difference methods for fourth-order fractional diffusion-wave and subdiffusion systems. Appl. Math. Comput. 218, 5019\u20135034 (2012)","journal-title":"Appl. Math. Comput."},{"key":"271_CR40","first-page":"96","volume":"28","author":"J Guo","year":"2014","unstructured":"Guo, J., Li, C.P., Ding, H.F.: Finite difference methods for time subdiffusion equation with space fourth order. Commun. Appl. Math. Comput. 28, 96\u2013108 (2014). in Chinese","journal-title":"Commun. Appl. Math. Comput."},{"issue":"7","key":"271_CR41","doi-asserted-by":"crossref","first-page":"1496","DOI":"10.1080\/00207160.2014.948430","volume":"92","author":"SS Siddiqi","year":"2014","unstructured":"Siddiqi, S.S., Arshed, S.: Numerical solution of time-fractional fourth-order partial differential equations. Int. J. Comput. Math. 92(7), 1496\u20131518 (2014)","journal-title":"Int. J. Comput. Math."},{"issue":"3","key":"271_CR42","first-page":"1148","volume":"66","author":"CC Ji","year":"2015","unstructured":"Ji, C.C., Sun, Z.Z., Hao, Z.P.: Numerical algorithms with high spatial accuracy for the fourth-order fractional sub-diffusion equations with the first Dirichlet boundary conditions. J. Sci. Comput. 66(3), 1148\u20131174 (2015)","journal-title":"J. Sci. Comput."},{"key":"271_CR43","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1002\/num.20414","volume":"26","author":"HL Liao","year":"2010","unstructured":"Liao, H.L., Sun, Z.Z.: Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations. Numer. Methods Partial Differ. Equ. 26, 37\u201360 (2010)","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"271_CR44","volume-title":"Finite Difference Methods for Elliptic Equation","author":"AA Samarskii","year":"1976","unstructured":"Samarskii, A.A., Andreev, V.B.: Finite Difference Methods for Elliptic Equation. Moscow, Nauka (1976). in Russian"},{"key":"271_CR45","volume-title":"Numerical Methods of Partial Differential Equations","author":"ZZ Sun","year":"2012","unstructured":"Sun, Z.Z.: Numerical Methods of Partial Differential Equations, 2D edn. Science Press, Beijing (2012). in Chinese","edition":"2D edn"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-017-0271-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-017-0271-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-017-0271-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,17]],"date-time":"2020-05-17T14:18:20Z","timestamp":1589725100000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-017-0271-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1,25]]},"references-count":45,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2017,10]]}},"alternative-id":["271"],"URL":"https:\/\/doi.org\/10.1007\/s11075-017-0271-7","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1,25]]}}}