{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,17]],"date-time":"2026-06-17T03:03:28Z","timestamp":1781665408845,"version":"3.54.5"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2013,1,24]],"date-time":"2013-01-24T00:00:00Z","timestamp":1358985600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2013,12]]},"DOI":"10.1007\/s11075-012-9689-0","type":"journal-article","created":{"date-parts":[[2013,1,23]],"date-time":"2013-01-23T05:40:36Z","timestamp":1358919636000},"page":"707-720","source":"Crossref","is-referenced-by-count":132,"title":["Two finite difference schemes for time fractional diffusion-wave equation"],"prefix":"10.1007","volume":"64","author":[{"given":"Jianfei","family":"Huang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yifa","family":"Tang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Luis","family":"V\u00e1zquez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiye","family":"Yang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2013,1,24]]},"reference":[{"issue":"4","key":"9689_CR1","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1002\/zamm.200310033","volume":"83","author":"OP Agrawal","year":"2003","unstructured":"Agrawal, O.P.: Response of a diffusion-wave system subjected to deterministic and stochastic fields. Z. Angew. Math. Mech. 83(4), 265\u2013274 (2003)","journal-title":"Z. Angew. Math. Mech."},{"key":"9689_CR2","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1239\/jap\/1082999078","volume":"41","author":"P Becker-Kern","year":"2004","unstructured":"Becker-Kern, P., Meerschaert, M.M., Scheffler, H.P.: Limit theorem for continuous-time random walks with two time scales. J. Appl. Prob. 41, 455\u2013466 (2004)","journal-title":"J. Appl. Prob."},{"issue":"2","key":"9689_CR3","doi-asserted-by":"crossref","first-page":"886","DOI":"10.1016\/j.jcp.2007.05.012","volume":"227","author":"C Chen","year":"2007","unstructured":"Chen, C., Liu, F., Turner, I., Anh, V.: A Fourier analysis method for the fractional diffusion equation describing sub-diffusion. J. Comput Phys. 227(2), 886\u2013897 (2007)","journal-title":"J. Comput Phys."},{"key":"9689_CR4","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1090\/S0025-5718-2011-02447-6","volume":"81","author":"C Chen","year":"2012","unstructured":"Chen, C., Liu, F., Anh, V., Turner, I.: Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation. Math. Comput. 81, 345\u2013366 (2012)","journal-title":"Math. Comput."},{"key":"9689_CR5","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-14574-2","volume-title":"The Analysis of Fractional Differential Equations","author":"K Diethelm","year":"2010","unstructured":"Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)"},{"issue":"10","key":"9689_CR6","doi-asserted-by":"crossref","first-page":"2998","DOI":"10.1016\/j.apm.2010.01.008","volume":"34","author":"R Du","year":"2010","unstructured":"Du, R., Cao, W.R., Sun, Z.Z.: A compact difference scheme for the fractional diffusion-wave equation. Appl. Math. Model. 34(10), 2998\u20133007 (2010)","journal-title":"Appl. Math. Model."},{"key":"9689_CR7","volume-title":"Fractional calculus and continuous-time finance. III, The diffusion limit. In: Mathematical Finance, Trends in Math","author":"R Gorenflo","year":"2001","unstructured":"Gorenflo, R., Mainardi, F., Scalas, E., Raberto, M.: Fractional calculus and continuous-time finance. III, The diffusion limit. In: Mathematical Finance, Trends in Math, pp. 171\u2013180. Birkh\u00e4user, Basel (2001)"},{"key":"9689_CR8","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1023\/A:1016547232119","volume":"29","author":"R Gorenflo","year":"2002","unstructured":"Gorenflo, R., Mainardi, F., Moretti, D., Paradisi, P.: Time fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 29, 129\u2013143 (2002)","journal-title":"Nonlinear Dyn."},{"key":"9689_CR9","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1115\/1.3167616","volume":"51","author":"RC Koeller","year":"1984","unstructured":"Koeller, R.C.: Application of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51, 229\u2013307 (1984)","journal-title":"J. Appl. Mech."},{"key":"9689_CR10","doi-asserted-by":"crossref","first-page":"855","DOI":"10.1016\/j.camwa.2011.02.045","volume":"62","author":"CP Li","year":"2011","unstructured":"Li, C.P., Zhao, Z.G., Chen, Y.Q.: Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Math. Appl. 62, 855\u2013875 (2011)","journal-title":"Comput. Math. Appl."},{"issue":"3","key":"9689_CR11","doi-asserted-by":"crossref","first-page":"2108","DOI":"10.1137\/080718942","volume":"47","author":"XJ Li","year":"2009","unstructured":"Li, X.J., Xu, C.J.: A space-time spectral method for the time fractional differential equation. SIAM J. Numer. Anal. 47(3), 2108\u20132131 (2009)","journal-title":"SIAM J. Numer. Anal."},{"key":"9689_CR12","doi-asserted-by":"crossref","first-page":"1533","DOI":"10.1016\/j.jcp.2007.02.001","volume":"225","author":"YM Lin","year":"2007","unstructured":"Lin, Y.M., Xu, C.J.: Finite difference\/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533\u20131552 (2007)","journal-title":"J. Comput. Phys."},{"issue":"E","key":"9689_CR13","doi-asserted-by":"crossref","first-page":"488","DOI":"10.21914\/anziamj.v46i0.973","volume":"46","author":"F Liu","year":"2005","unstructured":"Liu, F., Shen, S., Anh, V., Turner, I.: Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation. ANZIAM J. 46(E), 488\u2013504 (2005)","journal-title":"ANZIAM J."},{"key":"9689_CR14","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."},{"issue":"1","key":"9689_CR15","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1137\/0727002","volume":"27","author":"JC Lopez-Marcos","year":"1990","unstructured":"Lopez-Marcos, J.C.: A difference scheme for a nonlinear partial integrodifferential equation. SIAM J. Numer. Anal. 27(1), 20\u201331 (1990)","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"9689_CR16","doi-asserted-by":"crossref","first-page":"704","DOI":"10.1137\/0517050","volume":"17","author":"Ch Lubich","year":"1986","unstructured":"Lubich, Ch.: Discretized fractional calculus. SIAM J. Math. Anal. 17(3), 704\u2013719 (1986)","journal-title":"SIAM J. Math. Anal."},{"key":"9689_CR17","volume-title":"Fractals","author":"F Mainardi","year":"1997","unstructured":"Mainardi, F.: Some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals. Springer, Wien (1997)"},{"key":"9689_CR18","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.physa.2006.04.034","volume":"370","author":"MM Meerschaert","year":"2006","unstructured":"Meerschaert, M.M., Scalas, E.: Coupled continuous time random walks in finance. Physica A 370, 114\u2013118 (2006)","journal-title":"Physica A"},{"key":"9689_CR19","volume-title":"Fractional Differential Equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)"},{"key":"9689_CR20","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1016\/S0378-4371(02)01048-8","volume":"314","author":"M Raberto","year":"2002","unstructured":"Raberto, M., Scalas, E., Mainardi, F.: Waiting-times and returns in high-frequency financial data: an empirical study. Physica A 314, 749\u2013755 (2002)","journal-title":"Physica A"},{"key":"9689_CR21","volume-title":"Fractional Integrals and Derivative, Theory and Applications","author":"SG Samko","year":"1993","unstructured":"Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivative, Theory and Applications. Gordon and Breach, New York (1993)"},{"issue":"2","key":"9689_CR22","doi-asserted-by":"crossref","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 equations. SIAM J. Numer. Anal. 25(2), 319\u2013327 (1988)","journal-title":"SIAM J. Numer. Anal."},{"key":"9689_CR23","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":"9689_CR24","doi-asserted-by":"crossref","first-page":"1078","DOI":"10.1016\/j.camwa.2009.05.009","volume":"59","author":"MM Meerschaert","year":"2010","unstructured":"Meerschaert, M.M., Zhang, Y., Baeumerc, B.: Particle tracking for fractional diffusion with two time scales. Comput. Math. Appl. 59, 1078\u20131086 (2010)","journal-title":"Comput. Math. Appl."},{"key":"9689_CR25","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/0168-9274(93)90012-G","volume":"11","author":"T Tang","year":"1993","unstructured":"Tang, T.: A finite difference scheme for partial integro-differential equations with a weakly singular kernel. Appl. Numer. Math. 11, 309\u2013319 (1993)","journal-title":"Appl. Numer. Math."},{"key":"9689_CR26","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1016\/j.jcp.2005.12.006","volume":"216","author":"SB Yuste","year":"2006","unstructured":"Yuste, S.B.: Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216, 264\u2013274 (2006)","journal-title":"J. Comput. Phys."},{"issue":"6","key":"9689_CR27","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\u2013Nicolson-type difference schemes for the subdiffusion equation. SIAM J. Numer. Anal. 49(6), 2302\u20132322 (2011)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"9689_CR28","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 methods for the anomalous sub-diffusion equation. SIAM J. Numer. Anal. 46(2), 1079\u20131095 (2008)","journal-title":"SIAM J. Numer. Anal."},{"issue":"5","key":"9689_CR29","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1093\/imamat\/hxp015","volume":"74","author":"P Zhuang","year":"2009","unstructured":"Zhuang, P., Liu, F., Anh, V., Turner, I.: Stability and convergence of an implicit numerical method for the nonlinear fractional reaction-subdiffusion process. IMA J. Appl. Math. 74(5), 645\u2013667 (2009)","journal-title":"IMA J. Appl. Math."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-012-9689-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-012-9689-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-012-9689-0","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,1]],"date-time":"2019-06-01T01:08:20Z","timestamp":1559351300000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-012-9689-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,1,24]]},"references-count":29,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2013,12]]}},"alternative-id":["9689"],"URL":"https:\/\/doi.org\/10.1007\/s11075-012-9689-0","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,1,24]]}}}