{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,19]],"date-time":"2025-10-19T16:04:08Z","timestamp":1760889848355,"version":"3.37.3"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T00:00:00Z","timestamp":1558137600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T00:00:00Z","timestamp":1558137600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"name":"the University Natural Science Research Key Project of Anhui Province","award":["KJ2018A0523"],"award-info":[{"award-number":["KJ2018A0523"]}]},{"DOI":"10.13039\/501100001809","name":"the National Nature Science Foundation of China","doi-asserted-by":"crossref","award":["11571181"],"award-info":[{"award-number":["11571181"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"NUAA Scientific Research Starting Fund of Introduced Talent","award":["1008-56SYAH18037"],"award-info":[{"award-number":["1008-56SYAH18037"]}]},{"name":"333 High-level Personal Training Project of Jiangsu Province","award":["DRA2015518"],"award-info":[{"award-number":["DRA2015518"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2020,3]]},"DOI":"10.1007\/s11075-019-00722-w","type":"journal-article","created":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T21:15:29Z","timestamp":1558214129000},"page":"1223-1248","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Sharp H1-norm error estimate of a cosine pseudo-spectral scheme for 2D reaction-subdiffusion equations"],"prefix":"10.1007","volume":"83","author":[{"given":"Xin","family":"Li","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luming","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hong-lin","family":"Liao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,5,18]]},"reference":[{"key":"722_CR1","volume-title":"Fractional Differential Equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)"},{"key":"722_CR2","doi-asserted-by":"crossref","unstructured":"Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific (2000)","DOI":"10.1142\/3779"},{"key":"722_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0370-1573(00)00070-3","volume":"339","author":"R Metzler","year":"2000","unstructured":"Metzler, R., Klafter, J.: The random walk\u2019s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1\u201377 (2000)","journal-title":"Phys. Rep."},{"key":"722_CR4","doi-asserted-by":"publisher","first-page":"1112","DOI":"10.1103\/PhysRevE.63.021112","volume":"63","author":"MM Meerschaert","year":"2001","unstructured":"Meerschaert, M.M., Benson, D., Baeumer, B.: Operator L\u00e9vy motion and multiscaling anomalous diffusion. Phys. Rev. E 63, 1112\u20131117 (2001)","journal-title":"Phys. Rev. E"},{"key":"722_CR5","first-page":"1022","volume":"39","author":"R Schumer","year":"2003","unstructured":"Schumer, R., Benson, D.A., Meerschaert, M.M., Baeumer, B.: Multiscaling fractional advection-dispersion equations and their solutions. Water Resour. Res. 39, 1022\u20131032 (2003)","journal-title":"Water Resour. Res."},{"key":"722_CR6","doi-asserted-by":"publisher","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":"722_CR7","doi-asserted-by":"publisher","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."},{"key":"722_CR8","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1016\/j.jcp.2013.11.017","volume":"259","author":"GH Gao","year":"2014","unstructured":"Gao, G.H., Sun, Z.Z., Zhang, H.W.: 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":"722_CR9","doi-asserted-by":"publisher","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":"722_CR10","doi-asserted-by":"publisher","first-page":"6613","DOI":"10.1016\/j.jcp.2010.05.015","volume":"229","author":"H Brunner","year":"2010","unstructured":"Brunner, H., Ling, L., Yamamoto, M.: Numerical simulations of 2D frational subdiffusion problems. J. Comput. Phys. 229, 6613\u20136622 (2010)","journal-title":"J. Comput. Phys."},{"key":"722_CR11","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1017\/S1446181111000617","volume":"52","author":"W McLean","year":"2010","unstructured":"McLean, W.: Regularity of solutions to a time-fractional diffusion equation. ANZIAM J. 52, 123\u2013138 (2010)","journal-title":"ANZIAM J."},{"issue":"1","key":"722_CR12","first-page":"197","volume":"36","author":"BT Jin","year":"2016","unstructured":"Jin, B.T., Lazarov, R., Zhou, Z.: An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36(1), 197\u2013221 (2016)","journal-title":"IMA J. Numer. Anal."},{"key":"722_CR13","doi-asserted-by":"publisher","first-page":"426","DOI":"10.1016\/j.jmaa.2011.04.058","volume":"382","author":"K Sakamoto","year":"2011","unstructured":"Sakamoto, K., Yamamoto, M.: Initial value\/boundary value prolems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, 426\u2013447 (2011)","journal-title":"J. Math. Anal. Appl."},{"key":"722_CR14","doi-asserted-by":"publisher","first-page":"1057","DOI":"10.1137\/16M1082329","volume":"55","author":"M Stynes","year":"2017","unstructured":"Stynes, M., O\u2019Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55, 1057\u20131079 (2017)","journal-title":"SIAM J. Numer. Anal."},{"key":"722_CR15","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/j.jcp.2014.02.008","volume":"265","author":"YN Zhang","year":"2014","unstructured":"Zhang, Y.N., Sun, Z.Z., Liao, H. -L.: Finite difference methods for the time fractional diffusion equation on nonuniform meshes. J. Comput. Phys. 265, 195\u2013210 (2014)","journal-title":"J. Comput. Phys."},{"key":"722_CR16","doi-asserted-by":"publisher","first-page":"1112","DOI":"10.1137\/17M1131829","volume":"56","author":"H-L Liao","year":"2018","unstructured":"Liao, H.-L., Li, D.F., Zhang, J.W.: Sharp error estimate of the nonuniform L1 formula for linear reaction-subdiffusion equations. SIAM J. Numer. Anal. 56, 1112\u20131133 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"722_CR17","doi-asserted-by":"publisher","first-page":"218","DOI":"10.1137\/16M1175742","volume":"57","author":"H-L Liao","year":"2019","unstructured":"Liao, H.-L., McLean, W., Zhang, J.W.: A discrete Gr\u00f6nwall inequality with application to numerical schemes for subdiffusion problems. SIAM J. Numer. Anal. 57, 218\u2013237 (2019)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"722_CR18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-019-00927-0","volume":"80","author":"Hong-lin Liao","year":"2019","unstructured":"Liao, H.-L., Yan, Y.G., Zhang, J.W.: Unconditional convergence of a two-level linearized fast algorithm for semilinear subdiffusion equations. J. Sci. Comput. (2019), online, \nhttps:\/\/doi.org\/10.1007\/s10915-019-00927-0","journal-title":"Journal of Scientific Computing"},{"key":"722_CR19","unstructured":"Liao, H.-L., McLean, W., Zhang, J.W.: A second-order scheme with nonuniform time steps for a linear reaction-subdiffusion problem. (2018) arXiv:\n1803.09873v2"},{"key":"722_CR20","doi-asserted-by":"publisher","first-page":"593","DOI":"10.1007\/s11071-009-9566-0","volume":"59","author":"Y Povstenko","year":"2010","unstructured":"Povstenko, Y.: Signaling problem for time-fractional diffusion-wave equation in a half-space in the case of angular symmetry. Nonlinear Dyn. 59, 593\u2013605 (2010)","journal-title":"Nonlinear Dyn."},{"key":"722_CR21","doi-asserted-by":"publisher","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":"722_CR22","first-page":"6601","volume":"230","author":"X Zhao","year":"2011","unstructured":"Zhao, X., Sun, Z.Z.: A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 230, 6601\u20136074 (2011)","journal-title":"J. Comput. Phys."},{"key":"722_CR23","doi-asserted-by":"publisher","first-page":"456","DOI":"10.1016\/j.jcp.2012.08.026","volume":"232","author":"JC Ren","year":"2013","unstructured":"Ren, J.C., Sun, Z.Z., Zhao, X.: Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 232, 456\u2013467 (2013)","journal-title":"J. Comput. Phys."},{"key":"722_CR24","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1080\/00207160.2014.887702","volume":"92","author":"JX Cao","year":"2015","unstructured":"Cao, J.X., Li, C.P., Chen, Y.Q.: Compact difference method for solving the fractional reaction subdiffusion equation with Neumann boundary value condition. Int. J. Comput. Math. 92, 167\u2013180 (2015)","journal-title":"Int. J. Comput. Math."},{"key":"722_CR25","doi-asserted-by":"publisher","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 sub-diffusion. J. Comput. Phys. 227, 886\u2013897 (2007)","journal-title":"J. Comput. Phys."},{"key":"722_CR26","doi-asserted-by":"publisher","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 diffusion equation. SIAM J. Numer. Anal. 47, 2108\u20132131 (2009)","journal-title":"SIAM J. Numer. Anal."},{"key":"722_CR27","doi-asserted-by":"publisher","first-page":"A2699","DOI":"10.1137\/15M102664X","volume":"38","author":"CW Lv","year":"2016","unstructured":"Lv, C.W., Xu, C.J.: Error analysis of a high order method for time-fractional diffusion equations. SIAM J. Sci. Comput. 38, A2699\u2013A2724 (2016)","journal-title":"SIAM J. Sci. Comput."},{"key":"722_CR28","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1007\/s11075-015-0087-2","volume":"73","author":"AH Bhrawy","year":"2016","unstructured":"Bhrawy, A.H.: A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations. Numer. Algo. 73, 91\u2013113 (2016)","journal-title":"Numer. Algo."},{"key":"722_CR29","doi-asserted-by":"publisher","first-page":"35","DOI":"10.4208\/cicp.090313.041113a","volume":"16","author":"YZ Gong","year":"2014","unstructured":"Gong, Y.Z., Cai, J.X., Wang, Y.S.: Multi-symplectic Fourier pseudospectral method for the Kawahara equation. Commun. Comput. Phys. 16, 35\u201355 (2014)","journal-title":"Commun. Comput. Phys."},{"key":"722_CR30","doi-asserted-by":"publisher","first-page":"354","DOI":"10.1016\/j.jcp.2016.10.022","volume":"328","author":"YZ Gong","year":"2017","unstructured":"Gong, Y.Z., Wang, Q., Wang, Y.S., Cai, J.X.: A conservative Fourier pseudo-spectral method for the nonlinear Schr\u00f6dinger equation. J. Comput. Phys. 328, 354\u2013370 (2017)","journal-title":"J. Comput. Phys."},{"key":"722_CR31","doi-asserted-by":"publisher","first-page":"650","DOI":"10.4208\/cicp.OA-2016-0136","volume":"21","author":"SD Jiang","year":"2017","unstructured":"Jiang, S.D., Zhang, J.W., Zhang, Q., Zhang, Z.M.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations. Commun. Comput. Phys. 21, 650\u2013678 (2017)","journal-title":"Commun. Comput. Phys."},{"key":"722_CR32","unstructured":"Ren, J.C., Liao, H.-L., Zhang, J.W., Zhang, Z.M.: Sharp H1-norm error estimates of two time-stepping schemes for reaction-subdiffusion problems. (2018) arXiv:\n1811.08059v1"},{"key":"722_CR33","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1090\/S0025-5718-1982-0637287-3","volume":"38","author":"C Canuto","year":"1982","unstructured":"Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38, 67\u201386 (1982)","journal-title":"Math. Comput."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00722-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-019-00722-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00722-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,16]],"date-time":"2020-05-16T23:50:49Z","timestamp":1589673049000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-019-00722-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,18]]},"references-count":33,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,3]]}},"alternative-id":["722"],"URL":"https:\/\/doi.org\/10.1007\/s11075-019-00722-w","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"type":"print","value":"1017-1398"},{"type":"electronic","value":"1572-9265"}],"subject":[],"published":{"date-parts":[[2019,5,18]]},"assertion":[{"value":"23 November 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 May 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 May 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}