{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,6,25]],"date-time":"2024-06-25T11:15:50Z","timestamp":1719314150056},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2012,2,10]],"date-time":"2012-02-10T00:00:00Z","timestamp":1328832000000},"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":[[2012,12]]},"DOI":"10.1007\/s11075-012-9547-0","type":"journal-article","created":{"date-parts":[[2012,2,9]],"date-time":"2012-02-09T11:35:31Z","timestamp":1328787331000},"page":"525-543","source":"Crossref","is-referenced-by-count":36,"title":["A finite difference method for an anomalous sub-diffusion equation, theory and applications"],"prefix":"10.1007","volume":"61","author":[{"given":"Kassem","family":"Mustapha","sequence":"first","affiliation":[]},{"given":"Jaafar","family":"AlMutawa","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2012,2,10]]},"reference":[{"key":"9547_CR1","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. Physica A 132, 569\u2013580 (1985)","journal-title":"Physica A"},{"key":"9547_CR2","doi-asserted-by":"crossref","first-page":"1740","DOI":"10.1137\/090771715","volume":"32","author":"C-M 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 equation, SIAM J. Sci. Comput. 32, 1740\u20131760 (2010)","journal-title":"SIAM J. Sci. Comput."},{"key":"9547_CR3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11075-009-9320-1","volume":"54","author":"C-M Chen","year":"2010","unstructured":"Chen, C-M., Liu, F., Turner, I., Anh, V.: Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation. Numer. Algor. 54, 1\u201321 (2010)","journal-title":"Numer. Algor."},{"key":"9547_CR4","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1090\/S0025-5718-2011-02447-6","volume":"81","author":"C-M Chen","year":"2012","unstructured":"Chen, C-M. Liu, F., Anh, V., Turner, I.: Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation. Math. Comp. 81, 345\u2013366 (2012)","journal-title":"Math. Comp."},{"key":"9547_CR5","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1090\/S0025-5718-06-01788-1","volume":"75","author":"E Cuesta","year":"2006","unstructured":"Cuesta, E., Lubich, C., Palencia, C.: Convolution quadrature time discretization of fractional diffusive-wave equations. Math. Comp. 75, 673\u2013696 (2006)","journal-title":"Math. Comp."},{"key":"9547_CR6","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":"9547_CR7","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1016\/S0378-4371(99)00469-0","volume":"276","author":"BI Henry","year":"2000","unstructured":"Henry, B.I., Wearne, S.L.: Fractional reaction-diffusion. Physica A 276, 448\u2013455 (2000)","journal-title":"Physica A"},{"key":"9547_CR8","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\u2013936 (2005)","journal-title":"J. Comput. Phys."},{"key":"9547_CR9","doi-asserted-by":"crossref","unstructured":"Liu, F., Yang, Q., Turner, I.: Two new implicit numerical methods for the fractional cable equation. J. Comput. Nonlinear Dynamics 6, (2011). doi: 10.1115\/1.4002269","DOI":"10.1115\/1.4002269"},{"key":"9547_CR10","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. Comput. Appl. Math. 231, 160\u2013176 (2009)","journal-title":"Comput. Appl. Math."},{"key":"9547_CR11","doi-asserted-by":"crossref","first-page":"1332","DOI":"10.1137\/050629653","volume":"44","author":"M L\u00f3pez-Fern\u00e1ndez","year":"2006","unstructured":"L\u00f3pez-Fern\u00e1ndez, M., Palencia, C., Sch\u00e4dle, A.: A spectral order method for inverting sectorial Laplace transforms. SIAM J. Numer. Anal. 44, 1332\u20131350 (2006)","journal-title":"SIAM J. Numer. Anal."},{"key":"9547_CR12","doi-asserted-by":"crossref","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."},{"key":"9547_CR13","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1017\/S0334270000007268","volume":"35","author":"W McLean","year":"1993","unstructured":"McLean, W., Thom\u00e9e, V.: Numerical solution of an evolution equation with a positive-type memory term. J. Aust. Math. Soc. Ser. B 35, 23\u201370 (1993)","journal-title":"J. Aust. Math. Soc. Ser. B"},{"key":"9547_CR14","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1007\/s00211-006-0045-y","volume":"105","author":"W McLean","year":"2007","unstructured":"McLean, W., Mustapha, K.: A second-order accurate numerical method for a fractional wave equation. Numer. Math. 105, 481\u2013510 (2007)","journal-title":"Numer. Math."},{"key":"9547_CR15","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/s11075-008-9258-8","volume":"52","author":"W McLean","year":"2009","unstructured":"McLean, W., Mustapha, K.: Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation. Numer. Algor. 52, 69\u201388 (2009)","journal-title":"Numer. Algor."},{"key":"9547_CR16","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1216\/JIE-2010-22-1-57","volume":"22","author":"W McLean","year":"2010","unstructured":"McLean, W., Thom\u00e9e, V.: Numerical solution via Laplace transforms of a fractional order evolution equation. J. Integral Equ. Appl. 22, 57\u201394 (2010)","journal-title":"J. Integral Equ. Appl."},{"key":"9547_CR17","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1093\/imanum\/drp004","volume":"30","author":"W McLean","year":"2010","unstructured":"McLean, W., Thom\u00e9e, V.: Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional order evolution equation. IMA J. Numer. Anal. 30, 208\u2013230 (2010)","journal-title":"IMA J. Numer. Anal."},{"key":"9547_CR18","doi-asserted-by":"crossref","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":"9547_CR19","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1093\/imanum\/drp057","volume":"31","author":"K Mustapha","year":"2011","unstructured":"Mustapha, K.: An implicit finite difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements. IMA J. Numer. Anal. 31, 719\u2013739 (2011)","journal-title":"IMA J. Numer. Anal."},{"key":"9547_CR20","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1007\/s11075-010-9379-8","volume":"56","author":"K Mustapha","year":"2011","unstructured":"Mustapha, K., McLean, W.: Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation. Numer. Algor. 56, 159\u2013184 (2011)","journal-title":"Numer. Algor."},{"key":"9547_CR21","doi-asserted-by":"crossref","unstructured":"Mustapha, K., McLean, W.: Uniform convergence for a discontinuous Galerkin, time stepping method applied to a fractional diffusion equation. IMA J. Numer. Anal. doi: 10.1093\/imanum\/drr027","DOI":"10.1093\/imanum\/drr027"},{"key":"9547_CR22","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1137\/050623139","volume":"28","author":"A Sch\u00e4dle","year":"2006","unstructured":"Sch\u00e4dle, A., L\u00f3pez-Fernandez, M., Lubich, C.: Fast and oblivious convolution quadrature. SIAM J. Sci. Comput. 28, 421\u2013438 (2006)","journal-title":"SIAM J. Sci. Comput."},{"key":"9547_CR23","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":"9547_CR24","doi-asserted-by":"crossref","first-page":"2782","DOI":"10.1063\/1.527251","volume":"27","author":"W Wyss","year":"1986","unstructured":"Wyss, W.: Fractional diffusion equation. J. Math. Phys. 27, 2782\u20132785 (1986)","journal-title":"J. Math. Phys."},{"key":"9547_CR25","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":"9547_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."},{"key":"9547_CR27","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, 645\u2013667 (2009)","journal-title":"IMA J. Appl. Math."},{"key":"9547_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, 1079\u20131095 (2008)","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-012-9547-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-012-9547-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-012-9547-0","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,1]],"date-time":"2019-06-01T05:08:17Z","timestamp":1559365697000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-012-9547-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,2,10]]},"references-count":28,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2012,12]]}},"alternative-id":["9547"],"URL":"https:\/\/doi.org\/10.1007\/s11075-012-9547-0","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,2,10]]}}}