{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,19]],"date-time":"2025-10-19T15:48:39Z","timestamp":1760888919281},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2014,3,21]],"date-time":"2014-03-21T00:00:00Z","timestamp":1395360000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2014,12]]},"DOI":"10.1007\/s10915-014-9841-1","type":"journal-article","created":{"date-parts":[[2014,3,20]],"date-time":"2014-03-20T14:28:52Z","timestamp":1395325732000},"page":"629-648","source":"Crossref","is-referenced-by-count":26,"title":["Stability and Convergence of Modified Du Fort\u2013Frankel Schemes for Solving Time-Fractional Subdiffusion Equations"],"prefix":"10.1007","volume":"61","author":[{"given":"Hong-lin","family":"Liao","sequence":"first","affiliation":[]},{"given":"Ya-nan","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Ying","family":"Zhao","sequence":"additional","affiliation":[]},{"given":"Han-sheng","family":"Shi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2014,3,21]]},"reference":[{"issue":"865\u2013882","key":"9841_CR1","first-page":"965","volume":"2","author":"RL Bagley","year":"2000","unstructured":"Bagley, R.L., Torvik, P.J.: On the existence of the order domain and the solution of distributed order equations (Parts I, II). Int. J. Appl. Mech. 2(865\u2013882), 965\u2013987 (2000)","journal-title":"Int. J. Appl. Mech."},{"key":"9841_CR2","doi-asserted-by":"crossref","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 fractional subdiffusion problems. J. Comput. Phys. 229, 6613\u20136622 (2010)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"9841_CR3","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/BF02826009","volume":"41","author":"M Caputo","year":"1995","unstructured":"Caputo, M.: Mean fractional-order-derivatives differential equations and filters. Annali dell\u2019Universita di Ferrara 41(1), 73\u201384 (1995)","journal-title":"Annali dell\u2019Universita di Ferrara"},{"key":"9841_CR4","first-page":"421","volume":"4","author":"M Caputo","year":"2001","unstructured":"Caputo, M.: Distributed order differential equations modelling dielectric induction and diffusion. Fract. Calc. Appl. Anal. 4, 421\u2013442 (2001)","journal-title":"Fract. Calc. Appl. Anal."},{"issue":"046129","key":"9841_CR5","first-page":"1","volume":"66","author":"AV Chechkin","year":"2002","unstructured":"Chechkin, A.V., Gorenflo, R., Sokolov, I.M.: Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E 66(046129), 1\u20136 (2002)","journal-title":"Phys. Rev. E"},{"key":"9841_CR6","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 equation. SIAM J. Sci. Comput. 32, 1740\u20131760 (2010)","journal-title":"SIAM J. Sci. Comput."},{"key":"9841_CR7","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1090\/S0025-5718-2011-02447-6","volume":"81","author":"C Chen","year":"2011","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 (2011)","journal-title":"Math. Comput."},{"key":"9841_CR8","unstructured":"Ciesielski, M., Leszczynski, J.: Numerical simulations of anomalous diffusion. In: Proceedings of the 15th Conference on Computer Methods in Mechanics. Wisla, Polonia. (2003). arXiv:math-ph\/0309007v1"},{"key":"9841_CR9","doi-asserted-by":"crossref","first-page":"692","DOI":"10.1002\/andp.200310032","volume":"12","author":"CFM Coimbra","year":"2003","unstructured":"Coimbra, C.F.M.: Mechanics with variable-order differential operators. Annalen der Physik 12, 692\u2013703 (2003)","journal-title":"Annalen der Physik"},{"key":"9841_CR10","first-page":"531","volume":"4","author":"K Diethelm","year":"2001","unstructured":"Diethelm, K., Ford, N.J.: Numerical solution methods for distributed order differential equations. Fract. Calc. Appl. Anal. 4, 531\u2013542 (2001)","journal-title":"Fract. Calc. Appl. Anal."},{"key":"9841_CR11","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1016\/j.cma.2004.06.006","volume":"194","author":"K Diethelm","year":"2005","unstructured":"Diethelm, K., Ford, N.J., Freed, A.D., Luchko, Yu.: Algorithms for the fractional calculus: a selection of numerical methods. Comput. Methods Appl. Mech. Eng. 194, 743\u2013773 (2005)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"1","key":"9841_CR12","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.cam.2008.07.018","volume":"225","author":"K Diethelm","year":"2009","unstructured":"Diethelm, K., Ford, N.J.: Numerical analysis for distributed-order differential equations. J. Comput. Appl. Math. 225(1), 96\u2013104 (2009)","journal-title":"J. Comput. Appl. Math."},{"key":"9841_CR13","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1023\/A:1016547232119","volume":"29","author":"R Gorenflo","year":"2002","unstructured":"Gorenflo, R., Mainardi, F., et al.: Time-fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 29, 129 (2002)","journal-title":"Nonlinear Dyn."},{"key":"9841_CR14","doi-asserted-by":"crossref","unstructured":"Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann\u2013Liouville fractional derivatives. Rheologica Acta (2005). doi: 10.1007\/s00397-005-0043-5","DOI":"10.1007\/s00397-005-0043-5"},{"key":"9841_CR15","volume-title":"Applications of Fractional Calculus in Physics","year":"2000","unstructured":"Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)"},{"key":"9841_CR16","doi-asserted-by":"crossref","first-page":"1367","DOI":"10.1016\/j.camwa.2006.02.001","volume":"51","author":"G Jumarie","year":"2006","unstructured":"Jumarie, G.: Modified Riemann\u2013Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput. Math. Appl. 51, 1367\u20131376 (2006)","journal-title":"Comput. Math. Appl."},{"key":"9841_CR17","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/BF02832299","volume":"24","author":"G Jumarie","year":"2007","unstructured":"Jumarie, G.: Fractional partial differential equations and modified Riemann\u2013Liouville derivative new methods for solution. J. Appl. Math. Comput. 24, 31\u201348 (2007)","journal-title":"J. Appl. Math. Comput."},{"key":"9841_CR18","doi-asserted-by":"crossref","first-page":"836","DOI":"10.1016\/j.apm.2007.02.020","volume":"32","author":"G Jumarie","year":"2008","unstructured":"Jumarie, G.: Modeling fractional stochastic systems as non-random fractional dynamics driven by Brownian motions. Appl. Math. Model. 32, 836\u2013859 (2008)","journal-title":"Appl. Math. Model."},{"key":"9841_CR19","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1016\/j.aml.2008.06.003","volume":"22","author":"G Jumarie","year":"2009","unstructured":"Jumarie, G.: Table of some basic fractional calculus formulae derived from a modiffed Riemann\u2013Liouville derivative for non-differentiable functions. Appl. Math. Lett. 22, 378\u2013385 (2009)","journal-title":"Appl. Math. Lett."},{"key":"9841_CR20","doi-asserted-by":"crossref","first-page":"1444","DOI":"10.1016\/j.aml.2010.08.001","volume":"23","author":"G Jumarie","year":"2010","unstructured":"Jumarie, G.: Cauchys integral formula via modified Riemann\u2013Liouville derivative for analytic functions of fractional order. Appl. Math. Lett. 23, 1444\u20131450 (2010)","journal-title":"Appl. Math. Lett."},{"key":"9841_CR21","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":"9841_CR22","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/j.physa.2005.12.012","volume":"367","author":"TAM Langlands","year":"2006","unstructured":"Langlands, T.A.M.: Solution of a modified fractional diffusion equation. Phys. A 367, 136\u2013144 (2006)","journal-title":"Phys. A"},{"key":"9841_CR23","doi-asserted-by":"crossref","first-page":"1369","DOI":"10.1090\/S0025-5718-2010-02438-X","volume":"80","author":"Y Lin","year":"2011","unstructured":"Lin, Y., Li, X., Xu, C.: Finite difference\/spectral approximations for the fractional cable equation. Math. Comput. 80, 1369\u20131396 (2011)","journal-title":"Math. Comput."},{"key":"9841_CR24","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1023\/A:1016586905654","volume":"29","author":"CF Lorenzo","year":"2002","unstructured":"Lorenzo, C.F., Hartley, T.T.: Variable-order and distributed order fractional operators. Nonlinear Dyn. 29, 57\u201398 (2002)","journal-title":"Nonlinear Dyn."},{"key":"9841_CR25","doi-asserted-by":"crossref","first-page":"014025","DOI":"10.1088\/0031-8949\/2009\/T136\/014025","volume":"136","author":"JQ Murillo","year":"2009","unstructured":"Murillo, J.Q., Yuste, S.B.: On three explicit difference schemes for fractional diffusion and diffusion-wave equations. Phys. Scr. T 136, 014025 (2009)","journal-title":"Phys. Scr. T"},{"key":"9841_CR26","unstructured":"Murillo, J.Q., Yuste, S.B.: On an explicit difference method for fractional diffusion and diffusion-wave equations. In: Proceedings of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference. San Diego, California, (2009)"},{"key":"9841_CR27","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1007\/s11075-012-9547-0","volume":"61","author":"K Mustapha","year":"2012","unstructured":"Mustapha, K., AlMutawa, J.: A finite difference method for an anomalous sub-diffusion equation, theory and applications. Numer. Algorithm 61, 525\u2013543 (2012)","journal-title":"Numer. Algorithm"},{"key":"9841_CR28","doi-asserted-by":"crossref","first-page":"1975","DOI":"10.1090\/S0025-5718-09-02234-0","volume":"78","author":"K Mustapha","year":"2009","unstructured":"Mustapha, K., McLean, W.: Discontinuous Galerkin method for an evolution equation with a memory term of positive type. Math. Comput. 78, 1975\u20131995 (2009)","journal-title":"Math. Comput."},{"key":"9841_CR29","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1137\/120880719","volume":"51","author":"K Mustapha","year":"2013","unstructured":"Mustapha, K., McLean, W.: Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations. SIAM J. Numer. Anal. 51, 491\u2013515 (2013)","journal-title":"SIAM J. Numer. Anal."},{"key":"9841_CR30","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1142\/S0218348X04002410","volume":"12","author":"M Naber","year":"2004","unstructured":"Naber, M.: Distributed order fractional subdiffusion. Fractals 12, 23\u201332 (2004)","journal-title":"Fractals"},{"key":"9841_CR31","unstructured":"Oldham, K., Spanier, J.: The fractional calculus: theory and applications of differentiation and integration to arbitrary order. In: Mathematics in Science and Engineering, vol. 111. Academic Press, New York (1974)"},{"key":"9841_CR32","volume-title":"Fractional Differential Equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)"},{"key":"9841_CR33","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/BF01911126","volume":"21","author":"SG Samko","year":"1995","unstructured":"Samko, S.G.: Fractional integration and differentiation of variable order. Anal. Math. 21, 213\u2013236 (1995)","journal-title":"Anal. Math."},{"issue":"22","key":"9841_CR34","doi-asserted-by":"crossref","first-page":"10861","DOI":"10.1016\/j.amc.2012.04.047","volume":"218","author":"S Shen","year":"2012","unstructured":"Shen, S., Liu, F., Chen, J., Turner, I., Anh, V.: Numerical techniques for the variable order time fractional diffusion equation. Appl. Math. Comput. 218(22), 10861\u201310870 (2012)","journal-title":"Appl. Math. Comput."},{"key":"9841_CR35","first-page":"1323","volume":"35","author":"IM Sokolov","year":"2004","unstructured":"Sokolov, I.M., Chechkin, A.V., Klafter, J.: Distributed-order fractional kinetics. Acta Physica Polonica 35, 1323\u20131341 (2004)","journal-title":"Acta Physica Polonica"},{"key":"9841_CR36","doi-asserted-by":"crossref","first-page":"4586","DOI":"10.1016\/j.physa.2009.07.024","volume":"388","author":"H Sun","year":"2009","unstructured":"Sun, H., Chen, W., Chen, Y.: Variable-order fractional differential operators in anomalous diffusion modeling. Phys. A 388, 4586\u20134592 (2009)","journal-title":"Phys. A"},{"issue":"4","key":"9841_CR37","doi-asserted-by":"crossref","first-page":"1250085","DOI":"10.1142\/S021812741250085X","volume":"22","author":"H Sun","year":"2012","unstructured":"Sun, H., Chen, W., Li, C., Chen, Y.: Finite difference schemes for variable-order time fractional diffusion equation. Int. J. Bifurc. Chaos 22(4), 1250085 (2012)","journal-title":"Int. J. Bifurc. Chaos"},{"key":"9841_CR38","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1214\/074921706000000798","volume":"51","author":"S Umarov","year":"2006","unstructured":"Umarov, S., Steinberg, S.: Random walk model sassociated with distributed fractional order differential equations. Lect. Notes Monogr. Ser. 51, 117\u2013127 (2006)","journal-title":"Lect. Notes Monogr. Ser."},{"issue":"5","key":"9841_CR39","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 Neumann-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42(5), 1862\u20131874 (2005)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"9841_CR40","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(1), 264\u2013274 (2006)","journal-title":"J. Comput. Phys."},{"key":"9841_CR41","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 sub-diffusion equation. SIAM J. Numer. Anal. 49, 2302\u20132322 (2011)","journal-title":"SIAM J. Numer. Anal."},{"key":"9841_CR42","doi-asserted-by":"crossref","first-page":"1535","DOI":"10.1137\/110840959","volume":"50","author":"YN Zhang","year":"2012","unstructured":"Zhang, Y.N., Sun, Z.Z., Zhao, X.: Compact alternating direction implicit schemes for the two-dimensional fractional diffusion-wave equation. SIAM J. Numer. Anal. 50, 1535\u20131555 (2012)","journal-title":"SIAM J. Numer. Anal."},{"key":"9841_CR43","doi-asserted-by":"crossref","first-page":"1760","DOI":"10.1137\/080730597","volume":"47","author":"P Zhuang","year":"2009","unstructured":"Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical method for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47, 1760\u20131781 (2009)","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-014-9841-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-014-9841-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-014-9841-1","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T16:15:03Z","timestamp":1648656903000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-014-9841-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,3,21]]},"references-count":43,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2014,12]]}},"alternative-id":["9841"],"URL":"https:\/\/doi.org\/10.1007\/s10915-014-9841-1","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,3,21]]}}}