{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T03:07:45Z","timestamp":1774494465058,"version":"3.50.1"},"reference-count":53,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2015,1,29]],"date-time":"2015-01-29T00:00:00Z","timestamp":1422489600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2015,11]]},"DOI":"10.1007\/s11075-015-9965-x","type":"journal-article","created":{"date-parts":[[2015,1,28]],"date-time":"2015-01-28T01:54:54Z","timestamp":1422410094000},"page":"625-651","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":35,"title":["A compact finite difference method for a class of time fractional convection-diffusion-wave equations with variable coefficients"],"prefix":"10.1007","volume":"70","author":[{"given":"Yuan-Ming","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2015,1,29]]},"reference":[{"key":"9965_CR1","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1023\/A:1016539022492","volume":"29","author":"OP Agrawal","year":"2002","unstructured":"Agrawal, O.P.: Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dynam. 29, 145\u2013155 (2002)","journal-title":"Nonlinear Dynam."},{"key":"9965_CR2","doi-asserted-by":"crossref","unstructured":"Bueno-Orovio, A., Kay, D., Burrage, K.: Fourier spectral methods for fractional-in-space reaction-diffusion equations. BIT. doi: 10.1007\/s10543-014-0484-2","DOI":"10.1007\/s10543-014-0484-2"},{"key":"9965_CR3","doi-asserted-by":"crossref","first-page":"20140352","DOI":"10.1098\/rsif.2014.0352","volume":"11","author":"A Bueno-Orovio","year":"2014","unstructured":"Bueno-Orovio, A., Kay, D., Grau, V., Rodriguez, B., Burrage, K.: Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization. J. R. Soc. Interface. 11, 20140352\u201320140352 (2014)","journal-title":"J. R. Soc. Interface."},{"key":"9965_CR4","doi-asserted-by":"crossref","first-page":"A2145","DOI":"10.1137\/110847007","volume":"34","author":"K Burrage","year":"2012","unstructured":"Burrage, K., Hale, N., Kay, D.: An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations. SIAM, J. Sci. Comput. 34, A2145\u2013A2172 (2012)","journal-title":"SIAM, J. Sci. Comput."},{"key":"9965_CR5","doi-asserted-by":"crossref","first-page":"754","DOI":"10.1016\/j.amc.2007.09.020","volume":"198","author":"C Chen","year":"2008","unstructured":"Chen, C., Liu, F., Burrage, K.: Finite difference methods and a Fourier analysis for the fractional reaction-diffusion equation. Appl. Math. Comput. 198, 754\u2013769 (2008)","journal-title":"Appl. Math. Comput."},{"key":"9965_CR6","doi-asserted-by":"crossref","unstructured":"Chen, C., Liu, F., Turner, I., Anh, V.: Numerical schemes with high spatial accuracy for a variable order anomalous subdiffusion equation. SIAM J. Sci. Comput. 32, 1740\u20131760 (2010)","DOI":"10.1137\/090771715"},{"key":"9965_CR7","doi-asserted-by":"crossref","first-page":"5729","DOI":"10.1016\/j.amc.2010.12.049","volume":"217","author":"C Chen","year":"2011","unstructured":"Chen, C., Liu, F., Turner, I., Anh, V.: Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term. Appl. Math. Comput. 217, 5729\u20135742 (2011)","journal-title":"Appl. Math. Comput."},{"key":"9965_CR8","doi-asserted-by":"crossref","first-page":"1737","DOI":"10.1016\/j.amc.2012.08.014","volume":"219","author":"J Chen","year":"2012","unstructured":"Chen, J., Liu, F., Anh, V., Shen, S., Liu, Q., Liao, C.: The analytical solution and numerical solution of the fractional diffusion-wave equation with damping. Appl. Math. Comput. 219, 1737\u20131748 (2012)","journal-title":"Appl. Math. Comput."},{"key":"9965_CR9","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1007\/s12190-007-0013-4","volume":"26","author":"S Chen","year":"2008","unstructured":"Chen, S., Liu, F.: ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation. J. Appl. Math. Comput. 26, 295\u2013311 (2008)","journal-title":"J. Appl. Math. Comput."},{"key":"9965_CR10","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1016\/j.apm.2007.11.005","volume":"33","author":"S Chen","year":"2009","unstructured":"Chen, S., Liu, F., Zhuang, P., Anh, V.: Finite difference approximations for the fractional Fokker-Planck equation. Appl. Math. Model 33, 256\u2013273 (2009)","journal-title":"Appl. Math. Model"},{"key":"9965_CR11","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1016\/j.cam.2013.06.001","volume":"255","author":"M Cui","year":"2014","unstructured":"Cui, M.: A high-order compact exponential scheme for the fractional convection-diffusion equation. J. Comput. Appl. Math. 255, 404\u2013416 (2014)","journal-title":"J. Comput. Appl. Math."},{"key":"9965_CR12","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, 2998\u20133007 (2010)","journal-title":"Appl. Math. Model."},{"key":"9965_CR13","first-page":"309","volume":"27","author":"Y Fujita","year":"1990","unstructured":"Fujita, Y.: Integrodifferential equation which interpolates the heat equation and the wave equation. I. Osaka J. Math. 27, 309\u2013321 (1990)","journal-title":"I. Osaka J. Math."},{"key":"9965_CR14","first-page":"797","volume":"27","author":"Y Fujita","year":"1990","unstructured":"Fujita, Y.: Integrodifferential equation which interpolates the heat equation and the wave equation. II. Osaka J. Math. 27, 797\u2013804 (1990)","journal-title":"II. Osaka J. Math."},{"key":"9965_CR15","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 Dynam. 29, 129\u2013143 (2002)","journal-title":"Nonlinear Dynam."},{"key":"9965_CR16","first-page":"303","volume":"56","author":"Y Gu","year":"2010","unstructured":"Gu, Y., Zhuang, P., Liu, F.: An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation. Comput. Model. Eng. Sci 56, 303\u2013334 (2010)","journal-title":"Comput. Model. Eng. Sci"},{"key":"9965_CR17","doi-asserted-by":"crossref","DOI":"10.1142\/8072","volume-title":"Fractional Calculus: An Introduction for Physicists","author":"R Herrmann","year":"2011","unstructured":"Herrmann, R.: Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore (2011)"},{"key":"9965_CR18","doi-asserted-by":"crossref","DOI":"10.1142\/3779","volume-title":"Applications of Fractional Calculus in Physics","author":"R Hilfer","year":"2000","unstructured":"Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)"},{"key":"9965_CR19","doi-asserted-by":"crossref","first-page":"5019","DOI":"10.1016\/j.amc.2011.10.069","volume":"218","author":"X Hu","year":"2012","unstructured":"Hu, X., Zhang, L.: 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":"9965_CR20","doi-asserted-by":"crossref","unstructured":"Huang, J., Tang, Y., V\u00e1zquez, L., Yang, J.: Two finite difference schemes for time fractional diffusion-wave equation. Numer. Algor. doi: 10.1007\/s11075-012-9689-0","DOI":"10.1007\/s11075-012-9689-0"},{"key":"9965_CR21","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-VCH, Weinheim (2008)"},{"key":"9965_CR22","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":"9965_CR23","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."},{"key":"9965_CR24","doi-asserted-by":"crossref","first-page":"2108","DOI":"10.1137\/080718942","volume":"47","author":"X Li","year":"2009","unstructured":"Li, X., Xu, C.: 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":"9965_CR25","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":"9965_CR26","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.amc.2006.08.162","volume":"191","author":"F Liu","year":"2007","unstructured":"Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput. 191, 12\u201320 (2007)","journal-title":"Appl. Math. Comput."},{"key":"9965_CR27","doi-asserted-by":"crossref","first-page":"2990","DOI":"10.1016\/j.camwa.2012.01.020","volume":"64","author":"F Liu","year":"2012","unstructured":"Liu, F., Zhuang, P., Burrage, K.: Numerical methods and analysis for a class of fractional advection-dispersion models. Comput. Math. Appl. 64, 2990\u20133007 (2012)","journal-title":"Comput. Math. Appl."},{"key":"9965_CR28","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1016\/j.amc.2013.10.008","volume":"226","author":"Q Liu","year":"2014","unstructured":"Liu, Q., Liu, F., Turner, I., Anh, V., Gu, Y.T.: A RBF meshless approach for modeling a fractal mobile\/immobile transport model. Appl. Math. Comput. 226, 336\u2013347 (2014)","journal-title":"Appl. Math. Comput."},{"issue":"2","key":"9965_CR29","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1007\/s13137-010-0012-8","volume":"1","author":"Yu Luchko","year":"2011","unstructured":"Luchko, Yu., Punzi, A.: Modeling anomalous heat transport in geothermal reservoirs via fractional diffusion equations. Int. J. Geomath. 1(2), 257\u2013276 (2011)","journal-title":"Int. J. Geomath."},{"key":"9965_CR30","doi-asserted-by":"crossref","DOI":"10.1142\/p614","volume-title":"Fractional Calculus and Waves in Linear Viscoelasticity","author":"F Mainardi","year":"2010","unstructured":"Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010)"},{"key":"9965_CR31","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":"9965_CR32","doi-asserted-by":"crossref","first-page":"R161","DOI":"10.1088\/0305-4470\/37\/31\/R01","volume":"37","author":"R Metzler","year":"2004","unstructured":"Metzler, R., Klafter, J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A Math. Gen. 37, R161\u2013R208 (2004)","journal-title":"J. Phys. A Math. Gen."},{"key":"9965_CR33","doi-asserted-by":"crossref","DOI":"10.1142\/2933","volume-title":"Fitted Numerical Methods for Singular Perturbation Problems","author":"JJH Miller","year":"1996","unstructured":"Miller, J.J.H., O\u2019Riordan, E., Shishkin, G.I.: Fitted Numerical Methods for Singular Perturbation Problems. World Scientific, Singapore (1996)"},{"key":"9965_CR34","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1007\/s11075-012-9631-5","volume":"63","author":"A Mohebbi","year":"2013","unstructured":"Mohebbi, A., Abbaszadeh, M.: Compact finite difference scheme for the solution of time fractional advection-dispersion equation. Numer. Algor. 63, 431\u2013452 (2013)","journal-title":"Numer. Algor."},{"key":"9965_CR35","volume-title":"The Fractional Calculus","author":"KB Oldham","year":"1974","unstructured":"Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)"},{"key":"9965_CR36","volume-title":"Fractional Differential Equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)"},{"key":"9965_CR37","volume-title":"Numerical Approximation of Partial Differential Equations","author":"A Quarteroni","year":"1997","unstructured":"Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer-Verlag, New York (1997)"},{"key":"9965_CR38","doi-asserted-by":"crossref","first-page":"4125","DOI":"10.1016\/j.cnsns.2012.03.003","volume":"17","author":"A Saadatmandi","year":"2012","unstructured":"Saadatmandi, A., Dehghan, M., Azizi, M.R.: The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients. Commun. Nonlinear Sci. Numer. Simulat. 17, 4125\u20134136 (2012)","journal-title":"Commun. Nonlinear Sci. Numer. Simulat."},{"key":"9965_CR39","doi-asserted-by":"crossref","DOI":"10.1201\/9780203908518","volume-title":"The Theory of Difference Schemes","author":"AA Samarskii","year":"2001","unstructured":"Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker Inc., New York (2001)"},{"key":"9965_CR40","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":"9965_CR41","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1007\/s11075-010-9393-x","volume":"56","author":"S Shen","year":"2011","unstructured":"Shen, S., Liu, F., Anh, V.: Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation. Numer. Algor. 56, 383\u2013404 (2011)","journal-title":"Numer. Algor."},{"key":"9965_CR42","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":"9965_CR43","volume-title":"Method of Fractional Derivatives","author":"VV Uchaikin","year":"2008","unstructured":"Uchaikin, V.V.: Method of Fractional Derivatives. Artishok, Ul\u2019janovsk (2008)"},{"key":"9965_CR44","doi-asserted-by":"crossref","first-page":"4208","DOI":"10.1016\/j.cnsns.2011.03.021","volume":"16","author":"M Uddin","year":"2011","unstructured":"Uddin, M., Hag, S.: RBFs approximation method for time fractional partial differential equations. Commun. Nonlinear Sci. Numer. Simulat. 16, 4208\u20134214 (2011)","journal-title":"Commun. Nonlinear Sci. Numer. Simulat."},{"key":"9965_CR45","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-05156-2","volume-title":"Matrix Iterative Analysis","author":"RS Varga","year":"2000","unstructured":"Varga, R.S.: Matrix Iterative Analysis. Springer-Verlag, Berlin (2000)"},{"key":"9965_CR46","doi-asserted-by":"crossref","first-page":"170","DOI":"10.4208\/eajam.110312.240412a","volume":"2","author":"SW Vong","year":"2012","unstructured":"Vong, S.W., Pang, H.K., Jin, X.Q.: A high-order difference scheme for the generalized Cattaneo equation. East Asian J. Appl. Math. 2, 170\u2013184 (2012)","journal-title":"East Asian J. Appl. Math."},{"key":"9965_CR47","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":"9965_CR48","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":"9965_CR49","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1016\/j.camwa.2013.01.031","volume":"66","author":"H Zhang","year":"2013","unstructured":"Zhang, H., Liu, F., Phanikumar, M.S., Meerschaert, M.M.: A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model. Comput. Math. Appl. 66, 693\u2013701 (2013)","journal-title":"Comput. Math. Appl."},{"key":"9965_CR50","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1016\/j.advwatres.2009.01.008","volume":"32","author":"Y Zhang","year":"2009","unstructured":"Zhang, Y., Benson, D.A., Reeves, D.M.: Time and space nonlocalities underlying fractional-derivative models: distinction and literature review of field applications. Adv. Water Resour. 32, 561\u2013581 (2009)","journal-title":"Adv. Water Resour."},{"key":"9965_CR51","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."},{"key":"9965_CR52","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":"9965_CR53","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 methods 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":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-015-9965-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-015-9965-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-015-9965-x","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,3]],"date-time":"2019-06-03T14:01:15Z","timestamp":1559570475000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-015-9965-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1,29]]},"references-count":53,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2015,11]]}},"alternative-id":["9965"],"URL":"https:\/\/doi.org\/10.1007\/s11075-015-9965-x","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,1,29]]}}}