{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,31]],"date-time":"2026-07-31T07:52:39Z","timestamp":1785484359690,"version":"3.56.0"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,4,14]],"date-time":"2025-04-14T00:00:00Z","timestamp":1744588800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,4,14]],"date-time":"2025-04-14T00:00:00Z","timestamp":1744588800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2025,8]]},"DOI":"10.1007\/s12190-025-02448-6","type":"journal-article","created":{"date-parts":[[2025,4,14]],"date-time":"2025-04-14T07:18:27Z","timestamp":1744615107000},"page":"5977-6011","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A high-order $$L2$$-$$1_{\\sigma}$$ compact exponential difference method for multi-term time-fractional convection-reaction-diffusion equations"],"prefix":"10.1007","volume":"71","author":[{"given":"Yuan-Ming","family":"Wang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xin","family":"Wen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zi-Yun","family":"Zheng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,4,14]]},"reference":[{"key":"2448_CR1","first-page":"12","volume":"268","author":"A.A. Alikhanov","year":"2015","unstructured":"Alikhanov, A.A.: Numerical methods of solutions of boundary value problems for the multi-term variable-distributed order diffusion equation. Appl. Math. Comput. 268, 12\u201322 (2015)","journal-title":"Appl. Math. Comput."},{"key":"2448_CR2","doi-asserted-by":"publisher","first-page":"294","DOI":"10.1115\/1.3167615","volume":"51","author":"R.L. Bagley","year":"1984","unstructured":"Bagley, R.L., Torvik, P.J.: On the appearance of the fractional derivative in the behaviour of real materials. J. Appl. Mech. 51, 294\u2013298 (1984)","journal-title":"J. Appl. Mech."},{"key":"2448_CR3","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511543234","volume-title":"Collocation Methods for Volterra Integral and Related Functional Differential Equations","author":"H. Brunner","year":"2004","unstructured":"Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press, Cambridge (2004)"},{"key":"2448_CR4","doi-asserted-by":"publisher","first-page":"624","DOI":"10.1007\/s10915-018-0863-y","volume":"79","author":"H. Chen","year":"2019","unstructured":"Chen, H., Stynes, M.: Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem. J. Sci. Comput. 79, 624\u2013647 (2019)","journal-title":"J. Sci. Comput."},{"key":"2448_CR5","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1515\/cmam-2021-0053","volume":"22","author":"H. Chen","year":"2022","unstructured":"Chen, H., Stynes, M.: Using complete monotonicity to deduce local error estimates for discretizations of a multi-term time-fractional diffusion equation. Comput. Methods Appl. Math. 22, 15\u201329 (2022)","journal-title":"Comput. Methods Appl. Math."},{"key":"2448_CR6","doi-asserted-by":"publisher","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":"2448_CR7","doi-asserted-by":"publisher","first-page":"174","DOI":"10.1016\/j.cam.2015.04.037","volume":"290","author":"M. Dehghan","year":"2015","unstructured":"Dehghan, M., Safarpoor, M., Abbaszadeh, M.: Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations. J. Comput. Appl. Math. 290, 174\u2013195 (2015)","journal-title":"J. Comput. Appl. Math."},{"key":"2448_CR8","doi-asserted-by":"publisher","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-Verlag, Berlin (2010)"},{"key":"2448_CR9","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1007\/s10915-017-0407-x","volume":"73","author":"G. Gao","year":"2017","unstructured":"Gao, G., Alikhanov, A.A., Sun, Z.Z.: The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations. J. Sci. Comput. 73, 93\u2013121 (2017)","journal-title":"J. Sci. Comput."},{"key":"2448_CR10","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1515\/cmam-2017-0019","volume":"18","author":"J.L. Gracia","year":"2018","unstructured":"Gracia, J.L., O\u2019Riordan, E., Stynes, M.: Convergence in positive time for a finite difference method applied to a fractional convection-diffusion problem. Comput. Methods Appl. Math. 18, 33\u201342 (2018)","journal-title":"Comput. Methods Appl. Math."},{"key":"2448_CR11","doi-asserted-by":"publisher","first-page":"815","DOI":"10.1515\/cmam-2019-0042","volume":"20","author":"C. Huang","year":"2020","unstructured":"Huang, C., Liu, X., Meng, X., Stynes, M.: Error analysis of a finite difference method on graded meshes for a multiterm time-fractional initial-boundary value problem. Comput. Methods Appl. Math. 20, 815\u2013825 (2020)","journal-title":"Comput. Methods Appl. Math."},{"key":"2448_CR12","doi-asserted-by":"publisher","first-page":"10","DOI":"10.1007\/s10915-019-01115-w","volume":"82","author":"C. Huang","year":"2020","unstructured":"Huang, C., Stynes, M.: Superconvergence of a finite element method for the multi-term time-fractional diffusion problem. J. Sci. Comput. 82, 10 (2020)","journal-title":"J. Sci. Comput."},{"key":"2448_CR13","doi-asserted-by":"publisher","first-page":"3377","DOI":"10.1016\/j.camwa.2012.02.042","volume":"64","author":"H. Jiang","year":"2012","unstructured":"Jiang, H., Liu, F., Turner, I., Burrage, K.: Analytical solutions for the multi-term time-fractional diffusion-wave\/diffusion equations in a finite domain. Comput. Math. Appl. 64, 3377\u20133388 (2012)","journal-title":"Comput. Math. Appl."},{"key":"2448_CR14","doi-asserted-by":"publisher","first-page":"1117","DOI":"10.1016\/j.jmaa.2011.12.055","volume":"389","author":"H. Jiang","year":"2012","unstructured":"Jiang, H., Liu, F., Turner, I., Burrage, K.: Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain. J. Math. Anal. Appl. 389, 1117\u20131127 (2012)","journal-title":"J. Math. Anal. Appl."},{"key":"2448_CR15","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1016\/j.jcp.2014.10.051","volume":"281","author":"B. Jin","year":"2015","unstructured":"Jin, B., Lazarov, R., Liu, Y., Zhou, Z.: The Galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys. 281, 825\u2013843 (2015)","journal-title":"J. Comput. Phys."},{"key":"2448_CR16","doi-asserted-by":"publisher","first-page":"2135","DOI":"10.1090\/mcom\/3410","volume":"88","author":"N. Kopteva","year":"2019","unstructured":"Kopteva, N.: Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions. Math. Comp. 88, 2135\u20132155 (2019)","journal-title":"Math. Comp."},{"key":"2448_CR17","doi-asserted-by":"crossref","unstructured":"Kopteva, N., Stynes, M.: A posteriori error analysis for variable-coefficient multiterm time-fractional subdiffusion equations. J. Sci. Comput. 92, 73 (2022)","DOI":"10.1007\/s10915-022-01936-2"},{"key":"2448_CR18","first-page":"381","volume":"257","author":"Z. Li","year":"2015","unstructured":"Li, Z., Liu, Y., Yamamoto, M.: Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients. Appl. Math. Comput. 257, 381\u2013397 (2015)","journal-title":"Appl. Math. Comput."},{"key":"2448_CR19","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., Zhang, J.: 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":"2448_CR20","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.: A discrete Gr\u00f6nwall inequality with applications to numerical schemes for subdiffusion problems. SIAM J. Numer. Anal. 57, 218\u2013237 (2019)","journal-title":"SIAM J. Numer. Anal."},{"key":"2448_CR21","doi-asserted-by":"publisher","first-page":"567","DOI":"10.4208\/cicp.OA-2020-0124","volume":"30","author":"H.-L. Liao","year":"2021","unstructured":"Liao, H.-L., McLean, W., Zhang, J.: A second-order scheme with nonuniform time steps for a linear reaction-subdiffusion problem. Commun. Comput. Phys. 30, 567\u2013601 (2021)","journal-title":"Commun. Comput. Phys."},{"key":"2448_CR22","doi-asserted-by":"publisher","first-page":"9","DOI":"10.2478\/s13540-013-0002-2","volume":"16","author":"F. Liu","year":"2013","unstructured":"Liu, F., Meerschaert, M.M., McGough, R.J., Zhuang, P., Liu, Q.: Numerical methods for solving the multi-term time-fractional wave-diffusion equation. Fract. Calc. Appl. Anal. 16, 9\u201325 (2013)","journal-title":"Fract. Calc. Appl. Anal."},{"key":"2448_CR23","doi-asserted-by":"publisher","first-page":"538","DOI":"10.1016\/j.jmaa.2010.08.048","volume":"374","author":"Y. Luchko","year":"2011","unstructured":"Luchko, Y.: Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation. J. Math. Anal. Appl. 374, 538\u2013548 (2011)","journal-title":"J. Math. Anal. Appl."},{"key":"2448_CR24","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1007\/978-3-7091-2664-6_7","volume-title":"Fractals and Fractional Calculus in Continuum Mechanics","author":"F. Mainardi","year":"1997","unstructured":"Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.): Fractals and Fractional Calculus in Continuum Mechanics, pp. 291\u2013348. Springer, Vienna (1997)"},{"key":"2448_CR25","doi-asserted-by":"crossref","unstructured":"Meng, X., Stynes, M.: Barrier function local and global analysis of an L1 finite element method for a multiterm time-fractional initial-boundary value problem. J. Sci. Comput. 84, 5 (2020)","DOI":"10.1007\/s10915-020-01250-9"},{"key":"2448_CR26","doi-asserted-by":"publisher","first-page":"1621","DOI":"10.1103\/PhysRevE.58.1621","volume":"58","author":"R. Metzler","year":"1998","unstructured":"Metzler, R., Klafter, J., Sokolov, I.M.: Anomalous transport in external fields: continuous time random walks and fractional diffusion equations extended. Phys. Rev. E 58, 1621\u20131633 (1998)","journal-title":"Phys. Rev. E"},{"key":"2448_CR27","doi-asserted-by":"publisher","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 subdiffusion equation, theory and applications. Numer. Algorithms 61, 525\u2013543 (2012)","journal-title":"Numer. Algorithms"},{"key":"2448_CR28","doi-asserted-by":"publisher","first-page":"242","DOI":"10.4208\/eajam.181113.280514a","volume":"4","author":"J.C. Ren","year":"2014","unstructured":"Ren, J.C., Sun, Z.Z.: Efficient and stable numerical methods for multi-term time fractional sub-diffusion equations. East Asian J. Appl. Math. 4, 242\u2013266 (2014)","journal-title":"East Asian J. Appl. Math."},{"key":"2448_CR29","doi-asserted-by":"publisher","first-page":"1296","DOI":"10.1029\/2003WR002141","volume":"39","author":"R. Schumer","year":"2003","unstructured":"Schumer, R., Benson, D.A., Meerschaert, M.M., Baeumer, B.: Fractal mobile\/immobile solute transport. Water. Res. Res. 39, 1296 (2003)","journal-title":"Water. Res. Res."},{"key":"2448_CR30","doi-asserted-by":"crossref","unstructured":"She, M., Li, D., Sun, H.-W.: A transformed [CDATA[]] method for solving the multi-term time-fractional diffusion problem. Math. Comput. Simul. 193, 584\u2013606 (2022)","DOI":"10.1016\/j.matcom.2021.11.005"},{"key":"2448_CR31","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1007\/s11071-015-2326-4","volume":"83","author":"S.C. Shiralashetti","year":"2016","unstructured":"Shiralashetti, S.C., Deshi, A.B.: An efficient Haar wavelet collocation method for the numerical solution of multi-term fractional differential equations. Nonlinear Dynam. 83, 293\u2013303 (2016)","journal-title":"Nonlinear Dynam."},{"key":"2448_CR32","doi-asserted-by":"publisher","first-page":"616","DOI":"10.1016\/j.mcm.2009.11.002","volume":"51","author":"V. Srivastava","year":"2010","unstructured":"Srivastava, V., Rai, K.N.: A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues. Math. Comput. Model. 51, 616\u2013624 (2010)","journal-title":"Math. Comput. Model."},{"key":"2448_CR33","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":"2448_CR34","doi-asserted-by":"publisher","first-page":"268","DOI":"10.1016\/j.cam.2005.12.005","volume":"198","author":"Z.F. Tian","year":"2007","unstructured":"Tian, Z.F., Ge, Y.B.: A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problems. J. Comput. Appl. Math. 198, 268\u2013286 (2007)","journal-title":"J. Comput. Appl. Math."},{"key":"2448_CR35","doi-asserted-by":"publisher","first-page":"1023","DOI":"10.1007\/s10543-020-00841-0","volume":"61","author":"Y.-M. Wang","year":"2021","unstructured":"Wang, Y.-M.: A high-order compact finite difference method on nonuniform time meshes for variable coefficient reaction-subdiffusion problems with a weak initial singularity. BIT Numer. Math. 61, 1023\u20131059 (2021)","journal-title":"BIT Numer. Math."},{"key":"2448_CR36","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jcp.2014.08.012","volume":"277","author":"Z. Wang","year":"2014","unstructured":"Wang, Z., Vong, S.: Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation. J. Comput. Phys. 277, 1\u201315 (2014)","journal-title":"J. Comput. Phys."},{"key":"2448_CR37","first-page":"125316","volume":"381","author":"Y.-M. Wang","year":"2020","unstructured":"Wang, Y.-M., Wen, X.: A compact exponential difference method for multi-term time-fractional convection-reaction-diffusion problems with non-smooth solutions. Appl. Math. Comput. 381, 125316 (2020)","journal-title":"Appl. Math. Comput."},{"key":"2448_CR38","doi-asserted-by":"crossref","unstructured":"Xu, T., L\u00fc, S., Chen, W., Chen, H.: Finite difference scheme for multi-term variable-order fractional diffusion equation. Adv. Difference Equations 103, 13 (2018)","DOI":"10.1186\/s13662-018-1544-8"},{"key":"2448_CR39","doi-asserted-by":"publisher","first-page":"113118","DOI":"10.1016\/j.cma.2020.113118","volume":"367","author":"Z. Yang","year":"2020","unstructured":"Yang, Z., Zheng, X., Wang, H.: A variably distributed-order time-fractional diffusion equation: analysis and approximation. Comput. Methods Appl. Mech. Engrg. 367, 113118 (2020)","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"key":"2448_CR40","first-page":"531","volume":"227","author":"H. Ye","year":"2014","unstructured":"Ye, H., Liu, F., Turner, I., Anh, V., Burrage, K.: Maximum principle and numerical method for the multi-term time-space riesz-caputo fractional differential equations. Appl. Math. Comput. 227, 531\u2013540 (2014)","journal-title":"Appl. Math. Comput."},{"key":"2448_CR41","doi-asserted-by":"publisher","first-page":"3525","DOI":"10.1007\/s40314-017-0530-1","volume":"37","author":"M. Zaky","year":"2018","unstructured":"Zaky, M.: A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations. Comput. Appl. Math. 37, 3525\u20133538 (2018)","journal-title":"Comput. Appl. Math."},{"key":"2448_CR42","doi-asserted-by":"publisher","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":"2448_CR43","first-page":"857205","volume":"2013","author":"J. Zhao","year":"2013","unstructured":"Zhao, J., Xiao, J., Xu, Y.: Stability and convergence of an effective finite element method for multiterm fractional partial differential equations. Abstr. Appl. Anal. 2013, 857205 (2013)","journal-title":"Abstr. Appl. Anal."},{"key":"2448_CR44","doi-asserted-by":"publisher","first-page":"1087","DOI":"10.1016\/j.camwa.2016.05.005","volume":"73","author":"Y. Zhao","year":"2017","unstructured":"Zhao, Y., Zhang, Y., Liu, F., Turner, I., Tang, Y., Anh, V.: Convergence and superconvergence of a fully-discrete scheme for multi-term time fractional diffusion equations. Comput. Math. Appl. 73, 1087\u20131099 (2017)","journal-title":"Comput. Math. Appl."},{"key":"2448_CR45","doi-asserted-by":"publisher","first-page":"4970","DOI":"10.1016\/j.apm.2015.12.011","volume":"40","author":"M. Zheng","year":"2016","unstructured":"Zheng, M., Liu, F., Anh, V., Turner, I.: A high-order spectral method for the multi-term time-fractional diffusion equations. Appl. Math. Model. 40, 4970\u20134985 (2016)","journal-title":"Appl. Math. Model."},{"key":"2448_CR46","doi-asserted-by":"crossref","unstructured":"Zheng, Z.-Y., Wang, Y.-M.: An averaged L1-type compact difference method for time-fractional mobile\/immobile diffusion equations with weakly singular solutions. Appl. Math. Lett. 131, 108076 (2022)","DOI":"10.1016\/j.aml.2022.108076"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02448-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-025-02448-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02448-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,6]],"date-time":"2025-09-06T12:16:52Z","timestamp":1757161012000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-025-02448-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,14]]},"references-count":46,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,8]]}},"alternative-id":["2448"],"URL":"https:\/\/doi.org\/10.1007\/s12190-025-02448-6","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,14]]},"assertion":[{"value":"12 September 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 February 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 March 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 April 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}},{"value":"The authors declare that they have no competing interests.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}