{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T22:40:37Z","timestamp":1783118437971,"version":"3.54.6"},"reference-count":28,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2018,1,22]],"date-time":"2018-01-22T00:00:00Z","timestamp":1516579200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"The National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51574219."],"award-info":[{"award-number":["51574219."]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"The Initiation Fund of Doctor\u2019s Research","award":["No. 107-451117008."],"award-info":[{"award-number":["No. 107-451117008."]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, an anomalous advection-dispersion model involving a new general Liouville\u2013Caputo fractional-order derivative is addressed for the first time. The series solutions of the general fractional advection-dispersion equations are obtained with the aid of the Laplace transform. The results are given to demonstrate the efficiency of the proposed formulations to describe the anomalous advection dispersion processes.<\/jats:p>","DOI":"10.3390\/e20010078","type":"journal-article","created":{"date-parts":[[2018,1,22]],"date-time":"2018-01-22T13:40:39Z","timestamp":1516628439000},"page":"78","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Anomalous Advection-Dispersion Equations within General Fractional-Order Derivatives: Models and Series Solutions"],"prefix":"10.3390","volume":"20","author":[{"given":"Xin","family":"Liang","sequence":"first","affiliation":[{"name":"State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China"},{"name":"School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5564-9590","authenticated-orcid":false,"given":"Yu-Gui","family":"Yang","sequence":"additional","affiliation":[{"name":"State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China"},{"name":"School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Feng","family":"Gao","sequence":"additional","affiliation":[{"name":"State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China"},{"name":"School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0009-4599","authenticated-orcid":false,"given":"Xiao-Jun","family":"Yang","sequence":"additional","affiliation":[{"name":"State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China"},{"name":"School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China"},{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yi","family":"Xue","sequence":"additional","affiliation":[{"name":"Institute of Geotechnical Engineering, Xi\u2019an University of Technology, Xi\u2019an 710048, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,1,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1016\/j.physa.2017.04.054","article-title":"A new fractional operator of variable order: Application in the description of anomalous diffusion","volume":"481","author":"Yang","year":"2017","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"S11","DOI":"10.2298\/TSCI170310194G","article-title":"General fractional calculus in nonsingular power-law kernel applied to model anomalous diffusion phenomena in heat-transfer problems","volume":"21","author":"Gao","year":"2017","journal-title":"Therm. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/j.cnsns.2017.04.001","article-title":"The role of fractional calculus in modelling biological phenomena: A review","volume":"51","author":"Ionescu","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1061\/(ASCE)EM.1943-7889.0001318","article-title":"Solution of one-dimensional time fractional Advection dispersion equation by homotopy analysis method","volume":"143","author":"Singh","year":"2017","journal-title":"J. Eng. Mech."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"857","DOI":"10.1111\/gwat.12532","article-title":"Can a Time Fractional-Derivative Model Capture Scale-Dependent Dispersion in Saturated Soils?","volume":"55","author":"Garrard","year":"2017","journal-title":"Groundwater"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1007\/BF00616936","article-title":"Experimental investigation of solute transport in large, homogeneous and heterogeneous, saturated soil columns","volume":"18","author":"Huang","year":"1995","journal-title":"Transp. Porous Media"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0009-2509(53)80001-1","article-title":"Continuous flow systems: Distribution of residence times","volume":"2","author":"Danckwerts","year":"1953","journal-title":"Chem. Eng. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"S233","DOI":"10.2298\/TSCI17S1233L","article-title":"Applications of a novel integral transform to the convection-dispersion equations","volume":"21","author":"Liang","year":"2017","journal-title":"Therm. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1016\/j.camwa.2013.01.031","article-title":"A novel numerical method for the time variable fractional order mobile\u2013immobile advection\u2014Dispersion model","volume":"66","author":"Zhang","year":"2013","journal-title":"Comput. Math. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1403","DOI":"10.1029\/2000WR900031","article-title":"Application of a fractional advection-dispersion equation","volume":"36","author":"Benson","year":"2000","journal-title":"Water Resour."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"L17403","DOI":"10.1029\/2008GL034899","article-title":"Tempered anomalous diffusion in heterogeneous systems","volume":"35","author":"Meerschaert","year":"2008","journal-title":"Geophys. Res. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/S0378-4371(01)00647-1","article-title":"Fractional diffusion: Probability distributions and random walk models","volume":"305","author":"Gorenflo","year":"2002","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"032117","DOI":"10.1103\/PhysRevE.96.032117","article-title":"Continuous-time random-walk model for anomalous diffusion in expanding media","volume":"96","author":"Vot","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"428","DOI":"10.3182\/20120711-3-BE-2027.00063","article-title":"Entropy and information in a fractional order model of anomalous diffusion","volume":"45","author":"Magin","year":"2012","journal-title":"IFAC Proc. Vol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"4028","DOI":"10.3390\/e17064028","article-title":"Generalized boundary conditions for the time-fractional advection diffusion equation","volume":"17","author":"Povstenko","year":"2015","journal-title":"Entropy"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"7488","DOI":"10.1002\/2017WR020522","article-title":"Entropy-based critical reaction time for mixing-controlled reactive transport","volume":"53","author":"Chiogna","year":"2017","journal-title":"Water Resour. Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1016\/j.cnsns.2017.08.002","article-title":"Prabhakar-like fractional viscoelasticity","volume":"56","author":"Giusti","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"S317","DOI":"10.2298\/TSCI170510196Y","article-title":"General fractional calculus operators containing the generalized Mittag-Leffler functions applied to anomalous relaxation","volume":"21","author":"Yang","year":"2017","journal-title":"Therm. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1161","DOI":"10.2298\/TSCI161216326Y","article-title":"Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems","volume":"21","author":"Yang","year":"2017","journal-title":"Therm. Sci."},{"key":"ref_20","first-page":"118","article-title":"New general fractional-order rheological models with kernels of Mittag-Leffler functions","volume":"69","author":"Yang","year":"2017","journal-title":"Rom. Rep. Phys."},{"key":"ref_21","first-page":"115","article-title":"Anomalous diffusion models with general fractional derivatives within the kernels of the extended Mittag-Leffler type functions","volume":"69","author":"Yang","year":"2017","journal-title":"Rom. Rep. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Gorenflo, R., Kilbas, A.A., Mainardi, F., and Rogosin, S.V. (2014). Mittag-Leffler Functions, Related Topics and Applications, Springer.","DOI":"10.1007\/978-3-662-43930-2"},{"key":"ref_23","first-page":"554","article-title":"Sur la nouvelle function E\u03b1x","volume":"137","year":"1903","journal-title":"C. R. Acad. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/BF02403202","article-title":"\u00dcber den Fundamentalsatz in der Teorie der Funktionen E\u03b1x","volume":"29","author":"Wiman","year":"1905","journal-title":"Acta Math."},{"key":"ref_25","first-page":"7","article-title":"A Singular Integral Equation with a Generalized Mittag-Leffler Function in the Kernel","volume":"19","author":"Prabhakar","year":"1971","journal-title":"Yokohama Math. J."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"6324","DOI":"10.22436\/jnsa.010.12.16","article-title":"Classifications and duality relations for several integral transforms","volume":"10","author":"Liang","year":"2017","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Mathai, A.M., and Haubold, H.J. (2008). Special Functions for Applied Scientists, Springer.","DOI":"10.1007\/978-0-387-75894-7"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"478","DOI":"10.1016\/j.scitotenv.2015.09.030","article-title":"A review on risk assessment techniques for hydraulic fracturing water and produced water management implemented in onshore unconventional oil and gas production","volume":"539","author":"Torres","year":"2016","journal-title":"Sci. Total Environ."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/78\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:52:05Z","timestamp":1760194325000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/78"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,22]]},"references-count":28,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1]]}},"alternative-id":["e20010078"],"URL":"https:\/\/doi.org\/10.3390\/e20010078","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1,22]]}}}