{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T16:53:19Z","timestamp":1786985599452,"version":"build-2736575974"},"reference-count":45,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100004001","name":"Guizhou Provincial Science and Technology Department","doi-asserted-by":"publisher","award":["MS (2026) 798"],"award-info":[{"award-number":["MS (2026) 798"]}],"id":[{"id":"10.13039\/501100004001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004735","name":"Hunan Provincial Natural Science Foundation","doi-asserted-by":"publisher","award":["2026JJ60005"],"award-info":[{"award-number":["2026JJ60005"]}],"id":[{"id":"10.13039\/501100004735","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12101527"],"award-info":[{"award-number":["12101527"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12471391"],"award-info":[{"award-number":["12471391"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Computers &amp; Mathematics with Applications"],"published-print":{"date-parts":[[2026,8]]},"DOI":"10.1016\/j.camwa.2026.04.030","type":"journal-article","created":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T19:03:42Z","timestamp":1778094222000},"page":"122-134","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":1,"special_numbering":"C","title":["A lattice Boltzmann method for time-fractional Allen-Cahn equation"],"prefix":"10.1016","volume":"215","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7422-7158","authenticated-orcid":false,"given":"Zhi","family":"Mao","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yi","family":"Huang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5125-2492","authenticated-orcid":false,"given":"Yuan","family":"Yu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2075-5040","authenticated-orcid":false,"given":"Aiguo","family":"Xiao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Haizhuan","family":"Yuan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.camwa.2026.04.030_bib0001","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1017\/S0022112004000370","article-title":"A diffuse-interface method for simulating two-phase flows of complex fluids","volume":"515","author":"Yue","year":"2004","journal-title":"J. Fluid Mech."},{"key":"10.1016\/j.camwa.2026.04.030_bib0002","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.85.026704","article-title":"Additional interfacial force in lattice Boltzmann models for incompressible multiphase flows","volume":"85","author":"Li","year":"2012","journal-title":"Phys. Rev. E"},{"key":"10.1016\/j.camwa.2026.04.030_bib0003","doi-asserted-by":"crossref","DOI":"10.1088\/0960-1317\/19\/5\/055010","article-title":"Reduction of droplet volume by controlling actuating waveforms in inkjet printing for micro-pattern formation","volume":"19","author":"Gan","year":"2009","journal-title":"J. Micromech. Microeng."},{"key":"10.1016\/j.camwa.2026.04.030_bib0004","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.jcis.2005.03.067","article-title":"Vibration-induced mobilization of trapped oil ganglia in porous media: experimental validation of a capillary-physics mechanism","volume":"289","author":"Li","year":"2005","journal-title":"J. Colloid Interface Sci."},{"key":"10.1016\/j.camwa.2026.04.030_bib0005","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1039\/b715524g","article-title":"Droplet microfluidics","volume":"8","author":"Teh","year":"2008","journal-title":"Lab. Chip."},{"key":"10.1016\/j.camwa.2026.04.030_bib0006","article-title":"Multiphase flow in mircrofluidic devices","volume":"45","author":"Chen","year":"2015","journal-title":"Adv. Mech."},{"key":"10.1016\/j.camwa.2026.04.030_bib0007","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1146\/annurev.fluid.30.1.139","article-title":"Diffuse-interface methods in fluid mechanics","volume":"30","author":"Anderson","year":"1998","journal-title":"Annu. Rev. Fluid Mech."},{"key":"10.1016\/j.camwa.2026.04.030_bib0008","doi-asserted-by":"crossref","first-page":"1085","DOI":"10.1016\/0001-6160(79)90196-2","article-title":"A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening","volume":"27","author":"Allen","year":"1979","journal-title":"Acta Metall."},{"key":"10.1016\/j.camwa.2026.04.030_bib0009","doi-asserted-by":"crossref","first-page":"258","DOI":"10.1063\/1.1744102","article-title":"Free energy of a nonuniform system. I. Interfacial free energy","volume":"28","author":"Cahn","year":"1958","journal-title":"J. Chem. Phys."},{"key":"10.1016\/j.camwa.2026.04.030_bib0010","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1016\/0001-6160(76)90063-8","article-title":"Mechanisms of phase transformations within the miscibility gap of Fe-rich Fe-Al alloys","volume":"24","author":"Allen","year":"1976","journal-title":"Acta Metall."},{"key":"10.1016\/j.camwa.2026.04.030_bib0011","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1137\/17M1150153","article-title":"A new class of efficient and robust energy stable schemes for gradient flows","volume":"61","author":"Shen","year":"2019","journal-title":"SIAM Rev."},{"key":"10.1016\/j.camwa.2026.04.030_bib0012","doi-asserted-by":"crossref","first-page":"613","DOI":"10.4208\/cicp.301110.040811a","article-title":"Phase-field models for multi-component fluid flows","volume":"12","author":"Kim","year":"2012","journal-title":"Commun. Comput. Phys."},{"key":"10.1016\/j.camwa.2026.04.030_bib0013","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1016\/j.apm.2019.03.009","article-title":"Lattice Boltzmann modeling of wall-bounded ternary fluid flows","volume":"73","author":"Liang","year":"2019","journal-title":"Appl. Math. Model."},{"key":"10.1016\/j.camwa.2026.04.030_bib0014","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1006\/jcph.1999.6332","article-title":"Calculation of two-phase Navier-Stokes flows using phase-field modeling","volume":"155","author":"Jacqmin","year":"1999","journal-title":"J. Comput. Phys."},{"issue":"5","key":"10.1016\/j.camwa.2026.04.030_bib0015","doi-asserted-by":"crossref","first-page":"2157","DOI":"10.1137\/22M1520050","article-title":"Discrete gradient structure of a second-order variable-step method for nonlinear integro-differential models","volume":"61","author":"Liao","year":"2023","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"10.1016\/j.camwa.2026.04.030_bib0016","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/s11425-022-2229-5","article-title":"Positive definiteness of real quadratic forms resulting from the variable-step L1-type approximations of convolution operators","volume":"67","author":"Liao","year":"2024","journal-title":"Sci. China Math."},{"issue":"2","key":"10.1016\/j.camwa.2026.04.030_bib0017","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/s00366-025-02267-0","article-title":"Magnetohydrodynamic transient flow of dual-parameter fractional maxwell nanofluids past a vertical plate with generalised dual-phase-lagging heat conduction under ramped wall temperature conditions","volume":"42","author":"Mao","year":"2026","journal-title":"Eng. Comput."},{"key":"10.1016\/j.camwa.2026.04.030_bib0018","doi-asserted-by":"crossref","first-page":"584","DOI":"10.1016\/j.cjph.2024.12.025","article-title":"Transient free convective flow of viscoelastic nanofluids governed by fractional integrodifferential equations under Newtonian heating and thermal radiation","volume":"93","author":"Mao","year":"2025","journal-title":"Chin. J. Phys."},{"issue":"1","key":"10.1016\/j.camwa.2026.04.030_bib0019","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1186\/s13662-020-02616-x","article-title":"A numerical investigation of Caputo time fractional Allen-Cahn equation using redefined cubic B-spline functions","volume":"2020","author":"Khalid","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"10.1016\/j.camwa.2026.04.030_bib0020","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1080\/00207160.2024.2420681","article-title":"A numerical scheme for time-fractional Allen-Cahn equation with application in phase separation","volume":"3","author":"Sohaib","year":"2025","journal-title":"Int. J. Comput. Math."},{"key":"10.1016\/j.camwa.2026.04.030_bib0021","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/s10444-020-09782-2","article-title":"Simple maximum principle preserving time-stepping methods for time-fractional Allen-Cahn equation","volume":"46","author":"Ji","year":"2020","journal-title":"Adv. Comput. Math."},{"key":"10.1016\/j.camwa.2026.04.030_bib0022","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.jcp.2013.11.017","article-title":"A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications","volume":"259","author":"Gao","year":"2014","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.camwa.2026.04.030_bib0023","doi-asserted-by":"crossref","first-page":"6811","DOI":"10.1103\/PhysRevE.56.6811","article-title":"Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation","volume":"56","author":"He","year":"1997","journal-title":"Phys. Rev. E"},{"key":"10.1016\/j.camwa.2026.04.030_bib0024","article-title":"Time-fractional Allen-Cahn equations: analysis and numerical methods","volume":"42","author":"Du","year":"2020","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.camwa.2026.04.030_bib0025","doi-asserted-by":"crossref","first-page":"1077","DOI":"10.1007\/s11075-021-01068-y","article-title":"Highly efficient for time fractional Allen-Cahn equation using extended SAV approach","volume":"88","author":"Hou","year":"2021","journal-title":"Numer. Algor."},{"key":"10.1016\/j.camwa.2026.04.030_bib0026","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2020.109473","article-title":"A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations","volume":"414","author":"Liao","year":"2020","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.camwa.2026.04.030_bib0027","doi-asserted-by":"crossref","first-page":"1876","DOI":"10.1016\/j.camwa.2018.07.036","article-title":"Time-fractional Allen-Cahn and Cahn-Hilliard phase-field models and their numerical investigation","volume":"76","author":"Liu","year":"2018","journal-title":"Comput. Math. Appl."},{"key":"10.1016\/j.camwa.2026.04.030_bib0028","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1016\/j.apnum.2021.02.016","article-title":"An efficient second order stabilized scheme for the two dimensional time fractional Allen-Cahn equation","volume":"165","author":"Jia","year":"2021","journal-title":"Appl. Numer. Math."},{"key":"10.1016\/j.camwa.2026.04.030_bib0029","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s10915-022-02085-2","article-title":"The exponential SAV approach for the time-fractional Allen-Cahn and Cahn-Hilliard phase-field models","volume":"94","author":"Yu","year":"2023","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.camwa.2026.04.030_bib0030","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/s10915-024-02667-2","article-title":"Linearly implicit schemes preserve the maximum bound principle and energy dissipation for the time-fractional Allen-Cahn equation","volume":"101","author":"Jiang","year":"2024","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.camwa.2026.04.030_bib0031","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2026.114667","article-title":"An improved lattice Boltzmann method with a novel conservative boundary scheme for viscoelastic fluid flows","volume":"550","author":"Yu","year":"2026","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.camwa.2026.04.030_bib0032","doi-asserted-by":"crossref","first-page":"489","DOI":"10.4208\/aamm.OA-2024-0203","article-title":"Two-relaxation-time regularized lattice Boltzmann model for Navier-Stokes equations","volume":"17","author":"Yu","year":"2025","journal-title":"Adv. Appl. Math. Mech."},{"key":"10.1016\/j.camwa.2026.04.030_bib0033","doi-asserted-by":"crossref","first-page":"650","DOI":"10.4208\/cicp.OA-2016-0136","article-title":"Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations","volume":"21","author":"Jiang","year":"2017","journal-title":"Commun. Comput. Phys."},{"key":"10.1016\/j.camwa.2026.04.030_bib0034","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.amc.2019.04.014","article-title":"Lattice Boltzmann model for time sub-diffusion equation in Caputo sense","volume":"358","author":"Du","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"10.1016\/j.camwa.2026.04.030_bib0035","doi-asserted-by":"crossref","DOI":"10.1016\/j.cnsns.2020.105443","article-title":"Lattice Boltzmann method for fractional Cahn-Hilliard equation","volume":"91","author":"Liang","year":"2020","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"4","key":"10.1016\/j.camwa.2026.04.030_bib0036","doi-asserted-by":"crossref","first-page":"914","DOI":"10.4208\/aamm.OA-2021-0018","article-title":"Lattice Boltzmann model for time-fractional nonlinear wave equations","volume":"14","author":"Wang","year":"2021","journal-title":"Adv. Appl. Math. Mech."},{"issue":"4","key":"10.1016\/j.camwa.2026.04.030_bib0037","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.87.043301","article-title":"Phase-field-based lattice Boltzmann model for incompressible binary fluid systems with density and viscosity contrasts","volume":"87","author":"Zu","year":"2013","journal-title":"Phys. Rev. E"},{"key":"10.1016\/j.camwa.2026.04.030_bib0038","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.81.036707","article-title":"Phase-field modeling by the method of lattice Boltzmann equations","volume":"81","author":"Fakhari","year":"2010","journal-title":"Phys. Rev. E"},{"key":"10.1016\/j.camwa.2026.04.030_bib0039","doi-asserted-by":"crossref","first-page":"428","DOI":"10.1016\/j.apm.2020.01.012","article-title":"A bounce back-immersed boundary-lattice Boltzmann model for curved boundary","volume":"81","author":"Wang","year":"2020","journal-title":"Appl. Math. Model."},{"key":"10.1016\/j.camwa.2026.04.030_bib0040","series-title":"Lattice Boltzmann Method and Its Applications in Engineering","author":"Guo","year":"2013"},{"key":"10.1016\/j.camwa.2026.04.030_bib0041","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.102.023306","article-title":"Multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: modeling, analysis, and elements","volume":"102","author":"Chai","year":"2020","journal-title":"Phys. Rev. E"},{"key":"10.1016\/j.camwa.2026.04.030_bib0042","doi-asserted-by":"crossref","first-page":"A3757","DOI":"10.1137\/18M1203560","article-title":"On energy dissipation theory and numerical stability for time-fractional phase field equation","volume":"41","author":"Tang","year":"2019","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"10.1016\/j.camwa.2026.04.030_bib0043","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1007\/s10915-024-02515-3","article-title":"Asymptotically compatible energy and dissipation law of the nonuniform L2-1_\u03c3 scheme for time fractional Allen-Cahn model","volume":"99","author":"Liao","year":"2024","journal-title":"J. Sci. Comput."},{"issue":"5","key":"10.1016\/j.camwa.2026.04.030_bib0044","doi-asserted-by":"crossref","first-page":"A3503","DOI":"10.1137\/20M1384105","article-title":"An energy stable and maximum bound preserving scheme with variable time steps for time fractional Allen-Cahn equation","volume":"43","author":"Liao","year":"2021","journal-title":"SIAM J. Sci. Comput."},{"key":"10.1016\/j.camwa.2026.04.030_bib0045","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2025.113831","article-title":"Regularized lattice Boltzmann method based maximum principle and energy stability preserving finite-difference scheme for the Allen-Cahn equation","volume":"528","author":"Chen","year":"2025","journal-title":"J. Comput. Phys."}],"container-title":["Computers &amp; Mathematics with Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0898122126001951?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0898122126001951?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,7,4]],"date-time":"2026-07-04T09:22:30Z","timestamp":1783156950000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0898122126001951"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8]]},"references-count":45,"alternative-id":["S0898122126001951"],"URL":"https:\/\/doi.org\/10.1016\/j.camwa.2026.04.030","relation":{},"ISSN":["0898-1221"],"issn-type":[{"value":"0898-1221","type":"print"}],"subject":[],"published":{"date-parts":[[2026,8]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"A lattice Boltzmann method for time-fractional Allen-Cahn equation","name":"articletitle","label":"Article Title"},{"value":"Computers & Mathematics with Applications","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.camwa.2026.04.030","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}]}}