{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T06:17:02Z","timestamp":1764569822535,"version":"3.37.3"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,3,10]],"date-time":"2021-03-10T00:00:00Z","timestamp":1615334400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,3,10]],"date-time":"2021-03-10T00:00:00Z","timestamp":1615334400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2021,4]]},"DOI":"10.1007\/s40314-021-01474-x","type":"journal-article","created":{"date-parts":[[2021,3,10]],"date-time":"2021-03-10T13:03:28Z","timestamp":1615381408000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Development of a computational approach for a space\u2013time fractional moving boundary problem arising from drug release systems"],"prefix":"10.1007","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7310-6776","authenticated-orcid":false,"given":"M.","family":"Garshasbi","sequence":"first","affiliation":[]},{"given":"T.","family":"Nikazad","sequence":"additional","affiliation":[]},{"given":"F.","family":"Sanaei","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,3,10]]},"reference":[{"key":"1474_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.cam.2017.09.008","volume":"331","author":"J Bollati","year":"2018","unstructured":"Bollati J, Semitiel J, Tarzia DA (2018) Heat balance integral methods applied to the one-phase Stefan problem with a convective boundary condition at the fixed face. Appl Math Comput 331:1\u201319","journal-title":"Appl Math Comput"},{"key":"1474_CR2","doi-asserted-by":"publisher","unstructured":"Brambila F (ed) (2017) Fractal analysis: applications in physics, engineering and technology. https:\/\/doi.org\/10.5772\/65531","DOI":"10.5772\/65531"},{"key":"1474_CR3","first-page":"1","volume":"4","author":"M Caputo","year":"2020","unstructured":"Caputo M, Cametti C (2020) Diffusion through skin in the light of a fractional derivative approach: progress and challenges. J Pharmacokinet Pharmacodyn 4:1\u20137","journal-title":"J Pharmacokinet Pharmacodyn"},{"key":"1474_CR4","doi-asserted-by":"publisher","first-page":"138","DOI":"10.1016\/j.cnsns.2018.01.020","volume":"61","author":"JM Cruz-Duarte","year":"2018","unstructured":"Cruz-Duarte JM, Rosales-Garcia J, Correa-Cely CR, Garcia-Perez A, Avina-Cervantes JG (2018) A closed form expression for the Gaussian-based Caputo\u2013Fabrizio fractional derivative for signal processing applications. Commun Nonlin Sci Numer Sim 61:138\u2013148","journal-title":"Commun Nonlin Sci Numer Sim"},{"issue":"5","key":"1474_CR5","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1515\/zna-2011-0503","volume":"66","author":"S Das","year":"2011","unstructured":"Das S, Kumar R, Gupta PK (2011) Analytical approximate solution of space-time fractional diffusion equation with a moving boundary condition. Z Naturforschung A 66(5):281\u2013288","journal-title":"Z Naturforschung A"},{"key":"1474_CR6","doi-asserted-by":"publisher","first-page":"172","DOI":"10.1016\/j.apm.2018.06.052","volume":"63","author":"F Font","year":"2018","unstructured":"Font F (2018) A one-phase Stefan problem with size-dependent thermal conductivity. Appl Math Model 63:172\u2013178","journal-title":"Appl Math Model"},{"key":"1474_CR7","doi-asserted-by":"publisher","first-page":"316","DOI":"10.1016\/j.jcp.2019.03.030","volume":"88","author":"H Fu","year":"2019","unstructured":"Fu H, Liu H, Wang H (2019) A finite volume method for two-dimensional Riemann\u2013Liouville space-fractional diffusion equation and its efficient implementation. J Comput Phys 88:316\u2013334","journal-title":"J Comput Phys"},{"issue":"8","key":"1474_CR8","doi-asserted-by":"publisher","first-page":"2385","DOI":"10.1016\/j.apm.2014.10.053","volume":"39","author":"X Gao","year":"2015","unstructured":"Gao X, Jiang X, Chen S (2015) The numerical method for the moving boundary problem with space-fractional derivative in drug release devices. Appl Math Model 39(8):2385\u20132391","journal-title":"Appl Math Model"},{"issue":"6","key":"1474_CR9","doi-asserted-by":"publisher","first-page":"3754","DOI":"10.1002\/mma.6153","volume":"43","author":"M Garshasbi","year":"2020","unstructured":"Garshasbi M, Malek Bagomghaleh S (2020) An iterative approach to solve a nonlinear moving boundary problem describing the solvent diffusion within glassy polymers. Math Meth Appl Sci 43(6):3754\u20133772","journal-title":"Math Meth Appl Sci"},{"key":"1474_CR10","doi-asserted-by":"publisher","DOI":"10.1002\/jnm.2852","author":"M Garshasbi","year":"2020","unstructured":"Garshasbi M, Sanaei F (2020) A variable time-step method for a space fractional diffusion moving boundary problem: an application to planar drug release devices. Int J Numer Model Electron Netw Dev Field. https:\/\/doi.org\/10.1002\/jnm.2852","journal-title":"Int J Numer Model Electron Netw Dev Field"},{"key":"1474_CR11","volume-title":"Theory and applications of fractional differential equations","author":"AA Kilbas","year":"2006","unstructured":"Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier Science, Amsterdam"},{"key":"1474_CR12","doi-asserted-by":"publisher","first-page":"951","DOI":"10.1016\/j.apm.2020.06.070","volume":"88","author":"A Kumar","year":"2020","unstructured":"Kumar A (2020) A moving boundary problem with space-fractional diffusion logistic population model and density-dependent dispersal rate. Appl Math Model 88:951\u201365","journal-title":"Appl Math Model"},{"issue":"5","key":"1474_CR13","doi-asserted-by":"publisher","first-page":"3589","DOI":"10.1016\/j.apm.2012.07.047","volume":"37","author":"MS Kushwaha","year":"2013","unstructured":"Kushwaha MS (2013) Homotopy perturbation method for a limit case Stefan problem governed by fractional diffusion equation. Appl Math Model 37(5):3589\u20133599","journal-title":"Appl Math Model"},{"key":"1474_CR14","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1016\/j.conbuildmat.2018.04.094","volume":"174","author":"Y Lei","year":"2018","unstructured":"Lei Y, Wang H, Chen X, Yang X, You Z, Dong S, Gao J (2018) Shear property, high-temperature rheological performance and low-temperature flexibility of asphalt mastics modified with bio-oil. Constr Build Mater 174:30\u20137","journal-title":"Constr Build Mater"},{"key":"1474_CR15","doi-asserted-by":"crossref","unstructured":"Li C, Cai M (2019) Theory and numerical approximations of fractional integrals and derivatives. Society for Industrial and Applied Mathematic","DOI":"10.1137\/1.9781611975888"},{"key":"1474_CR16","doi-asserted-by":"publisher","first-page":"614","DOI":"10.1016\/j.jcp.2016.04.039","volume":"316","author":"C Li","year":"2016","unstructured":"Li C, Yi Q, Chen A (2016) Finite difference methods with non-uniform meshes for nonlinear fractional differential equations. J Comput Phys 316:614\u2013631","journal-title":"J Comput Phys"},{"issue":"40","key":"1474_CR17","doi-asserted-by":"publisher","first-page":"12131","DOI":"10.1088\/1751-8113\/40\/40\/008","volume":"40","author":"X Li","year":"2007","unstructured":"Li X, Xu M, Wang S (2007) Analytical solutions to the moving boundary problems with space-time-fractional derivatives in drug release devices. J Phys A Math Theor 40(40):12131","journal-title":"J Phys A Math Theor"},{"issue":"15","key":"1474_CR18","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1088\/1751-8113\/41\/15\/155202","volume":"41","author":"X Li","year":"2008","unstructured":"Li X, Xu M, Wang S (2008) Scale-invariant solutions to partial differential equations of fractional order with a moving boundary condition. J Phys A Math Theor 41(15):155\u2013202","journal-title":"J Phys A Math Theor"},{"issue":"10","key":"1474_CR19","doi-asserted-by":"publisher","first-page":"3521","DOI":"10.1016\/j.camwa.2018.02.014","volume":"75","author":"N Liu","year":"2017","unstructured":"Liu N, Liu Y, Li H, Wang J (2017a) Time second-order finite difference\/finite element algorithm for nonlinear time-fractional diffusion problem with fourth-order derivative term. Comput Math Appl 75(10):3521\u20133536","journal-title":"Comput Math Appl"},{"key":"1474_CR20","doi-asserted-by":"publisher","first-page":"146","DOI":"10.1016\/j.apnum.2017.04.003","volume":"119","author":"Z Liu","year":"2017","unstructured":"Liu Z, Cheng A, Li X, Wang H (2017b) A fast solution technique for finite element discretization of the space-time fractional diffusion equation. Appl Numer Math 119:146\u2013163","journal-title":"Appl Numer Math"},{"key":"1474_CR21","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-69823-6_1","volume-title":"Acoustics and vibration of mechanical structures-AVMS-2017 springer proceedings in physics","author":"JAT Machado","year":"2018","unstructured":"Machado JAT (2018) Fractional calculus: fundamentals and applications. In: Herisanu N, Marinca V (eds) Acoustics and vibration of mechanical structures-AVMS-2017 springer proceedings in physics, vol 198. Springer, Cham. https:\/\/doi.org\/10.1007\/978-3-319-69823-6_1"},{"key":"1474_CR22","doi-asserted-by":"crossref","unstructured":"Matouk AE (ed) (2020) Advanced Applications of Fractional Differential Operators to Science and Technology. IGI Global","DOI":"10.4018\/978-1-7998-3122-8"},{"key":"1474_CR23","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-00895-6","volume-title":"Introduction to fractional differential equations","author":"C Milici","year":"2019","unstructured":"Milici C, Dr\u0103g\u0103nescu G, Machado JT (2019) Introduction to fractional differential equations. Springer, Berlin"},{"issue":"1","key":"1474_CR24","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1137\/080733036","volume":"52","author":"SL Mitchell","year":"2010","unstructured":"Mitchell SL, Myers TG (2010) Application of standard and refined heat balance integral methods to one-dimensional Stefan problems. SIAM Rev 52(1):57\u201386","journal-title":"SIAM Rev"},{"key":"1474_CR25","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1016\/j.cam.2015.12.021","volume":"300","author":"SL Mitchell","year":"2016","unstructured":"Mitchell SL, Vynnycky M (2016) On the accurate numerical solution of a two-phase Stefan problem with phase formation and depletion. J Comput Appl Math 300:259\u2013274","journal-title":"J Comput Appl Math"},{"key":"1474_CR26","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1016\/j.compbiomed.2019.04.013","volume":"109","author":"N Pachauri","year":"2019","unstructured":"Pachauri N, Yadav J, Rani A, Singh V (2019) Modified fractional order IMC design based drug scheduling for cancer treatment. Comput Biol Med 109:121\u201337","journal-title":"Comput Biol Med"},{"key":"1474_CR27","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1016\/S0376-7388(00)82256-5","volume":"1","author":"DR Paul","year":"1978","unstructured":"Paul DR, Mc Spadden SK (1978) Diffusional release of a solute from a polymer matrix. J Membr Sci 1:33\u201348","journal-title":"J Membr Sci"},{"key":"1474_CR28","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1016\/j.jconrel.2014.06.041","volume":"190","author":"NA Peppas","year":"2014","unstructured":"Peppas NA, Narasimhan B (2014) Mathematical models in drug delivery: how modeling has shaped the way we design new drug delivery systems. J Control Release 190:75\u201381","journal-title":"J Control Release"},{"key":"1474_CR29","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/s13318-012-0097-6","volume":"38","author":"JK Popovic","year":"2013","unstructured":"Popovic JK, Posa M, Popovic KJ, Popovic DJ, Milosevic N, Tepavcevic V (2013) Individualization of a pharmacokinetic model by fractional and nonlinear fit improvement. Eur J Drug Metab Pharmacokinet 38:69\u201376","journal-title":"Eur J Drug Metab Pharmacokinet"},{"issue":"6","key":"1474_CR30","doi-asserted-by":"publisher","first-page":"531","DOI":"10.1016\/j.apm.2005.06.003","volume":"30","author":"N Sadoun","year":"2006","unstructured":"Sadoun N, Si-Ahmed EK, Colinet P (2006) On the refined integral method for the one-phase Stefan problem with time-dependent boundary conditions. Appl Math Model 30(6):531\u2013544","journal-title":"Appl Math Model"},{"key":"1474_CR31","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1016\/j.cnsns.2018.04.019","volume":"64","author":"H Sun","year":"2018","unstructured":"Sun H, Zhang Y, Baleanu D, Chen W, Chen Y (2018) A new collection of real world applications of fractional calculus in science and engineering. Commun Nonlinear Sci Numer Simul 64:213\u201331","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"2","key":"1474_CR32","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1016\/j.jconrel.2011.10.006","volume":"161","author":"J Siepmann","year":"2012","unstructured":"Siepmann J, Siepmann F (2012) Modeling of diffusion controlled drug delivery. J Control Release 161(2):351\u2013362","journal-title":"J Control Release"},{"key":"1474_CR33","doi-asserted-by":"publisher","unstructured":"Singh BK, Kumar P (2017) Fractional variational iteration method for solving fractional partial differential equations with proportional delay. Int J Diff Equ 2017:5206380. https:\/\/doi.org\/10.1155\/2017\/5206380","DOI":"10.1155\/2017\/5206380"},{"issue":"4","key":"1474_CR34","doi-asserted-by":"publisher","first-page":"1937","DOI":"10.1016\/j.apm.2010.11.005","volume":"35","author":"J Singh","year":"2011","unstructured":"Singh J, Gupta PK, Rai KN (2011) Homotopy perturbation method to space-time fractional solidification in a finite slab. Appl Math Model 35(4):1937\u20131945","journal-title":"Appl Math Model"},{"issue":"1","key":"1474_CR35","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1007\/s10928-017-9547-8","volume":"45","author":"P Sopasakis","year":"2018","unstructured":"Sopasakis P, Sarimveis H, Macheras P, Dokoumetzidis A (2018) Fractional calculus in pharmacokinetics. J Pharmacokinet Pharmacodyn 45(1):107\u201325","journal-title":"J Pharmacokinet Pharmacodyn"},{"issue":"9","key":"1474_CR36","doi-asserted-by":"publisher","first-page":"6183","DOI":"10.1016\/j.apm.2012.12.018","volume":"37","author":"G Wu","year":"2013","unstructured":"Wu G, Baleanu D (2013) Variational iteration method for the Burgers\u2019 flow with fractional derivatives-new Lagrange multipliers. Appl Math Model 37(9):6183\u20136190","journal-title":"Appl Math Model"},{"issue":"2","key":"1474_CR37","doi-asserted-by":"publisher","first-page":"66","DOI":"10.1007\/s40314-019-0835-3","volume":"38","author":"R Yan","year":"2019","unstructured":"Yan R, Han M, Ma Q, Ding X (2019) A spectral collocation method for nonlinear fractional initial value problems with a variable-order fractional derivative. Comput Appl Math 38(2):66","journal-title":"Comput Appl Math"},{"issue":"1","key":"1474_CR38","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1016\/j.ijpharm.2010.12.009","volume":"418","author":"C Yin","year":"2011","unstructured":"Yin C, Li X (2011) Anomalous diffusion of drug release from a slab matrix: fractional diffusion models. Int J Pharm 418(1):78\u201387","journal-title":"Int J Pharm"},{"issue":"11","key":"1474_CR39","doi-asserted-by":"publisher","first-page":"115210","DOI":"10.1088\/1751-8113\/42\/11\/115210","volume":"42","author":"C Yin","year":"2009","unstructured":"Yin C, Xu M (2009) An asymptotic analytical solution to the problem of two moving boundaries with fractional diffusion in one-dimensional drug release devices. J Phys A Math Theor 42(11):115210","journal-title":"J Phys A Math Theor"},{"key":"1474_CR40","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1016\/j.apnum.2020.04.002","volume":"154","author":"MA Zaky","year":"2020","unstructured":"Zaky MA (2020) An accurate spectral collocation method for nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions. Appl Numer Math 154:205\u2013222","journal-title":"Appl Numer Math"},{"issue":"1","key":"1474_CR41","doi-asserted-by":"publisher","first-page":"360","DOI":"10.1137\/16M1076083","volume":"39","author":"F Zeng","year":"2017","unstructured":"Zeng F, Mao Z, Karniadakis GE (2017) A generalized spectral collocation method with tunable accuracy for fractional differential equations with end-point singularities. SIAM J Sci Comput 39(1):360\u2013383","journal-title":"SIAM J Sci Comput"},{"key":"1474_CR42","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1016\/j.amc.2014.06.003","volume":"242","author":"H Zhang","year":"2014","unstructured":"Zhang H, Liu F, Zhuang P, Turner I, Anh V (2014) Numerical analysis of a new space-time variable fractional order advection-dispersion equation. Appl Math Comput 242:541\u2013550","journal-title":"Appl Math Comput"},{"issue":"4","key":"1474_CR43","doi-asserted-by":"publisher","first-page":"4877","DOI":"10.1007\/s40314-018-0609-3","volume":"37","author":"GA Zou","year":"2018","unstructured":"Zou GA (2018) Galerkin finite element method for time-fractional stochastic diffusion equations. Comput Appl Math 37(4):4877\u201398","journal-title":"Comput Appl Math"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01474-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s40314-021-01474-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01474-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,31]],"date-time":"2023-10-31T11:11:32Z","timestamp":1698750692000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s40314-021-01474-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,10]]},"references-count":43,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,4]]}},"alternative-id":["1474"],"URL":"https:\/\/doi.org\/10.1007\/s40314-021-01474-x","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2021,3,10]]},"assertion":[{"value":"16 November 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 February 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 March 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 March 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"80"}}