{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T06:17:03Z","timestamp":1764569823098},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2021,5,3]],"date-time":"2021-05-03T00:00:00Z","timestamp":1620000000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,5,3]],"date-time":"2021-05-03T00:00:00Z","timestamp":1620000000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2021,6]]},"DOI":"10.1007\/s40314-021-01513-7","type":"journal-article","created":{"date-parts":[[2021,5,3]],"date-time":"2021-05-03T22:27:49Z","timestamp":1620080869000},"update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A robust computational framework for analyzing the Bloch\u2013Torrey equation of fractional order"],"prefix":"10.1007","volume":"40","author":[{"given":"K.","family":"Sayevand","sequence":"first","affiliation":[]},{"given":"N.","family":"Ghanbari","sequence":"additional","affiliation":[]},{"given":"I.","family":"Masti","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,5,3]]},"reference":[{"key":"1513_CR1","doi-asserted-by":"publisher","first-page":"253","DOI":"10.2298\/TSCI170715293A","volume":"22","author":"EK Akgul","year":"2018","unstructured":"Akgul EK (2018) A novel method for the space and time fractional Bloch\u2013Torrey equations. Therm Sci 22:253\u2013258","journal-title":"Therm Sci"},{"key":"1513_CR2","doi-asserted-by":"publisher","first-page":"4793","DOI":"10.1007\/s40314-018-0593-7","volume":"37","author":"A Azizi","year":"2018","unstructured":"Azizi A, Abdi S, Saeidian J (2018) Applying Legendre wavelet method with Tikhonov regularization for one-dimensional time-fractional diffusion equations. Comput Appl Math 37:4793\u20134804","journal-title":"Comput Appl Math"},{"key":"1513_CR3","doi-asserted-by":"publisher","DOI":"10.1142\/8180","volume-title":"Fractional calculus models and numerical methods","author":"D Baleanu","year":"2012","unstructured":"Baleanu D, Diethelm K, Scalas E, Trujillo JJ (2012) Fractional calculus models and numerical methods. World Scientific, Singapore"},{"key":"1513_CR4","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1016\/0021-9045(77)90055-7","volume":"19","author":"PL Butzer","year":"1972","unstructured":"Butzer PL, Wefon R (1972) On the Lax equivalence theorem equipped with orders. J Approx Theory 19:239\u2013252","journal-title":"J Approx Theory"},{"key":"1513_CR5","doi-asserted-by":"publisher","first-page":"1743","DOI":"10.1016\/j.jcp.2011.11.008","volume":"231","author":"C Celik","year":"2012","unstructured":"Celik C, Duman M (2012) Crank\u2013Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J Comput Phys 231:1743\u20131750","journal-title":"J Comput Phys"},{"key":"1513_CR6","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1007\/s11075-016-0103-1","volume":"4","author":"M Dehghan","year":"2016","unstructured":"Dehghan M, Abbaszadeh M, Mohebbi A (2016) Analysis of a meshless method for the time fractional diffusion-wave equation. Numer Algorithms 4:445\u2013476","journal-title":"Numer Algorithms"},{"issue":"1","key":"1513_CR7","doi-asserted-by":"publisher","first-page":"614","DOI":"10.1002\/num.22543","volume":"37","author":"R Erfanifar","year":"2021","unstructured":"Erfanifar R, Sayevand K, Ghanbari N, Esmaeili H (2021) A modified Chebyshev $$\\vartheta $$-weighted Crank\u2013Nicolson method for analyzing fractional sub-diffusion equations. Numer Methods Partial Differ Equ 37(1):614\u2013625","journal-title":"Numer Methods Partial Differ Equ"},{"key":"1513_CR8","doi-asserted-by":"publisher","first-page":"1214","DOI":"10.1007\/s10915-017-0396-9","volume":"72","author":"T Hou","year":"2017","unstructured":"Hou T, Tang T, Yang J (2017) Numerical analysis of fully discretized Crank\u2013Nicolson scheme for fractional-in-space Allen\u2013Cahn equations. J Sci Comput 72:1214\u20131231","journal-title":"J Sci Comput"},{"key":"1513_CR9","doi-asserted-by":"publisher","first-page":"617","DOI":"10.1002\/mrm.24706","volume":"71","author":"C Ingo","year":"2014","unstructured":"Ingo C, Magin RL, Colon-Perez L, Triplett W, Mareci TH (2014) On random walks and entropy in diffusion weighted magnetic resonance imaging studies of neural tissue. Magn Reson Med 71:617\u2013627","journal-title":"Magn Reson Med"},{"issue":"5","key":"1513_CR10","doi-asserted-by":"publisher","first-page":"3213","DOI":"10.1016\/j.aej.2020.08.015","volume":"59","author":"H Jafari","year":"2020","unstructured":"Jafari H, Firoozjaee MA, Johnston SJ (2020) An effective approach to solve a system fractional differential equations. Alex Eng J 59(5):3213\u20133219","journal-title":"Alex Eng J"},{"issue":"1","key":"1513_CR11","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/j.econlet.2014.04.026","volume":"124","author":"N Jinji","year":"2014","unstructured":"Jinji N (2014) Comparative statics for oligopoly: a generalized result. Econ Lett 124(1):79\u201382","journal-title":"Econ Lett"},{"issue":"1","key":"1513_CR12","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1007\/s00366-018-0595-5","volume":"35","author":"S Kazem","year":"2019","unstructured":"Kazem S, Dehghan M (2019) Semi-analytical solution for time-fractional diffusion equation based on finite difference method of lines (MOL). Eng Comput 35(1):229\u2013241","journal-title":"Eng Comput"},{"key":"1513_CR13","volume-title":"Theory and applications of fractional differential equations","author":"A Kilbas","year":"2006","unstructured":"Kilbas A, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam"},{"key":"1513_CR14","volume-title":"Numerical analysis","author":"D Kincaid","year":"1991","unstructured":"Kincaid D, Cheney W (1991) Numerical analysis. Brooks\/Cole Publishing, California"},{"issue":"4","key":"1513_CR15","first-page":"273","volume":"11","author":"S Kumar","year":"2014","unstructured":"Kumar S, Faraz N, Sayevand K (2014) A Fractional model of Bloch equation in nuclear magnetic resonance and its analytic approximate solution. Walailak J Sci Technol 11(4):273\u2013285","journal-title":"Walailak J Sci Technol"},{"key":"1513_CR16","first-page":"77","volume":"363","author":"FR Lin","year":"2020","unstructured":"Lin FR, Wang QY, Jin XQ (2020) Crank\u2013Nicolson-weighted-shifted-Gr\u00fcnwald-difference schemes for space Riesz variable-order fractional diffusion equations. Numer Algorithms 363:77\u201391","journal-title":"Numer Algorithms"},{"issue":"5","key":"1513_CR17","doi-asserted-by":"publisher","first-page":"1637","DOI":"10.1016\/j.camwa.2019.01.007","volume":"78","author":"F Liu","year":"2019","unstructured":"Liu F, Feng L, Anh V, Li J (2019) Unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time-space fractional Bloch\u2013Torrey equations on irregular convex domains. Comput Math Appl 78(5):1637\u20131650","journal-title":"Comput Math Appl"},{"key":"1513_CR18","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1016\/j.jmr.2007.11.007","volume":"190","author":"RL Magin","year":"2008","unstructured":"Magin RL, Abdullah O, Baleanu D, Zhou XJ (2008) Anomalous diffusion expressed through fractional order differential operators in the Bloch\u2013Torrey equation. J Magn Reson 190:255\u2013270","journal-title":"J Magn Reson"},{"key":"1513_CR19","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1615\/CritRevBiomedEng.2020033925","volume":"48","author":"RL Magin","year":"2020","unstructured":"Magin RL, Hall MG, Karaman MM, Vegh V (2020) Fractional calculus models of magnetic resonance phenomena: relaxation and diffusion. Crit Rev Biomed Eng 48:285\u2013326","journal-title":"Crit Rev Biomed Eng"},{"issue":"1","key":"1513_CR20","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1016\/j.cam.2004.01.033","volume":"172","author":"MM Meerschaert","year":"2004","unstructured":"Meerschaert MM, Tadjeran C (2004) Finite difference approximations for fractional advection-dispersion flow equations. J Comput Appl Math 172(1):65\u201377","journal-title":"J Comput Appl Math"},{"key":"1513_CR21","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1016\/j.enganabound.2020.08.017","volume":"120","author":"O Nikan","year":"2020","unstructured":"Nikan O, Avazzadeh Z, Tenreiro Machado JA (2020) Numerical investigation of fractional nonlinear sine-Gordon and Klein\u2013Gordon models arising in relativistic quantum mechanics. Eng Anal Bound Elem 120:223\u2013237","journal-title":"Eng Anal Bound Elem"},{"key":"1513_CR22","doi-asserted-by":"publisher","DOI":"10.1016\/j.jksus.2020.101243","author":"O Nikan","year":"2021","unstructured":"Nikan O, Avazzadeh Z, Tenreiro Machado JA (2021a) An efficient local meshless approach for solving nonlinear time-fractional fourth-order diffusion model. J King Saud Univ Sci. https:\/\/doi.org\/10.1016\/j.jksus.2020.101243","journal-title":"J King Saud Univ Sci"},{"key":"1513_CR23","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2021.105755","author":"O Nikan","year":"2021","unstructured":"Nikan O, Avazzadeh Z, Tenreiro Machado JA (2021b) Numerical approximation of the nonlinear time-fractional telegraph equation arising in neutron transport. Commun Nonlinear Sci Numer Simul. https:\/\/doi.org\/10.1016\/j.cnsns.2021.105755","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"1513_CR24","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.109983","author":"O Nikan","year":"2021","unstructured":"Nikan O, Tenreiro Machado JA, Golbabai A, Rashidinia J (2021c) Numerical evaluation of the fractional Klein\u2013Kramers model arising in molecular dynamics. J Comput Phys. https:\/\/doi.org\/10.1016\/j.jcp.2020.109983","journal-title":"J Comput Phys"},{"key":"1513_CR25","doi-asserted-by":"publisher","unstructured":"Ortigueira MD (2006) Riesz potential operators and inverses via fractional centred derivatives. Int J Math Math Sci 2006:048391. https:\/\/doi.org\/10.1155\/IJMMS\/2006\/48391","DOI":"10.1155\/IJMMS\/2006\/48391"},{"key":"1513_CR26","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-007-0747-4","volume-title":"Fractional calculus for scientists and engineers","author":"MD Ortigueira","year":"2011","unstructured":"Ortigueira MD (2011) Fractional calculus for scientists and engineers. Springer, Berlin"},{"issue":"1","key":"1513_CR27","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1016\/j.camwa.2017.08.032","volume":"75","author":"S Qin","year":"2018","unstructured":"Qin S, Liu F, Turner IW, Yang Q, Yu Q (2018) Modelling anomalous diffusion using fractional Bloch\u2013Torrey equations on approximate irregular domains. Comput Math Appl 75(1):7\u201321","journal-title":"Comput Math Appl"},{"key":"1513_CR28","volume-title":"Fractional integrals and derivatives: theory and applications","author":"SG Samko","year":"1993","unstructured":"Samko SG, Kilbas A, Marichev O (1993) Fractional integrals and derivatives: theory and applications. Gordon and Breach, Langhorne"},{"issue":"5","key":"1513_CR29","doi-asserted-by":"publisher","first-page":"1681","DOI":"10.1016\/j.camwa.2018.12.016","volume":"78","author":"K Sayevand","year":"2019","unstructured":"Sayevand K, Machado JT, Moradi V (2019) A new non-standard finite difference method for analyzing the fractional Navier\u2013Stokes equations. Comput Math Appl 78(5):1681\u20131694","journal-title":"Comput Math Appl"},{"issue":"2","key":"1513_CR30","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1016\/j.apnum.2005.03.003","volume":"56","author":"Z Sun","year":"2006","unstructured":"Sun Z, Wu X (2006) A fully discrete difference scheme for a diffusion-wave system. Appl Numer Math 56(2):193\u2013209","journal-title":"Appl Numer Math"},{"key":"1513_CR31","doi-asserted-by":"crossref","first-page":"356","DOI":"10.1016\/j.amc.2016.01.044","volume":"281","author":"H Sun","year":"2016","unstructured":"Sun H, Sun ZZ, Gao GH (2016) Some high order difference schemes for the space and time fractional Bloch\u2013Torrey equations. Appl Math Comput 281:356\u2013380","journal-title":"Appl Math Comput"},{"issue":"1","key":"1513_CR32","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s12043-020-02004-w","volume":"94","author":"H Tajadodi","year":"2020","unstructured":"Tajadodi H, Kadkhoda N, Jafari H (2020) Approximate technique for solving fractional variational problems. Pramana 94(1):1\u20138","journal-title":"Pramana"},{"key":"1513_CR33","volume-title":"Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media","author":"VE Tarasov","year":"2011","unstructured":"Tarasov VE (2011) Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media. Springer, Berlin"},{"issue":"294","key":"1513_CR34","doi-asserted-by":"publisher","first-page":"1703","DOI":"10.1090\/S0025-5718-2015-02917-2","volume":"84","author":"W Tian","year":"2015","unstructured":"Tian W, Zhou H, Deng W (2015) A class of second order difference approximations for solving space fractional diffusion equations. Math Comput 84(294):1703\u20131727","journal-title":"Math Comput"},{"key":"1513_CR35","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1016\/j.jcp.2014.01.009","volume":"263","author":"D Van Nguyen","year":"2014","unstructured":"Van Nguyen D, Li JR, Grebenkov D, Le Bihan D (2014) A finite elements method to solve the Bloch\u2013Torrey equation applied to diffusion magnetic resonance imaging. J Comput Phys 263:283\u2013302","journal-title":"J Comput Phys"},{"issue":"5","key":"1513_CR36","doi-asserted-by":"publisher","first-page":"645","DOI":"10.1093\/imamat\/hxp015","volume":"74","author":"P Zhuang","year":"2009","unstructured":"Zhuang P, Liu F, Anh V, Turner I (2009) Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process. IMA J Appl Math 74(5):645\u2013667","journal-title":"IMA J Appl Math"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01513-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-021-01513-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01513-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,3]],"date-time":"2023-11-03T06:23:21Z","timestamp":1698992601000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-021-01513-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,3]]},"references-count":36,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,6]]}},"alternative-id":["1513"],"URL":"https:\/\/doi.org\/10.1007\/s40314-021-01513-7","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,3]]},"assertion":[{"value":"3 February 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 April 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 April 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 May 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"131"}}