{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,4]],"date-time":"2025-12-04T01:26:31Z","timestamp":1764811591887,"version":"3.37.3"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2019,8,14]],"date-time":"2019-08-14T00:00:00Z","timestamp":1565740800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,8,14]],"date-time":"2019-08-14T00:00:00Z","timestamp":1565740800000},"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":["Engineering with Computers"],"published-print":{"date-parts":[[2021,1]]},"DOI":"10.1007\/s00366-019-00843-9","type":"journal-article","created":{"date-parts":[[2019,8,14]],"date-time":"2019-08-14T23:11:30Z","timestamp":1565824290000},"page":"579-595","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":39,"title":["Chebyshev polynomials for generalized Couette flow of fractional Jeffrey nanofluid subjected to several thermochemical effects"],"prefix":"10.1007","volume":"37","author":[{"given":"R.","family":"Roohi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M. H.","family":"Heydari","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3539-0849","authenticated-orcid":false,"given":"O.","family":"Bavi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"H.","family":"Emdad","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,8,14]]},"reference":[{"key":"843_CR1","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1038\/nature05058","volume":"441","author":"GM Whitesides","year":"2006","unstructured":"Whitesides GM (2006) The origins and the future of microfluidics. Nature 441:368\u2013373","journal-title":"Nature"},{"key":"843_CR2","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.ijthermalsci.2018.02.013","volume":"128","author":"M Kiyasatfar","year":"2018","unstructured":"Kiyasatfar M (2018) Convective heat transfer and entropy generation analysis of non-newtonian power-law fluid flows in parallel-plate and circular microchannels under slip boundary conditions. Int J Therm Sci 128:15\u201327","journal-title":"Int J Therm Sci"},{"key":"843_CR3","first-page":"1","volume":"5","author":"FM Abbasi","year":"2015","unstructured":"Abbasi FM, Shehzad SA, Alsaedi A, Hayat T, Obid MA (2015) Influence of heat and mass flux conditions in hydromagnetic flow of jeffrey nanofluid. AIP Adv 5:1\u201312","journal-title":"AIP Adv"},{"key":"843_CR4","first-page":"1","volume":"126","author":"R Roohi","year":"2018","unstructured":"Roohi R, Emdad H, Jafarpur K, Mahmoudi M (2018) Determination of magnetic nanoparticles injection characteristics for optimal hyperthermia treatment of an arbitrary cancerous cells distribution. J Test Eval 126:1\u201317","journal-title":"J Test Eval"},{"key":"843_CR5","first-page":"1","volume":"127","author":"R Roohi","year":"2019","unstructured":"Roohi R, Emdad H, Jafarpur K (2019) A comprehensive study and optimization of magnetic nanoparticle drug delivery to cancerous tissues via external magnetic field. J Test Eval 127:1\u201323","journal-title":"J Test Eval"},{"key":"843_CR6","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1016\/j.ijheatmasstransfer.2018.05.087","volume":"126","author":"WM Hasona","year":"2018","unstructured":"Hasona WM, El-Shekhipy AA, Ibrahim MG (2018) Combined effects of magnetohydrodynamic and temperature dependent viscosity on peristaltic flow of jeffrey nanofluid through a porous medium: Applications to oil refinement. Int J Heat Mass Transf 126:700\u2013714","journal-title":"Int J Heat Mass Transf"},{"issue":"4","key":"843_CR7","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1016\/j.jppr.2018.11.008","volume":"7","author":"K Ramesh","year":"2018","unstructured":"Ramesh K (2018) Effects of viscous dissipation and joule heating on the couette and poiseuille flows of a jeffrey fluid with slip boundary conditions. Propuls Power Res 7(4):329\u2013341","journal-title":"Propuls Power Res"},{"key":"843_CR8","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1016\/j.ijheatmasstransfer.2016.11.102","volume":"108","author":"S Ahmed","year":"2017","unstructured":"Ahmed S, Zueco J, L\u00f3pez-Gonz\u00e1lez L (2017) Numerical and analytical solutions for magneto-hydrodynamic 3D flow through two parallel porous plates. Int J Heat Mass Transf 108:322\u2013331","journal-title":"Int J Heat Mass Transf"},{"key":"843_CR9","first-page":"349","volume":"3","author":"RS Raju","year":"2016","unstructured":"Raju RS, Reddy GJ, Rao JA, Rashidi MM (2016) Thermal diffusion and diffusion thermo effects on an unsteady heat and mass transfer magnetohydrodynamic natural convection couette flow using fem. J Comput Des Eng 3:349\u2013362","journal-title":"J Comput Des Eng"},{"key":"843_CR10","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1080\/10652469308819027","volume":"1","author":"SG Samko","year":"1993","unstructured":"Samko SG, Ross B (1993) Integration and differentiation to a variable fractional order. Integral Transform Spec Funct 1:277\u2013300","journal-title":"Integral Transform Spec Funct"},{"key":"843_CR11","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/BF01911126","volume":"21","author":"SG Samko","year":"1995","unstructured":"Samko SG (1995) Fractional integration and differentiation of variable order. Anal Math 21:213\u2013236","journal-title":"Anal Math"},{"key":"843_CR12","doi-asserted-by":"crossref","first-page":"653","DOI":"10.1007\/s11071-012-0485-0","volume":"71","author":"SG Samko","year":"2013","unstructured":"Samko SG (2013) Fractional integration and differentiation of variable order: an overview. Nonlinear Dyn 71:653\u2013662","journal-title":"Nonlinear Dyn"},{"key":"843_CR13","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1140\/epjp\/i2018-12021-3","volume":"133","author":"A Atangana","year":"2018","unstructured":"Atangana A, G\u00f3mez-Aguilar JF (2018) Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena. Eur Phys J Plus 133:166","journal-title":"Eur Phys J Plus"},{"key":"843_CR14","doi-asserted-by":"crossref","first-page":"516","DOI":"10.1016\/j.chaos.2018.07.033","volume":"114","author":"A Atangana","year":"2018","unstructured":"Atangana A, G\u00f3mez-Aguilar JF (2018) Fractional derivatives with no-index law property: application to chaos and statistics. Chaos Solitons Fractals 114:516\u2013535","journal-title":"Chaos Solitons Fractals"},{"key":"843_CR15","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1140\/epjp\/i2016-16373-2","volume":"131","author":"A Atangana","year":"2016","unstructured":"Atangana A (2016) Derivative with two fractional orders: a new avenue of investigation toward revolution in fractional calculus. Eur Phys J Plus 131:373","journal-title":"Eur Phys J Plus"},{"key":"843_CR16","doi-asserted-by":"crossref","first-page":"1111","DOI":"10.1016\/j.physd.2011.04.001","volume":"240","author":"LES Ramirez","year":"2011","unstructured":"Ramirez LES, Coimbra CFM (2011) On the variable order dynamics of the nonlinear wake caused by a sedimenting particle. Phys D 240:1111\u20131118","journal-title":"Phys D"},{"key":"843_CR17","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1140\/epjst\/e2011-01390-6","volume":"193","author":"HG Sun","year":"2011","unstructured":"Sun HG, Chen W, Wei H, Chen YQ (2011) A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems. Eur Phys J Spec Top 193:185","journal-title":"Eur Phys J Spec Top"},{"issue":"3","key":"843_CR18","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1016\/j.sigpro.2008.09.009","volume":"89","author":"J-J Shyu","year":"2009","unstructured":"Shyu J-J, Pei S-C, Chan C-H (2009) An iterative method for the design of variable fractional-order fir differintegrators. Signal Process 89(3):320\u2013327","journal-title":"Signal Process"},{"key":"843_CR19","doi-asserted-by":"crossref","first-page":"692","DOI":"10.1002\/andp.200310032","volume":"12","author":"C Coimbra","year":"2003","unstructured":"Coimbra C (2003) Mechanics with variable-order differential operators. Ann Phys 12:692\u2013703","journal-title":"Ann Phys"},{"key":"843_CR20","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1088\/0305-4470\/38\/42\/L03","volume":"38","author":"AV Chechkin","year":"2005","unstructured":"Chechkin AV, Gorenflo R, Sokolov IM (2005) Fractional diffusion in inhomogeneous media. J Phys A Math Gen 38:679\u2013684","journal-title":"J Phys A Math Gen"},{"key":"843_CR21","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1016\/j.neuron.2006.10.025","volume":"52","author":"F Santamaria","year":"2006","unstructured":"Santamaria F, Wils S, De Schutter E, Augustine GJ (2006) Anomalous diffusion in purkinjecell dendrites caused by spines. Neuron 52:635\u2013648","journal-title":"Neuron"},{"key":"843_CR22","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1016\/j.chaos.2019.06.014","volume":"127","author":"A Atangana","year":"2019","unstructured":"Atangana A, Shafiq A (2019) Differential and integral operators with constant fractional order and variable fractional dimension. Chaos Solitons Fractals 127:226\u2013243","journal-title":"Chaos Solitons Fractals"},{"key":"843_CR23","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1016\/j.chaos.2019.07.010","volume":"127","author":"LF\u00c1 Ruiz","year":"2019","unstructured":"Ruiz LF\u00c1, G\u00f3mez-Aguilar JF, Atangana A, Owolabi KM (2019) On the dynamics of fractional maps with power-law, exponential decay and mittag-leffler memory. Chaos Solitons Fractals 127:364\u2013388","journal-title":"Chaos Solitons Fractals"},{"issue":"1","key":"843_CR24","first-page":"79","volume":"16","author":"W Chen","year":"2013","unstructured":"Chen W, Zhang J, Zhang J (2013) A variable-order time-fractional derivative model for chloride ions sub-diffusion in concrete structures. Fract Calc Appl Anal 16(1):79\u201392","journal-title":"Fract Calc Appl Anal"},{"key":"843_CR25","first-page":"435","volume":"212","author":"R Lin","year":"2009","unstructured":"Lin R, Liu F, Anh V, Turner I (2009) Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl Math Comput 212:435\u2013445","journal-title":"Appl Math Comput"},{"issue":"1","key":"843_CR26","first-page":"101","volume":"80","author":"AH Bhrawy","year":"2016","unstructured":"Bhrawy AH, Zaky MA (2016) Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation. Nonlinear Dyn 80(1):101\u2013116","journal-title":"Nonlinear Dyn"},{"issue":"1","key":"843_CR27","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1016\/j.jcp.2014.12.001","volume":"80","author":"M Zayernouri","year":"2015","unstructured":"Zayernouri M, Karniadakis GE (2015) Fractional spectral collocation methods for linear and nonlinear variable order FPDES. J Comput Phys 80(1):312\u2013338","journal-title":"J Comput Phys"},{"key":"843_CR28","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.aml.2014.12.012","volume":"43","author":"XY Li","year":"2015","unstructured":"Li XY, Wu BY (2015) A numerical technique for variable fractional functional boundary value problems. Appl Math Lett 43:108\u2013113","journal-title":"Appl Math Lett"},{"key":"843_CR29","first-page":"235","volume":"352","author":"MH Heydari","year":"2019","unstructured":"Heydari MH, Avazzadeh Z, Yang Y (2019) A computational method for solving variable-order fractional nonlinear diffusion-wave equation. Appl Math Comput 352:235\u2013248","journal-title":"Appl Math Comput"},{"key":"843_CR30","first-page":"215","volume":"341","author":"MH Heydari","year":"2019","unstructured":"Heydari MH, Avazzadeh Z, Haromi MF (2019) A wavelet approach for solving multi-term variable-order time fractional diffusion-wave equation. Appl Math Comput 341:215\u2013228","journal-title":"Appl Math Comput"},{"key":"843_CR31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jcp.2019.06.024","volume":"395","author":"M Hosseininia","year":"2019","unstructured":"Hosseininia M, Heydari MH, Roohi R, Avazzadeh Z (2019) A computational wavelet method for variable-order fractional model of dual phase lag bioheat equation. J Comput Phys 395:1\u201318","journal-title":"J Comput Phys"},{"key":"843_CR32","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1140\/epjp\/i2018-12204-x","volume":"133","author":"R Roohi","year":"2018","unstructured":"Roohi R, Heydari MH, Aslami M, Mahmoudi MR (2018) A comprehensive numerical study of space-time fractional bioheat equation using fractional-order Legendre functions. Eur Phys J Plus 133:412\u2013422","journal-title":"Eur Phys J Plus"},{"key":"843_CR33","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1016\/j.chaos.2018.04.028","volume":"112","author":"MH Heydari","year":"2018","unstructured":"Heydari MH, Avazzadeh Z (2018) Legendre wavelets optimization method for variable-order fractional Poisson equation. Chaos Solitons Fractals 112:180\u2013190","journal-title":"Chaos Solitons Fractals"},{"key":"843_CR34","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1016\/j.chaos.2019.07.015","volume":"127","author":"M Hosseininia","year":"2019","unstructured":"Hosseininia M, Heydari MH (2019) Meshfree moving least squares method for nonlinear variable-order time fractional 2D telegraph equation involving Mittag\u2013Leffler non-singular kernel. Chaos Solitons Fractals 127:389\u2013399","journal-title":"Chaos Solitons Fractals"},{"issue":"1","key":"843_CR35","first-page":"27","volume":"22","author":"HG Sun","year":"2019","unstructured":"Sun HG, Chang A, Zhang Y, Chen W (2019) A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications. Eur Phys J Spec Top 22(1):27\u201359","journal-title":"Eur Phys J Spec Top"},{"issue":"5","key":"843_CR36","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1016\/j.camwa.2013.01.031","volume":"66","author":"H Zhang","year":"2013","unstructured":"Zhang H, Liu F, Phanikumar MS, Meerschaert MM (2013) A novel numerical method for the time variable fractional order mobile-immobile advection\u2013dispersion model. Comput Math Appl 66(5):693\u2013701","journal-title":"Comput Math Appl"},{"issue":"9","key":"843_CR37","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/j.camwa.2014.06.001","volume":"68","author":"MH Heydari","year":"2014","unstructured":"Heydari MH, Hooshmandasl MR, Ghaini FMM (2014) An efficient computational method for solving fractional biharmonic equation. Comput Math Appl 68(9):269\u2013287","journal-title":"Comput Math Appl"},{"issue":"9","key":"843_CR38","doi-asserted-by":"crossref","first-page":"1579","DOI":"10.1080\/00207160.2015.1067311","volume":"93","author":"MH Heydari","year":"2016","unstructured":"Heydari MH, Hooshmandasl MR, Loghmani GB, Cattani C (2016) Wavelets Galerkin method for solving stochastic heat equation. Int J Comput Math 93(9):1579\u20131596","journal-title":"Int J Comput Math"},{"key":"843_CR39","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1140\/epjp\/i2016-16268-2","volume":"131","author":"MR Hooshmandasl","year":"2016","unstructured":"Hooshmandasl MR, Heydari MH, Cattani C (2016) Numerical solution of fractional sub-diffusion and time-fractional diffusion-wave equations via fractional-order Legendre functions. Eur Phys J Plus 131:268","journal-title":"Eur Phys J Plus"},{"issue":"2","key":"843_CR40","doi-asserted-by":"crossref","first-page":"1185","DOI":"10.1007\/s11071-016-2753-x","volume":"85","author":"MH Heydari","year":"2016","unstructured":"Heydari MH, Hooshmandasl MR, Shakiba A, Cattani C (2016) Legendre wavelets Galerkin method for solving nonlinear stochastic integral equations. Nonlinear Dyn 85(2):1185\u20131202","journal-title":"Nonlinear Dyn"},{"key":"843_CR41","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-84108-8","volume-title":"Spectral methods in fluid dynamics","author":"C Canuto","year":"1988","unstructured":"Canuto C, Hussaini M, Quarteroni A, Zang T (1988) Spectral methods in fluid dynamics. Springer, Berlin"},{"key":"843_CR42","volume-title":"Fractional differential equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny I (1999) Fractional differential equations. Academic Press, San Diego"},{"key":"843_CR43","doi-asserted-by":"crossref","first-page":"2630","DOI":"10.1016\/j.apt.2017.07.015","volume":"28","author":"R Tripathi","year":"2017","unstructured":"Tripathi R, Seth GS, Mishra MK, El-Shekhipy AA, Ibrahim MG (2017) Double diffusive flow of a hydromagnetic nanofluid in a rotating channel with hall effect and viscous dissipation: active and passive control of nanoparticles. Adv Powder Technol 28:2630\u20132641","journal-title":"Adv Powder Technol"},{"key":"843_CR44","first-page":"10861","volume":"218","author":"S Shen","year":"2012","unstructured":"Shen S, Liu F, Chen J, Turner I, Anh V (2012) Numerical techniques for the variable order time fractional diffusion equation. Appl Math Comput 218:10861\u201310870","journal-title":"Appl Math Comput"},{"key":"843_CR45","first-page":"329","volume":"238","author":"Y Chen","year":"2014","unstructured":"Chen Y, Liu L, Li B, Sun Y (2014) Numerical solution for the variable order linear cable equation with bernstein polynomials. Appl Math Comput 238:329\u2013341","journal-title":"Appl Math Comput"},{"key":"843_CR46","doi-asserted-by":"crossref","first-page":"2364","DOI":"10.1016\/j.camwa.2011.07.024","volume":"62","author":"EH Doha","year":"2011","unstructured":"Doha EH, Bhrawy AH, Ezz-Eldien SS (2011) A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput Math Appl 62:2364\u20132373","journal-title":"Comput Math Appl"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-019-00843-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00366-019-00843-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-019-00843-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,1,21]],"date-time":"2021-01-21T10:29:20Z","timestamp":1611224960000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00366-019-00843-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,14]]},"references-count":46,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["843"],"URL":"https:\/\/doi.org\/10.1007\/s00366-019-00843-9","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"type":"print","value":"0177-0667"},{"type":"electronic","value":"1435-5663"}],"subject":[],"published":{"date-parts":[[2019,8,14]]},"assertion":[{"value":"14 May 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 August 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 August 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}