{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,23]],"date-time":"2025-12-23T00:29:14Z","timestamp":1766449754967,"version":"3.37.3"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T00:00:00Z","timestamp":1583107200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T00:00:00Z","timestamp":1583107200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11772121","11702083","11572112"],"award-info":[{"award-number":["11772121","11702083","11572112"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011402","name":"State Key Laboratory of Mechanics and Control of Mechanical Structures","doi-asserted-by":"crossref","award":["MCMS-0218G01"],"award-info":[{"award-number":["MCMS-0218G01"]}],"id":[{"id":"10.13039\/501100011402","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Engineering with Computers"],"published-print":{"date-parts":[[2021,10]]},"DOI":"10.1007\/s00366-020-00991-3","type":"journal-article","created":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T19:02:47Z","timestamp":1583175767000},"page":"3151-3166","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Improved singular boundary method and dual reciprocity method for fractional derivative Rayleigh\u2013Stokes problem"],"prefix":"10.1007","volume":"37","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4546-5704","authenticated-orcid":false,"given":"Farzaneh","family":"Safari","sequence":"first","affiliation":[]},{"given":"HongGuang","family":"Sun","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,3,2]]},"reference":[{"key":"991_CR1","first-page":"424","volume":"315","author":"P Assari","year":"2017","unstructured":"Assari P, Dehghan M (2017) A meshless discrete collocation method for the numerical solution of singular-logarithmic boundary integral equations utilizing radial basis functions. Appl Math Comput 315:424\u2013444","journal-title":"Appl Math Comput"},{"issue":"3","key":"991_CR2","first-page":"205","volume":"38","author":"Z Avazzadeh","year":"2014","unstructured":"Avazzadeh Z, Hosseini VR, Chen W (2014) Radial basis functions and FDM for solving fractional diffusion-wave equation. Iran J Sci Technol (Sciences) 38(3):205\u2013212","journal-title":"Iran J Sci Technol (Sciences)"},{"key":"991_CR3","doi-asserted-by":"crossref","unstructured":"Avazzadeh Z, Chen W, Reza Hosseini V (2014) The coupling of RBF and FDM for solving higher order fractional partial differential equations. In: Applied mechanics and materials. Trans Tech Publications Ltd, vol 598, pp 409\u2013413","DOI":"10.4028\/www.scientific.net\/AMM.598.409"},{"issue":"1","key":"991_CR4","first-page":"340","volume":"204","author":"CM Chen","year":"2008","unstructured":"Chen CM, Liu F, Anh V (2008) Numerical analysis of the Rayleigh\u2013Stokes problem for a heated generalized second grade fluid with fractional derivatives. Appl Math Comput 204(1):340\u2013351","journal-title":"Appl Math Comput"},{"issue":"6","key":"991_CR5","first-page":"592","volume":"30","author":"W Chen","year":"2009","unstructured":"Chen W (2009) Singular boundary method: a novel, simple, meshfree, boundary collocation numerical method. Chin J Solid Mech 30(6):592\u2013599","journal-title":"Chin J Solid Mech"},{"issue":"5","key":"991_CR6","doi-asserted-by":"crossref","first-page":"530","DOI":"10.1016\/j.enganabound.2009.12.002","volume":"34","author":"W Chen","year":"2010","unstructured":"Chen W, Wang FZ (2010) A method of fundamental solutions without fictitious boundary. Eng Anal Bound Elem 34(5):530\u2013532","journal-title":"Eng Anal Bound Elem"},{"key":"991_CR7","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1016\/j.enganabound.2014.02.007","volume":"44","author":"W Chen","year":"2014","unstructured":"Chen W, Zhang JY, Fu ZJ (2014) Singular boundary method for modified Helmholtz equations. Eng Anal Bound Elem 44:112\u2013119","journal-title":"Eng Anal Bound Elem"},{"issue":"5","key":"991_CR8","doi-asserted-by":"crossref","first-page":"543","DOI":"10.4208\/aamm.11-m11118","volume":"4","author":"W Chen","year":"2012","unstructured":"Chen W, Gu Y (2012) An improved formulation of singular boundary method. Adv Appl Math Mech 4(5):543\u2013558","journal-title":"Adv Appl Math Mech"},{"issue":"1","key":"991_CR9","first-page":"65","volume":"54","author":"W Chen","year":"2009","unstructured":"Chen W, Fu Z, Wei X (2009) Potential problems by singular boundary method satisfying moment condition. Comput Model Eng Sci 54(1):65","journal-title":"Comput Model Eng Sci"},{"issue":"20","key":"991_CR10","doi-asserted-by":"crossref","first-page":"7792","DOI":"10.1016\/j.jcp.2009.07.021","volume":"228","author":"M Cui","year":"2009","unstructured":"Cui M (2009) Compact finite difference method for the fractional diffusion equation. J Comput Phys 228(20):7792\u20137804","journal-title":"J Comput Phys"},{"key":"991_CR11","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/j.enganabound.2015.11.011","volume":"64","author":"M Dehghan","year":"2016","unstructured":"Dehghan M, Abbaszadeh M, Mohebbi A (2016) Analysis of two methods based on Galerkin weak form for fractional diffusion-wave: meshless interpolating element free Galerkin (IEFG) and finite element methods. Eng Anal Bound Elem 64:205\u2013221","journal-title":"Eng Anal Bound Elem"},{"issue":"3","key":"991_CR12","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1007\/s00366-016-0491-9","volume":"33","author":"M Dehghan","year":"2017","unstructured":"Dehghan M, Abbaszadeh M (2017) A finite element method for the numerical solution of Rayleigh\u2013Stokes problem for a heated generalized second grade fluid with fractional derivatives. Eng Comput 33(3):587\u2013605","journal-title":"Eng Comput"},{"issue":"2","key":"991_CR13","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1002\/num.20460","volume":"26","author":"M Dehghan","year":"2010","unstructured":"Dehghan M, Manafian J, Saadatmandi A (2010) Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer Methods Partial Differ Equ Int J 26(2):448\u2013479","journal-title":"Numer Methods Partial Differ Equ Int J"},{"issue":"10","key":"991_CR14","doi-asserted-by":"crossref","first-page":"2461","DOI":"10.1002\/mma.3707","volume":"39","author":"M Dehghan","year":"2016","unstructured":"Dehghan M, Safarpoor M (2016) The dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equations. Math Methods Appl Sci 39(10):2461\u20132476","journal-title":"Math Methods Appl Sci"},{"issue":"14","key":"991_CR15","doi-asserted-by":"crossref","first-page":"3979","DOI":"10.1002\/mma.3839","volume":"39","author":"M Dehghan","year":"2016","unstructured":"Dehghan M, Safarpoor M (2016) The dual reciprocity boundary elements method for the linear and nonlinear two-dimensional time-fractional partial differential equations. Math Methods Appl Sci 39(14):3979\u20133995","journal-title":"Math Methods Appl Sci"},{"key":"991_CR16","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.enganabound.2015.03.013","volume":"58","author":"M Dehghan","year":"2015","unstructured":"Dehghan M Shirzadi (2015) The modified dual reciprocity boundary elements method and its application for solving stochastic partial differential equations. Eng Anal Bound Elem 5:99\u2013111","journal-title":"Eng Anal Bound Elem"},{"issue":"4","key":"991_CR17","doi-asserted-by":"crossref","first-page":"522","DOI":"10.1016\/j.enganabound.2008.08.008","volume":"33","author":"M Dehghan","year":"2009","unstructured":"Dehghan M, Mirzaei D (2009) A numerical method based on the boundary integral equation and dual reciprocity methods for one-dimensional Cahn\u2013Hilliard equation. Eng Anal Bound Elem 33(4):522\u2013528","journal-title":"Eng Anal Bound Elem"},{"issue":"1","key":"991_CR18","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.enganabound.2009.07.002","volume":"34","author":"M Dehghan","year":"2010","unstructured":"Dehghan M, Ghesmati A (2010) Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method. Eng Anal Bound Elem 34(1):51\u201359","journal-title":"Eng Anal Bound Elem"},{"issue":"6\u20138","key":"991_CR19","doi-asserted-by":"crossref","first-page":"476","DOI":"10.1016\/j.cma.2007.08.016","volume":"197","author":"M Dehghan","year":"2008","unstructured":"Dehghan M, Mirzaei D (2008) The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation. Comput Methods Appl Mech Eng 197(6\u20138):476\u2013486","journal-title":"Comput Methods Appl Mech Eng"},{"key":"991_CR20","doi-asserted-by":"crossref","unstructured":"Ernst OG, Gander MJ (2012) Why it is difficult to solve Helmholtz problems with classical iterative methods. In: Numerical analysis of multiscale problems. Springer, Berlin, Heidelberg, pp 325\u2013363","DOI":"10.1007\/978-3-642-22061-6_10"},{"key":"991_CR21","first-page":"1","volume":"2020","author":"H Fazli","year":"2020","unstructured":"Fazli H, Sun H, Aghchi S (2020) Existence of extremal solutions of fractional Langevin equation involving nonlinear boundary conditions. Int J Comput Math 2020:1\u20133","journal-title":"Int J Comput Math"},{"issue":"6","key":"991_CR22","doi-asserted-by":"crossref","first-page":"1655","DOI":"10.2298\/FIL1706655F","volume":"31","author":"H Fazli","year":"2017","unstructured":"Fazli H, Bahrami F (2017) On the steady solutions of fractional reaction\u2013diffusion equations. Filomat 31(6):1655\u20131664","journal-title":"Filomat"},{"key":"991_CR23","doi-asserted-by":"crossref","first-page":"1011","DOI":"10.1016\/S0020-7462(00)00118-9","volume":"37","author":"C Fetecau","year":"2002","unstructured":"Fetecau C, Corina F (2002) The Rayleigh-Stokes problem for heated second grade fluids. Int J Nonlin Mech 37:1011\u20131015","journal-title":"Int J Nonlin Mech"},{"key":"991_CR24","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.enganabound.2013.10.009","volume":"38","author":"VR Hosseini","year":"2014","unstructured":"Hosseini VR, Chen W, Avazzadeh Z (2014) Numerical solution of fractional telegraph equation by using radial basis functions. Eng Anal Bound Elem 38:31\u201339","journal-title":"Eng Anal Bound Elem"},{"issue":"2","key":"991_CR25","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1140\/epjp\/i2015-15033-5","volume":"130","author":"VR Hosseini","year":"2015","unstructured":"Hosseini VR, Shivanian E, Chen W (2015) Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation. Eur Phys J Plus 130(2):33","journal-title":"Eur Phys J Plus"},{"key":"991_CR26","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1016\/j.jcp.2016.02.030","volume":"312","author":"VR Hosseini","year":"2016","unstructured":"Hosseini VR, Shivanian E, Chen W (2016) Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with damping. J Comput Phys 312:307\u2013332","journal-title":"J Comput Phys"},{"issue":"2","key":"991_CR27","first-page":"199","volume":"7","author":"H Jafari","year":"2019","unstructured":"Jafari H, Sadeghi J, Safari F, Kubeka A (2019) Factorization method for fractional Schr\u00f6dinger equation in D-dimensional fractional space and homogeneous manifold $$SL (2, c)\/GL (1, c)$$. CMDE 7(2):199\u2013205","journal-title":"CMDE"},{"key":"991_CR28","doi-asserted-by":"crossref","first-page":"S301","DOI":"10.2298\/TSCI170707033J","volume":"22","author":"H Jafari","year":"2018","unstructured":"Jafari H, Jassim HK, Vahidi J (2018) The reduced differential transform and variational iteration methods for 3D diffusion model in fractal heat transfer within local fractional operators. Therm Sci 22:S301\u2013S307","journal-title":"Therm Sci"},{"issue":"12","key":"991_CR29","doi-asserted-by":"crossref","first-page":"1767","DOI":"10.1016\/j.apnum.2012.05.011","volume":"62","author":"A Karageorghis","year":"2012","unstructured":"Karageorghis A, Johansson BT, Lesnic D (2012) The method of fundamental solutions for the identification of a sound-soft obstacle in inverse acoustic scattering. Appl Numer Math 62(12):1767\u20131780","journal-title":"Appl Numer Math"},{"key":"991_CR30","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/j.cma.2013.05.012","volume":"264","author":"A Mohebbi","year":"2013","unstructured":"Mohebbi A, Abbaszadeh M, Dehghan M (2013) Compact finite difference scheme and RBF meshless approach for solving 2D Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives. Comput Methods Appl Mech Eng 264:163\u2013177","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"5\u20136","key":"991_CR31","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/s004660050420","volume":"23","author":"AS Muleshkov","year":"1999","unstructured":"Muleshkov AS, Golberg MA, Chen CS (1999) Particular solutions of Helmholtz-type operators using higher order polyhrmonic splines. Comput Mech 23(5\u20136):411\u2013419","journal-title":"Comput Mech"},{"key":"991_CR32","unstructured":"Nowak AJ, Neves AC (1994) The multiple reciprocity boundary element method. Computational Mechanics Publication, pp 1\u201341"},{"issue":"5","key":"991_CR33","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1002\/nme.1620300502","volume":"30","author":"PW Partridge","year":"1990","unstructured":"Partridge PW, Wrobel LC (1990) The dual reciprocity boundary element method for spontaneous ignition. Int J Numer Meth Eng 30(5):953\u2013963","journal-title":"Int J Numer Meth Eng"},{"issue":"5","key":"991_CR34","doi-asserted-by":"crossref","first-page":"839","DOI":"10.1016\/S0022-460X(02)01006-4","volume":"261","author":"E Perrey-Debain","year":"2003","unstructured":"Perrey-Debain E, Trevelyan J, Bettess P (2003) Plane wave interpolation in direct collocation boundary element method for radiation and wave scattering: numerical aspects and applications. J Sound Vib 261(5):839\u2013858","journal-title":"J Sound Vib"},{"key":"991_CR35","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/0377-0257(84)80008-7","volume":"15","author":"KR Rajagopal","year":"1984","unstructured":"Rajagopal KR (1984) On the creeping flow of the second order fluid. J. Non-Newton Fluid Mech 15:239\u2013246","journal-title":"J. Non-Newton Fluid Mech"},{"issue":"2","key":"991_CR36","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1007\/BF01560464","volume":"19","author":"KR Rajagopal","year":"1984","unstructured":"Rajagopal KR, Gupta AS (1984) An exact solution for the flow of a non-Newtonian fluid past an infinite porous plate. Meccanica 19(2):158\u2013160","journal-title":"Meccanica"},{"key":"991_CR37","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1016\/j.oceaneng.2014.12.008","volume":"96","author":"Z Razafizana","year":"2015","unstructured":"Razafizana Z, Fu ZJ (2015) Singular boundary method for water wave problems. Ocean Eng 96:330\u2013337","journal-title":"Ocean Eng"},{"issue":"5","key":"991_CR38","doi-asserted-by":"crossref","first-page":"1594","DOI":"10.1016\/j.camwa.2019.02.001","volume":"78","author":"F Safari","year":"2019","unstructured":"Safari F, Chen W (2019) Coupling of the improved singular boundary method and dual reciprocity method for multi-term time-fractional mixed diffusion-wave equations. Comput Math Appl 78(5):1594\u20131607","journal-title":"Comput Math Appl"},{"issue":"2","key":"991_CR39","doi-asserted-by":"crossref","first-page":"847","DOI":"10.1002\/mma.5963","volume":"43","author":"F Safari","year":"2019","unstructured":"Safari F, Azarsa P (2019) Backward substitution method based on M\u00fcntz polynomials for solving the nonlinear space fractional partial differential equations. Math Methods Appl Sci 43(2):847\u2013864","journal-title":"Math Methods Appl Sci"},{"issue":"6","key":"991_CR40","doi-asserted-by":"crossref","first-page":"060301","DOI":"10.1088\/0256-307X\/34\/6\/060301","volume":"34","author":"F Safari","year":"2017","unstructured":"Safari F, Jafari H, Sadeghi J, Johnston SJ, Baleanu D (2017) Stability of Dirac equation in four-dimensional gravity. Chin Phys Lett 34(6):060301","journal-title":"Chin Phys Lett"},{"issue":"1","key":"991_CR41","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/s00366-017-0522-1","volume":"34","author":"E Shivanian","year":"2018","unstructured":"Shivanian E, Jafarabadi A (2018) Rayleigh\u2013Stokes problem for a heated generalized second grade fluid with fractional derivatives: a stable scheme based on spectral meshless radial point interpolation. Eng Comput 34(1):77\u201390","journal-title":"Eng Comput"},{"issue":"5","key":"991_CR42","doi-asserted-by":"crossref","first-page":"1072","DOI":"10.1016\/j.nonrwa.2005.09.007","volume":"7","author":"F Shen","year":"2006","unstructured":"Shen F, Tan W, Zhao Y, Masuoka T (2006) The Rayleigh\u2013Stokes problem for a heated generalized second grade fluid with fractional derivative model. Nonlinear Anal Real 7(5):1072\u20131080","journal-title":"Nonlinear Anal Real"},{"key":"991_CR43","doi-asserted-by":"crossref","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 Simulat 64:213\u2013231","journal-title":"Commun Nonlinear Sci Numer Simulat"},{"issue":"1","key":"991_CR44","doi-asserted-by":"crossref","first-page":"524","DOI":"10.1016\/j.apm.2007.11.015","volume":"33","author":"C Xue","year":"2009","unstructured":"Xue C, Nie J (2009) Exact solutions of the Rayleigh\u2013Stokes problem for a heated generalized second grade fluid in a porous half-space. Appl Math Model 33(1):524\u2013531","journal-title":"Appl Math Model"},{"key":"991_CR45","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.enganabound.2015.02.001","volume":"56","author":"C Yang","year":"2015","unstructured":"Yang C, Li X (2015) Meshless singular boundary methods for biharmonic problems. Eng Anal Bound Elem 56:39\u201348","journal-title":"Eng Anal Bound Elem"},{"issue":"4","key":"991_CR46","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1260\/174830109789621310","volume":"3","author":"Q Yu","year":"2009","unstructured":"Yu Q, Song J, Liu F, Anhc V et al (2009) An approximate solution for the Rayleigh\u2013Stokes problem for a heated generalized second grade fluid with fractional derivative model using the Adomian decomposition method. J Algorithm Comput Technol 3(4):553\u2013572","journal-title":"J Algorithm Comput Technol"},{"issue":"12","key":"991_CR47","doi-asserted-by":"crossref","first-page":"1533","DOI":"10.1007\/s10483-009-1205-7","volume":"30","author":"PH Zhuang","year":"2009","unstructured":"Zhuang PH, Liu QX (2009) Numerical method of Rayleigh\u2013Stokes problem for heated generalized second grade fluid with fractional derivative. Appl Math Mech 30(12):1533","journal-title":"Appl Math Mech"},{"key":"991_CR48","volume-title":"The finite element method","author":"OC Zienkiewicz","year":"1991","unstructured":"Zienkiewicz OC, Taylor RL (1991) The finite element method, 4th edn. McGraw-Hill, New York","edition":"4"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-020-00991-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-020-00991-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-020-00991-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,13]],"date-time":"2021-09-13T11:37:24Z","timestamp":1631533044000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-020-00991-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,2]]},"references-count":48,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,10]]}},"alternative-id":["991"],"URL":"https:\/\/doi.org\/10.1007\/s00366-020-00991-3","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"type":"print","value":"0177-0667"},{"type":"electronic","value":"1435-5663"}],"subject":[],"published":{"date-parts":[[2020,3,2]]},"assertion":[{"value":"28 November 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 February 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 March 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}