{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T06:11:12Z","timestamp":1780380672033,"version":"3.54.1"},"reference-count":71,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,11,10]],"date-time":"2020-11-10T00:00:00Z","timestamp":1604966400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,11,10]],"date-time":"2020-11-10T00:00:00Z","timestamp":1604966400000},"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":["Engineering with Computers"],"published-print":{"date-parts":[[2022,6]]},"DOI":"10.1007\/s00366-020-01195-5","type":"journal-article","created":{"date-parts":[[2020,11,10]],"date-time":"2020-11-10T08:02:56Z","timestamp":1604995376000},"page":"2245-2268","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["A POD reduced-order model based on spectral Galerkin method for solving the space-fractional Gray\u2013Scott model with error estimate"],"prefix":"10.1007","volume":"38","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6954-3896","authenticated-orcid":false,"given":"Mostafa","family":"Abbaszadeh","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mehdi","family":"Dehghan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ionel Michael","family":"Navon","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,11,10]]},"reference":[{"issue":"1","key":"1195_CR1","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1007\/BF00289234","volume":"12","author":"A Gierer","year":"1972","unstructured":"Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetik 12(1):30\u201339","journal-title":"Kybernetik"},{"issue":"2","key":"1195_CR2","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1007\/s11538-007-9264-3","volume":"70","author":"M Alber","year":"2008","unstructured":"Alber M, Glimm T, Hentschel H, Kazmierczak B, Zhang Y-T, Zhu J, Newman SA (2008) The morphostatic limit for a model of skeletal pattern formation in the vertebrate limb. Bull Math Biol 70(2):460\u2013483","journal-title":"Bull. Math. Biol."},{"issue":"3","key":"1195_CR3","doi-asserted-by":"crossref","first-page":"105","DOI":"10.3103\/S0278641911030071","volume":"35","author":"I Pavelchak","year":"2011","unstructured":"Pavelchak I (2011) A numerical method for determining the localized initial condition in the Fitzhugh\u2013Nagumo and Aliev\u2013Panfilov models. Mosc Univ Comput Math Cybern 35(3):105","journal-title":"Mosc. Univ. Comput. Math. Cybern."},{"issue":"2","key":"1195_CR4","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/s10915-004-4617-7","volume":"24","author":"A Madzvamuse","year":"2005","unstructured":"Madzvamuse A, Maini PK, Wathen AJ (2005) A moving grid finite element method for the simulation of pattern generation by turing models on growing domains. J Sci Comput 24(2):247\u2013262","journal-title":"J Sci Comput"},{"key":"1195_CR5","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1016\/j.physa.2017.09.014","volume":"491","author":"A Coronel-Escamilla","year":"2018","unstructured":"Coronel-Escamilla A, G\u00f3mez-Aguilar J, Torres L, Escobar-Jim\u00e9nez R (2018) A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel. Phys A Stat Mech Appl 491:406\u2013424","journal-title":"Phys A Stat Mech Appl"},{"issue":"4","key":"1195_CR6","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1007\/s10543-014-0484-2","volume":"54","author":"A Bueno-Orovio","year":"2014","unstructured":"Bueno-Orovio A, Kay D, Burrage K (2014) Fourier spectral methods for fractional-in-space reaction-diffusion equations. BIT Numer Math 54(4):937\u2013954","journal-title":"BIT Numer Math"},{"issue":"7\u20138","key":"1195_CR7","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1515\/ijnsns-2013-0124","volume":"15","author":"KM Owolabi","year":"2014","unstructured":"Owolabi KM, Patidar KC (2014) Numerical solution of singular patterns in one-dimensional Gray\u2013Scott-like models. Int J Nonlinear Sci Numer Simul 15(7\u20138):437\u2013462","journal-title":"Int J Nonlinear Sci Numer Simul"},{"key":"1195_CR8","doi-asserted-by":"crossref","first-page":"1030","DOI":"10.1016\/j.cma.2019.01.002","volume":"347","author":"T Wang","year":"2019","unstructured":"Wang T, Song F, Wang H, Karniadakis GE (2019) Fractional Gray\u2013Scott model: well-posedness, discretization, and simulations. Comput Methods Appl Mech Eng 347:1030\u20131049","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"3","key":"1195_CR9","doi-asserted-by":"crossref","first-page":"320","DOI":"10.3390\/axioms3030320","volume":"3","author":"RK Saxena","year":"2014","unstructured":"Saxena RK, Mathai AM, Haubold HJ (2014) Space-time fractional reaction-diffusion equations associated with a generalized Riemann\u2013Liouville fractional derivative. Axioms 3(3):320\u2013334","journal-title":"Axioms"},{"key":"1195_CR10","doi-asserted-by":"crossref","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"},{"key":"1195_CR11","doi-asserted-by":"publisher","DOI":"10.1007\/s00366-020-01095-8","author":"AS Hendy","year":"2020","unstructured":"Hendy AS, Zaky MA (2020) Graded mesh discretization for coupled system of nonlinear multi-term time-space fractional diffusion equations. Eng Comput. https:\/\/doi.org\/10.1007\/s00366-020-01095-8","journal-title":"Eng Comput"},{"key":"1195_CR12","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1002\/num.20112","volume":"22","author":"VJ Ervin","year":"2006","unstructured":"Ervin VJ, Roop JP (2006) Variational formulation for the stationary fractional advection dispersion equation. Numer Methods Partial Differ Equ 22:558\u2013576","journal-title":"Numer Methods Partial Differ Equ"},{"key":"1195_CR13","unstructured":"Roop JP (2004) Variational solution of the fractional advection dispersion equation, Ph.D thesis, Clemson University"},{"key":"1195_CR14","volume-title":"Numerical approximation of partial differential equations","author":"A Quarteroni","year":"1997","unstructured":"Quarteroni A, Valli A (1997) Numerical approximation of partial differential equations. Springer, New York"},{"issue":"1","key":"1195_CR15","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1137\/16M1103622","volume":"56","author":"Z Mao","year":"2018","unstructured":"Mao Z, Karniadakis GE (2018) A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative. SIAM J Numer Anal 56(1):24\u201349","journal-title":"SIAM J Numer Anal"},{"issue":"1","key":"1195_CR16","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1016\/j.jmaa.2011.06.051","volume":"385","author":"Z Luo","year":"2012","unstructured":"Luo Z, Li H, Zhou Y, Xie Z (2012) A reduced finite element formulation based on POD method for two-dimensional solute transport problems. J Math Anal Appl 385(1):371\u2013383","journal-title":"J Math Anal Appl"},{"key":"1195_CR17","doi-asserted-by":"publisher","first-page":"484","DOI":"10.1140\/epjp\/i2019-12871-y","volume":"134","author":"M Hamid","year":"2019","unstructured":"Hamid M, Usman M, Zubair T, Haq RU, Wang W (2019) Innovative operational matrices based computational scheme for fractional diffusion problems with the Riesz derivative. Eur Phys J Plus 134:484. https:\/\/doi.org\/10.1140\/epjp\/i2019-12871-y","journal-title":"Eur Phys J Plus"},{"key":"1195_CR18","doi-asserted-by":"crossref","first-page":"2323","DOI":"10.1016\/j.asej.2016.08.002","volume":"9","author":"M Hamid","year":"2018","unstructured":"Hamid M, Usman M, Zubair T, Mohyud-Din ST (2018) Comparison of Lagrange multipliers for telegraph equations. Ain Shams Eng J 9:2323\u20132328","journal-title":"Ain Shams Eng J"},{"issue":"5","key":"1195_CR19","doi-asserted-by":"crossref","first-page":"1416","DOI":"10.3934\/math.2019.5.1416","volume":"4","author":"M Hamid","year":"2019","unstructured":"Hamid M, Zubair T, Usma M, Haq RU (2019) Numerical investigation of fractional-order unsteady natural convective radiating flow of nanofluid in a vertical channel. AIMS Math 4(5):1416\u20131429","journal-title":"AIMS Math"},{"key":"1195_CR20","doi-asserted-by":"crossref","first-page":"124985","DOI":"10.1016\/j.amc.2019.124985","volume":"372","author":"M Usman","year":"2020","unstructured":"Usman M, Hamid M, Zubair T, Haq RU, Wang W, Liu MB (2020) Novel operational matrices-based method for solving fractional-order delay differential equations via shifted Gegenbauer polynomials. Appl Math Comput 372:124985","journal-title":"Appl Math Comput"},{"key":"1195_CR21","doi-asserted-by":"publisher","DOI":"10.1002\/num.22492","author":"M Usman","year":"2020","unstructured":"Usman M, Hamid M, Khalid MSU, Ul Haq R, Liu M (2020) A robust scheme based on novel-operational matrices for some classes of time-fractional nonlinear problems arising in mechanics and mathematical physics. Numer Methods Partial Differ Equ. https:\/\/doi.org\/10.1002\/num.22492 (In press)","journal-title":"Numer Methods Partial Differ Equ"},{"key":"1195_CR22","doi-asserted-by":"crossref","first-page":"124227","DOI":"10.1016\/j.physa.2020.124227","volume":"551","author":"M Hamid","year":"2020","unstructured":"Hamid M, Usman M, Haq RU, Wang W (2020) A Chelyshkov polynomial based algorithm to analyze the transport dynamics and anomalous diffusion in fractional model. Phys A Stat Mech Appl 551:124227","journal-title":"Phys A Stat Mech Appl"},{"key":"1195_CR23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10915-019-01102-1","volume":"82","author":"MA Zaky","year":"2020","unstructured":"Zaky MA, Hendy AS, Macias-Diaz JE (2020) Semi-implicit GalerkinLegendre spectral schemes for nonlinear time-space fractional diffusionreaction equations with smooth and nonsmooth solutions. J Sci Comput 82:1\u201327","journal-title":"J Sci Comput"},{"key":"1195_CR24","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/s11075-019-00743-5","volume":"84","author":"MA Zaky","year":"2020","unstructured":"Zaky MA, Ameen IG (2020) A priori error estimates of a Jacobi spectral method for nonlinear systems of fractional boundary value problems and related Volterra\u2013Fredholm integral equations. Numer Algorithms 84:63\u201389","journal-title":"Numer Algorithms"},{"issue":"8","key":"1195_CR25","doi-asserted-by":"crossref","first-page":"1657","DOI":"10.1364\/JOSAA.12.001657","volume":"12","author":"R Everson","year":"1995","unstructured":"Everson R, Sirovich L (1995) Karhunen\u2013Loeve procedure for gappy data. J Opt Soc Am A 12(8):1657\u20131664","journal-title":"J Opt Soc Am A"},{"issue":"1","key":"1195_CR26","first-page":"173","volume":"7","author":"G Dimitriu","year":"2015","unstructured":"Dimitriu G, Stefanescu R, Navon IM (2015) POD-DEIM approach on dimension reduction of a multi-species host-parasitoid system. Acad Rom Sci 7(1):173\u2013188","journal-title":"Acad Rom Sci"},{"issue":"4","key":"1195_CR27","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1023\/A:1011184714898","volume":"15","author":"S Ravindran","year":"2000","unstructured":"Ravindran S (2000) Reduced-order adaptive controllers for fluid flows using POD. J Sci Comput 15(4):457\u2013478","journal-title":"J Sci Comput"},{"issue":"10","key":"1195_CR28","doi-asserted-by":"crossref","first-page":"1571","DOI":"10.1002\/fld.1365","volume":"53","author":"Y Cao","year":"2007","unstructured":"Cao Y, Zhu J, Navon IM, Luo Z (2007) A reduced-order approach to four-dimensional variational data assimilation using proper orthogonal decomposition. Int J Numer Methods Fluids 53(10):1571\u20131583","journal-title":"Int J Numer Methods Fluids"},{"issue":"12","key":"1195_CR29","doi-asserted-by":"crossref","first-page":"909","DOI":"10.1002\/fld.4249","volume":"82","author":"Y Wang","year":"2016","unstructured":"Wang Y, Navon IM, Wang X, Cheng Y (2016) 2D Burgers equation with large Reynolds number using POD\/DEIM and calibration. Int J Numer Methods Fluids 82(12):909\u2013931","journal-title":"Int J Numer Methods Fluids"},{"key":"1195_CR30","first-page":"1","volume":"1","author":"D Xiao","year":"2015","unstructured":"Xiao D, Fang F, Pain CC, Navon IM, Salinas P, Muggeridge A (2015) Non-intrusive reduced order modeling of multi-phase flow in porous media using the POD-RBF method. J Comput Phys 1:1\u201325","journal-title":"J Comput Phys"},{"key":"1195_CR31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jcp.2014.01.011","volume":"263","author":"D Xiao","year":"2014","unstructured":"Xiao D, Fang F, Buchan AG, Pain CC, Navon IM, Du J, Datum GH (2014) Non-linear model reduction for the Navier\u2013Stokes equations using residual DEIM method. J Comput Phys 263:1\u201318","journal-title":"J Comput Phys"},{"key":"1195_CR32","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.cma.2012.11.002","volume":"255","author":"D Xiao","year":"2013","unstructured":"Xiao D, Fang F, Du J, Pain C, Navon I, Buchan A, ElSheikh A, Hu G (2013) Non-linear Petrov\u2013Galerkin methods for reduced order modelling of the Navier\u2013Stokes equations using a mixed finite element pair. Comput Methods Appl Mech Eng 255:147\u2013157","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1195_CR33","doi-asserted-by":"crossref","first-page":"580","DOI":"10.1002\/fld.4066","volume":"79","author":"D Xiao","year":"2015","unstructured":"Xiao D, Fang F, Pain CC, Datum GH (2015) Non-intrusive reduced-order modelling of the Navier\u2013Stokes equations based on RBF interpolation. Int J Numer Methods Fluids 79:580\u2013595","journal-title":"Int J Numer Methods Fluids"},{"issue":"4","key":"1195_CR34","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1080\/10407782.2016.1177327","volume":"70","author":"P Zhang","year":"2016","unstructured":"Zhang P, Zhang XH, Xiang H, Song L (2016) A fast and stabilized meshless method for the convection-dominated convection\u2013diffusion problems. Numer Heat Transf Part A Appl 70(4):420\u2013431","journal-title":"Numer Heat Transf Part A Appl"},{"key":"1195_CR35","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.matcom.2005.10.001","volume":"71","author":"M Dehghan","year":"2006","unstructured":"Dehghan M (2006) Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math Comput Simul 71:16\u201330","journal-title":"Math Comput Simul"},{"key":"1195_CR36","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1016\/j.cma.2019.06.018","volume":"355","author":"D Xiao","year":"2019","unstructured":"Xiao D (2019) Error estimation of the parametric non-intrusive reduced order model using machine learning. Comput Methods Appl Mech Eng 355:513\u2013534","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1195_CR37","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/j.aml.2018.08.024","volume":"88","author":"M Abbaszadeh","year":"2019","unstructured":"Abbaszadeh M (2019) Error estimate of second-order finite difference scheme for solving the Riesz space distributed-order diffusion equation. Appl Math Lett 88:179\u2013185","journal-title":"Appl Math Lett"},{"key":"1195_CR38","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/j.apnum.2020.07.020","volume":"158","author":"M Abbaszadeh","year":"2020","unstructured":"Abbaszadeh M, Dehghan M (2020) A POD-based reduced-order Crank\u2013Nicolson\/fourth-order alternating direction implicit (ADI) finite difference scheme for solving the two-dimensional distributed-order Riesz space-fractional diffusion equation. Appl Numer Math 158:271\u2013291","journal-title":"Appl Numer Math"},{"key":"1195_CR39","doi-asserted-by":"publisher","DOI":"10.1002\/fld.4834","author":"M Abbaszadeh","year":"2020","unstructured":"Abbaszadeh M, Dehghan M, Navon IM (2020) A proper orthogonal decomposition variational multiscale meshless interpolating element free Galerkin method for incompressible magnetohydrodynamics flow. Int J Numer Methods Fluids. https:\/\/doi.org\/10.1002\/fld.4834 (In press)","journal-title":"Int J Numer Methods Fluids"},{"key":"1195_CR40","doi-asserted-by":"crossref","first-page":"2642","DOI":"10.1108\/HFF-11-2018-0647","volume":"29","author":"M Dehghan","year":"2019","unstructured":"Dehghan M, Abbaszadeh M, Khodadadian A, Heitzinger C (2019) Galerkin proper orthogonal decomposition-reduced order method (POD-ROM) for solving generalized Swift\u2013Hohenberg equation. Int J Numer Methods Heat Fluid Flow 29:2642\u20132665","journal-title":"Int J Numer Methods Heat Fluid Flow"},{"key":"1195_CR41","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1016\/j.enganabound.2017.10.004","volume":"92","author":"M Dehghan","year":"2018","unstructured":"Dehghan M, Abbaszadeh M (2018) An upwind local radial basis functions-differential quadrature (RBF-DQ) method with proper orthogonal decomposition (POD) approach for solving compressible Euler equation. Eng Anal Bound Elem 92:244\u2013256","journal-title":"Eng Anal Bound Elem"},{"key":"1195_CR42","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.apnum.2017.12.004","volume":"126","author":"M Dehghan","year":"2018","unstructured":"Dehghan M, Abbaszadeh M (2018) A reduced proper orthogonal decomposition (POD) element free Galerkin (POD-EFG) method to simulate two-dimensional solute transport problems and error estimate. Appl Numer Math 126:92\u2013112","journal-title":"Appl Numer Math"},{"issue":"4","key":"1195_CR43","doi-asserted-by":"crossref","first-page":"1390","DOI":"10.1016\/j.camwa.2017.11.012","volume":"75","author":"M Dehghan","year":"2018","unstructured":"Dehghan M, Abbaszadeh M (2018) A combination of proper orthogonal decomposition discrete empirical interpolation method (PODDEIM) and meshless local RBF-DQ approach for prevention of groundwater contamination. Comput Math Appl 75(4):1390\u20131412","journal-title":"Comput Math Appl"},{"issue":"1","key":"1195_CR44","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1146\/annurev.fl.25.010193.002543","volume":"25","author":"G Berkooz","year":"1993","unstructured":"Berkooz G, Holmes P, Lumley JL (1993) The proper orthogonal decomposition in the analysis of turbulent flows. Ann Rev Fluid Mech 25(1):539\u2013575","journal-title":"Ann Rev Fluid Mech"},{"key":"1195_CR45","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"},{"key":"1195_CR46","doi-asserted-by":"crossref","first-page":"703","DOI":"10.1016\/j.amc.2014.06.023","volume":"243","author":"Y Liu","year":"2014","unstructured":"Liu Y, Fang Z, Li H, He S (2014) A mixed finite element method for a time-fractional fourth-order partial differential equation. Appl Math Comput 243:703\u2013717","journal-title":"Appl Math Comput"},{"key":"1195_CR47","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1016\/j.jcp.2014.07.023","volume":"276","author":"W Bu","year":"2014","unstructured":"Bu W, Tang Y, Yang J (2014) Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations. J Comput Phys 276:26\u201338","journal-title":"J Comput Phys"},{"key":"1195_CR48","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1016\/j.amc.2014.09.034","volume":"257","author":"W Bu","year":"2015","unstructured":"Bu W, Tang Y, Wu Y, Yang J (2015) Crank\u2013Nicolson ADI Galerkin finite element method for two-dimensional fractional FitzHugh\u2013Nagumo monodomain model. Appl Math Comput 257:355\u2013364","journal-title":"Appl Math Comput"},{"issue":"15\u201316","key":"1195_CR49","doi-asserted-by":"crossref","first-page":"3860","DOI":"10.1016\/j.apm.2013.10.008","volume":"38","author":"P Zhuang","year":"2014","unstructured":"Zhuang P, Liu F, Turner I, Gu Y (2014) Finite volume and finite element methods for solving a one-dimensional space-fractional Boussinesq equation. Appl Math Model 38(15\u201316):3860\u20133870","journal-title":"Appl Math Model"},{"issue":"6","key":"1195_CR50","doi-asserted-by":"crossref","first-page":"2975","DOI":"10.1016\/j.amc.2012.09.022","volume":"219","author":"Z Zhao","year":"2012","unstructured":"Zhao Z, Li C (2012) Fractional difference\/finite element approximations for the time-space fractional telegraph equation. Appl Math Comput 219(6):2975\u20132988","journal-title":"Appl Math Comput"},{"issue":"1","key":"1195_CR51","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1137\/080714130","volume":"47","author":"W Deng","year":"2008","unstructured":"Deng W (2008) Finite element method for the space and time fractional Fokker\u2013Planck equation. SIAM J Numer Anal 47(1):204\u2013226","journal-title":"SIAM J Numer Anal"},{"key":"1195_CR52","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 33:587\u2013605","journal-title":"Eng Comput"},{"key":"1195_CR53","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/BF02826009","volume":"41","author":"M Caputo","year":"1995","unstructured":"Caputo M (1995) Mean fractional-order-derivatives differential equations and filters. Annali dellUniversita di Ferrara 41:73\u201384","journal-title":"Annali dellUniversita di Ferrara"},{"key":"1195_CR54","doi-asserted-by":"crossref","first-page":"1249","DOI":"10.1177\/1077546307077468","volume":"13","author":"F Mainardi","year":"2007","unstructured":"Mainardi F, Mura A, Gorenflo R, Stojanovic M (2007) The two forms of fractional relaxation of distributed order. J Vib Control 13:1249\u20131268","journal-title":"J Vib Control"},{"key":"1195_CR55","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1016\/j.jcp.2014.09.031","volume":"280","author":"AA Alikhanov","year":"2015","unstructured":"Alikhanov AA (2015) A new difference scheme for the time fractional diffusion equation. J Comput Phys 280:424\u2013438","journal-title":"J Comput Phys"},{"key":"1195_CR56","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/j.apnum.2016.09.009","volume":"111","author":"A Bhrawy","year":"2017","unstructured":"Bhrawy A, Zaky M (2017) An improved collocation method for multi-dimensional space-time variable-order fractional Schr\u00f6dinger equations. Appl Numer Math 111:197\u2013218","journal-title":"Appl Numer Math"},{"key":"1195_CR57","doi-asserted-by":"publisher","DOI":"10.1109\/MESA.2010.5552095","volume-title":"On distributed order low-pass filter","author":"Y Li","year":"2010","unstructured":"Li Y, Sheng H, Chen YQ (2010) On distributed order low-pass filter. In: Proceedings of 2010 IEEE\/ASME international conference on mechatronic and embedded systems and applications IEEE, Qingdao, China. https:\/\/doi.org\/10.1109\/MESA.2010.5552095"},{"key":"1195_CR58","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1016\/j.cnsns.2016.04.020","volume":"40","author":"E Pindza","year":"2016","unstructured":"Pindza E, Owolabi KM (2016) Fourier spectral method for higher order space fractional reaction-diffusion equations. Commun Nonlinear Sci Numer Simul 40:112\u2013128","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"17\u201318","key":"1195_CR59","doi-asserted-by":"crossref","first-page":"7816","DOI":"10.1016\/j.apm.2016.03.036","volume":"40","author":"Z Yuan","year":"2016","unstructured":"Yuan Z, Nie Y, Liu F, Turner I, Zhang G, Gu Y (2016) An advanced numerical modeling for Riesz space fractional advection\u2013dispersion equations by a meshfree approach. Appl Math Model 40(17\u201318):7816\u20137829","journal-title":"Appl Math Model"},{"key":"1195_CR60","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/j.jcp.2016.01.015","volume":"310","author":"J Jia","year":"2016","unstructured":"Jia J, Wang H (2016) A fast finite volume method for conservative space-fractional diffusion equations in convex domains. J Comput Phys 310:63\u201384","journal-title":"J Comput Phys"},{"issue":"2","key":"1195_CR61","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1515\/fca-2017-0019","volume":"20","author":"W Fan","year":"2017","unstructured":"Fan W, Liu F, Jiang X, Turner I (2017) A novel unstructured mesh finite element method for solving the time-space fractional wave equation on a two-dimensional irregular convex domain. Fract Calculus Appl Anal 20(2):352\u2013383","journal-title":"Fract Calculus Appl Anal"},{"issue":"6","key":"1195_CR62","doi-asserted-by":"crossref","first-page":"A2865","DOI":"10.1137\/140961560","volume":"36","author":"X Zhao","year":"2014","unstructured":"Zhao X, Sun Z-Z, Hao Z-P (2014) A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrodinger equation. SIAM J Sci Comput 36(6):A2865\u2013A2886","journal-title":"SIAM J Sci Comput"},{"key":"1195_CR63","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1016\/j.jcp.2014.06.031","volume":"293","author":"W Bu","year":"2015","unstructured":"Bu W, Tang Y, Wu Y, Yang J (2015) Finite difference\/finite element method for two-dimensional space and time fractional Bloch\u2013Torrey equations. J Comput Phys 293:264\u2013279","journal-title":"J Comput Phys"},{"key":"1195_CR64","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"},{"issue":"8","key":"1195_CR65","doi-asserted-by":"crossref","first-page":"5984","DOI":"10.1016\/j.apm.2012.12.012","volume":"37","author":"Z-P Ma","year":"2013","unstructured":"Ma Z-P (2013) Stability and hopf bifurcation for a three-component reaction-diffusion population model with delay effect. Appl Math Model 37(8):5984\u20136007","journal-title":"Appl Math Model"},{"key":"1195_CR66","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.matcom.2012.11.012","volume":"89","author":"Z Luo","year":"2013","unstructured":"Luo Z, Li H, Sun P, An J, Navon IM (2013) A reduced-order finite volume element formulation based on POD method and numerical simulation for two-dimensional solute transport problems. Math Comput Simul 89:50\u201368","journal-title":"Math Comput Simul"},{"issue":"2","key":"1195_CR67","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1002\/fld.1452","volume":"55","author":"Z Luo","year":"2007","unstructured":"Luo Z, Chen J, Zhu J, Wang R, Navon I (2007) An optimizing reduced order FDS for the tropical Pacific Ocean reduced gravity model. Int J Numer Methods Fluids 55(2):143\u2013161","journal-title":"Int J Numer Methods Fluids"},{"key":"1195_CR68","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1016\/j.ijheatmasstransfer.2015.01.008","volume":"84","author":"X Zhang","year":"2015","unstructured":"Zhang X, Xiang H (2015) A fast meshless method based on proper orthogonal decomposition for the transient heat conduction problems. Int J Heat Mass Transfer 84:729\u2013739","journal-title":"Int J Heat Mass Transfer"},{"key":"1195_CR69","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1126\/science.261.5118.189","volume":"261","author":"JE Pearson","year":"1993","unstructured":"Pearson JE (1993) Complex patterns in a simple system. Science 261:189\u2013192","journal-title":"Science"},{"key":"1195_CR70","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1016\/j.jmaa.2007.08.024","volume":"340","author":"AN Kochubei","year":"2008","unstructured":"Kochubei AN (2008) Distributed order calculus and equations of ultraslow diffusion. J Math Anal Appl 340:252\u2013281","journal-title":"J Math Anal Appl"},{"key":"1195_CR71","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1016\/j.ijheatmasstransfer.2015.01.008","volume":"84","author":"X Zhang","year":"2015","unstructured":"Zhang X, Xiang H (2015) A fast meshless method based on proper orthogonal decomposition for the transient heat conduction problems. Int J Heat Mass Transf 84:729\u2013739","journal-title":"Int J Heat Mass Transf"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-020-01195-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-020-01195-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-020-01195-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,26]],"date-time":"2023-10-26T18:21:59Z","timestamp":1698344519000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-020-01195-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,10]]},"references-count":71,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,6]]}},"alternative-id":["1195"],"URL":"https:\/\/doi.org\/10.1007\/s00366-020-01195-5","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,11,10]]},"assertion":[{"value":"16 July 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 October 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 November 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}