分数阶扩散模型数值模拟的研究进展
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摘要
本文从分数阶导数概念的提出与反常扩散过程的数学刻画入手,简述有关分数阶扩散模型数学理论与数值模拟研究的进展与动态,其中也包括了作者近年来在这一领域所做的部分工作.
This paper starts with the concept of fractional derivative and the mathematical description of the anomalous diffusion process. It briefly describes the progress and dynamics of the mathematical theory and numerical simulation of the fractional diffusion model. It also includes the author′s recent work in this field.
This paper starts with the concept of fractional derivative and the mathematical description of the anomalous diffusion process. It briefly describes the progress and dynamics of the mathematical theory and numerical simulation of the fractional diffusion model. It also includes the author′s recent work in this field.
引文
[1]Abel N H.Aufl oSung Einer Mechanischen Aufgabe[M].Fu¨r Die Reine:Angewandte Mathematik,1826:153-157.
[2]Abel N H.Solution Dequelques Probl eˋMes aˋL'aide d'Integrales d eˊFinies[M].Oeuvres:Compl eˋtes,1881:16-18.
[3]Al-Khaled,Momani S.An approximate solution for a fractional diffusion-wave equation using the decomposition method[J].Appl Math Comput,2005,165(2):473-483.
[4]Basu T S,Wang H.A fast second-order finite difference method for space-fractional diffusion equations[J].Numer Anal Modeling,2012,(9):658-666.
[5]Berkowitz B,Emmanuel S,Scher H.Non-Fickian transport and multiple-rate mass transfer in porous media[J].Water Resour Res,2008,(44):W03402.
[6]Carreras B A,Lynch V E,Zaslavsky G M.Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence models[J].Phys Plasmas,2001,(8):5096-5103.
[7]Chen H,Wang H.Numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation[J].Comp Appl Math,2016,(296):480-498.
[8]Chen S,Liu F.ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersionequation[J].Appl Math Comput,2008,26(1):295-311.
[9]Daftardar-Gejji V,Jafari H.Solving a multi-order fractional differential equation using Adomian decomposition[J].Appl Math Comput,2007,(189):541-548.
[10]Deng W,Hesthaven J S.Discontinuous galerkin methods for fractional diffusion equations[J].Scientific Computing Group,2010,(19):120-125.
[11]Deng W,Hesthaven J S.Local discontinuous Galerkin methods for fractional diffusion equations[J].Model Numer Anal,2013,(47):1845-1864.
[12]Diethelm K,Freed A D.On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity[J].Computational Fluid Dynamics Reaction Engineering and Molecular Properties,1999,125(10):217-224.
[13]Ervin V J,Roop J P.Variational formulation for the stationary fractional advection dis-persion equation[J].Numer Methods Partial Differ Equ,2005,(22):558-576.
[14]Ervin V J,Roop J P.Variational solution of fractional advection dispersion equations on bounded domains in Rd[J].Methods Partial Differential Equations,2007,(23):256-281.
[15]Ervin V J,Heuer N,Roop J P.Regularity of the solution to 1-D fractional order diffusion equations[J].Comp App Math,2018,87(313):2273-2294.
[16]Fix G J,Roop J P.Least squares finite-element solution of a fractional order two-Point boundary value problem[J].Comput Math Appl,2004,(48):1017-1033.
[17]Gorenflo R,Fabritiis G D.Discrete random walk models for symmetric Lévy-Feller diffusion processes[J].Phys A,1999,269(1):79-89.
[18]Gorenflo R,Luchko,Mainardi F.Wright functions as scale-invariant solutions of the diffusion-wave equation[J].Comp Appl Math,2000,(118):175-191.
[19]Gorenflo R,Mainardi F.Random walk models for space fractional diffusion processes[J].Fract Calc ApplAnal,1998,1(2):167-191.
[20]Gorenflo R,Mainardi F.Approximation of Levy-Feller diffusion by random walk[J].Anal Appl,1999,18(2):231-246.
[21]Gorenflo R,Mainardi F,Moretti D.Discrete random walk models for space-time fractional diffusion[J].Chem Phys.,2002,(284):521-541.
[22]Gru¨nward A K.Uber begrenzte derivationen und deren anwendung[J].Z angew Math und Phys,1867,(12):441-480.
[23]郭柏灵,蒲学科,黄凤辉.分数阶偏微分方程及其数值解[M].北京:科学出版社,2011:80-85.
[24]He J.Variational iteration method for autonomous ordinary differential system[J].Appl Math Comput,2000,(114):115-123.
[25]Huang J F,Nie N M,Tang F.A second order finite difference-spectral method for space fractional diffusion equations[J].Sci Chin Math,2014,(57):1303-1317.
[26]Jia J H,Wang H.A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh[J].Comput Phys,2015,299(15):842-862.
[27]Jia L L,Chen H,Wang H.Mixed-type Galerkin variational principle and numerical simulation for a generalized nonlocal elastic model[J].Sci Comput,2017,(71):660-681.
[28]Jin B T,Lazarov R,Lu X.A simple finite element method for boundary value problems with a Riemann-Liouville derivative[J].Comput Appl Math,2016,(293):94-111.
[29]Jin B,Lazarov R,Pasciak J.Variational formulation of problems involving fractional order differential operators[J].Math Comp,2015,(84):2665-2700.
[30]Jin B T,Lazarov R,Zhou Z.A Petrov-Galerkin finite element method for fractional convection-diffusion equations[J].SIAM J Numer Anal,2016,(54):481-503.
[31]Jin B T,Zhou Z.A finite element method with singularity reconstruction for fractional boundary valueproblems[J].ESAIM Math Model Numer Anal,2015,(49):1261-1283.
[32]Klafter J,Sokolov I M.Anomalous diffusion spreads its wings[J].Phys World,2005,18(8):29-32.
[33]Lacroix S F.Traite du Cazcuz Diff eˊRentiez et du Cazcuz Int eˊgraz[M].Paris:Mme Ve Courcier Tome Troisi,1819:409-410.
[34]Laurent H.Sur lecalcul des d eˊriv eˊes aˋindicies quelconques[J].Nouv Annales de Mathematiques,1884,(3):240-252.
[35]Letnikov A V.Theory of differentiation with an arbitrary index(Russian)[J].Mat sb,1868,(3):1-66.
[36]Letnikov A V.An explanation of the concepts of the theory of differentiation of arbitrary index(Russian)[J].Moscow Matem Sbornik,1872,(6):413-445.
[37]Li S,Zhou Z J.Legendre pseudo-spectral method for optimal control problem governed by time fractional diffusion equation[J].Comput Math,2018,95(6-7):1308-1325.
[38]Li X,Xu C.A space-time spectral method for the time fractional diffusion equation.[J].Numer Anal,2009,(47):2108-2131.
[39]Li S,Chen H,Wang H.A mixed-type Galerkin variational formulation and fast algorithms for variable-coefficient fractional diffusion equations[J].Math Method Appl Sci,2017,(40):5018-5034.
[40]Lin Y,Xu C.Finite difference spectral approximations for the time-fractional diffusion equation[J].Comput Phys,2007,(225):1533-1552.
[41]Liu F,Anh V,Turner I.Numerical solution of the space fractional Fokker-Planck equation[J].Comput Appl Math,2004,(166):209-219.
[42]Liu F,Anh V,Turner I.Numerical simulation for solute transport in fractal porous media[J].ANZIAM,2004,(45):C461-C473.
[43]刘青霞.空间分数阶对流一扩散方程的数值解及其应用[D].厦门:厦门大学,2008:30-35.
[44]Mao Z P,Chen S,Shen J.Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations[J].Appl Numer Math,2016,(106):165-181.
[45]Mandelbrot B B.The Fractal Geometry of Nature[M].New York:W H Freeman,1982:80-85.
[46]Meerschaert M M,Mortensen J,Wheatcraft S W.Fractional vector calculus for fractional advection-dispersion[J].Physica A,2006,(367):181-190.
[47]Meerschaert M M,Sikorskii A.Stochastic models for fractional calculus[J].Walter de Gruyter Gmb H and Co KG,2012,36(7):25-30.
[48]Meerschaert M M,Tadjeran C.Finite difference approximation for two-sided space-fractional partial differential equations[J].Appl Math,2006,(56):80-90.
[49]Meerschaert M M,Tadjeran C.Finite difference approximations for two-dimensional fractional dispersion equation[J].Comp Phy,2006,(211):249-261.
[50]Meerschaert M M,Tadjeran C.Finite difference approximations for fractional advection-dispersion flow equations[J].Comp Appl Math,2004,(172):65-77.
[51]Momani S.Analytic and approximate solutions of the space-and time-fractional telegraph equations[J].Appl Math Comput,2005,170(2):1126-1134.
[52]Nie N,Huang J,Wang W,Solving spatial-fractional partial differential diffusion equations by spectral method[J].Stat Comput Simul,2014,(84):1173-1189.
[53]ksendal B.Stochastic Differential Equations:An Introduction with Applications[M].Heidelberg:Springer,2010:105-110.
[54]Oldham K B,Spanier J.The Fractional Calculus[M].New York:Academic Press,1974:205-210.
[55]Pang H K,Sun H W.Multigrid method for fractional diffusion equations[J].Comput Phys,2012,(231):693-703.
[56]Podlubny I.Fractional Differential Equations[M].New York:Academic Press,1999:56-60.
[57]Pozrikidis C.The Fractional Laplacian[M].Massachusetts Amherst:CRC Press,2016:20-25.
[58]Rawashdeh E A.Numerical solution of fractional integro-differential equations by collocation method[J].Appl Math Comput,2006,(176):1-6.
[59]Samko S,Kilbas A,Marichev O.Fractional Integrals and Derivatives:Theory and Applications[M].London:Gordon and Breach,1993:102-104.
[60]Schey H M.Div Grad Curl and All That[M].New York:W W Norton,1992:70-76.
[61]Schumer R,Benson D A,Meerschaert M M.Eulerian derivation of the fractional advection-dispersion equation[J].Contaminant Hydrol,2001,(48):69-88.
[62]Sonin N Y.On differentiation with arbitrary index[J].Moscow Matem Sbornik,1869,6(1):1-38.
[63]Tadjeran C,Meerschaert M M,Scheffler H.A second-order accurate numerical approximation for the fractional diffusion equation[J].Comp Phy,2006,(213):205-213.
[64]Tsallis C,Lenzi E K.Anomalous diffusion:Nonlinear fractional Fokker-Plancker equation[J].Chem Phys,2002,(284):341-347.
[65]Wang F,Chen H,Wang H.Least-squares mixedgalerkin formulation for variable-coefficient fractional differential equations with mixed boundary conditions[J].Math Method Appl Sci,2017,(250):70-75.
[66]Wang F,Chen,Wang H.Finite element simulation and efficient algorithm for fractional cahn-hilliard equation[J].Comput Appl Math,2018,(152):560-570.
[67]Wang F,Chen H,Wang H.Numerical analysis of finite element simulation for fractional cahn-hilliard equation[J].Completed Math,2015,(78):65.
[68]Wang H,Basu T S.A fast finite difference method for two-dimensional space-fractional diffusion equations[J].SIAM J Sci Comput,2012,(34):A2444-A2458.
[69]Wang H,Du N.A fast finite difference method for three-dimensional time-dependent space-fractional diffusion equations and its efficientimplementation[J].Comput Phys,2013,(253):50-63.
[70]Wang H,Du N.A superfast-preconditioned iterative method for steady-state space-fractional diffusionequations[J].Comput Phys,2013,(240):49-57.
[71]Wang H,Du N.Fast solution methods for space-fractional diffusion equations[J].Comput Appl Math,2014,(255):376-383.
[72]Wang H,Wang K X.An O(Nlog2N)alternating-direction finite difference method for two-dimensional fractional diffusion equations[J].Comput Phys,2011,(230):7830-7839.
[73]Wang H,Wang K X,Sircar T.A direct O(N log2N)finite difference method for fractional diffusion equations[J].Comput Phys,2010,(229):8095-8104.
[74]Wang H,Yang D P.Wellposedness of variable-coefficient conservative fractional elliptic differential equations[J].SIAM J Numer Anal,2013,51(2):1088-1107.
[75]Wang H,Yang D P,Zhu S F.Inhomogeneous Dirichlet boundary-value problems of space-fractional diffusion equations and their finite element approximations[J].SIAM J Numer Anal,2014,(52):1292-1310.
[76]Wang H,Yang D P,Zhu S F.A Petrov-Galerkin finite element method for variable-coefficient fractional diffusion equations[J].Comput Methods Appl Mech Engrg,2015,(290):45-56.
[77]Wang K X,Wang H.A fast characteristic finite difference method for fractional advection-diffusionequations[J].Adv Water Resour,2011,(34):810-816.
[78]Wheatcraft S W,Meerschaert M M.Fractional conservation of mass[J].Adv Water Resour,2008,(31):1377-1381.
[79]徐强.分数阶渗流方程的数值研究[D].北京:中国工程物理研究院,2015:25-30.
[80]Yang D P,Wang H.Wellposedness and regularity of steady-state two-sided variable-coefficient conservative space-fractional diffusion equations[J].ArXiv:Math J,2016,(35):720-724.
[81]Yang Q,Turner I,Moroney T.A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction-diffusion equations[J].Appl Math Model,2014,(38):3755-3762.
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[84]Yuan Q,Chen H.An expanded mixed finite element simulation for two-sided time-dependent fractional diffusion problem[J].Adv Differ Equ,2018,(34):310-316.
[85]Zaslavsky G M,Stevens D,Weitzner H.Self-similar transport in incomplete chaos[J].Phys Rev E,1993,48(3):1683-1694.
[86]Zhang Y,Benson D A,Meerschaert M M.Space-fractional advection-dispersion equations with variable parameters:Diverse formulas,numerical solutions,and application to the macrodispersion experiment site data[J].Adv Water Resour,2007,43(5):325-327.
[87]Zayernouri M,Karniadakis G.Fractional spectral collocation method[J].SIAM J Sci Comput,2014,(36):A40-A62.
[88]Zeng F,Liu F,Li C.Crank-Nicolson ADI spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation[J].SIAM J Numer Anal,2014,(52):2599-2622.
[89]Zhang Y,Benson D A,Meerschaert M M.On using random walks to solve the space-fractional advection-dispersion equations[J].Stat Phys,2006,(123):89-110.
[90]Zhou Z J,Gong W.Finite element approximation of optimal control problems governed by time fractional diffusion equations[J].Math Appl,2016,(71):301-308.
[91]Zhou Z J,Zhang C Y.Time stepping discontinuous Galerkin approximation of optimal control problem governed by time fractional diffusion equation[J].Numer Algorithms,2018,79(2):437-455.
[92]Zhuang P,Liu F.Implicit difference approximation for the time fractional diffusion equation[J].Appl Math Comput,2006,22(3):87-99.
[2]Abel N H.Solution Dequelques Probl eˋMes aˋL'aide d'Integrales d eˊFinies[M].Oeuvres:Compl eˋtes,1881:16-18.
[3]Al-Khaled,Momani S.An approximate solution for a fractional diffusion-wave equation using the decomposition method[J].Appl Math Comput,2005,165(2):473-483.
[4]Basu T S,Wang H.A fast second-order finite difference method for space-fractional diffusion equations[J].Numer Anal Modeling,2012,(9):658-666.
[5]Berkowitz B,Emmanuel S,Scher H.Non-Fickian transport and multiple-rate mass transfer in porous media[J].Water Resour Res,2008,(44):W03402.
[6]Carreras B A,Lynch V E,Zaslavsky G M.Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence models[J].Phys Plasmas,2001,(8):5096-5103.
[7]Chen H,Wang H.Numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation[J].Comp Appl Math,2016,(296):480-498.
[8]Chen S,Liu F.ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersionequation[J].Appl Math Comput,2008,26(1):295-311.
[9]Daftardar-Gejji V,Jafari H.Solving a multi-order fractional differential equation using Adomian decomposition[J].Appl Math Comput,2007,(189):541-548.
[10]Deng W,Hesthaven J S.Discontinuous galerkin methods for fractional diffusion equations[J].Scientific Computing Group,2010,(19):120-125.
[11]Deng W,Hesthaven J S.Local discontinuous Galerkin methods for fractional diffusion equations[J].Model Numer Anal,2013,(47):1845-1864.
[12]Diethelm K,Freed A D.On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity[J].Computational Fluid Dynamics Reaction Engineering and Molecular Properties,1999,125(10):217-224.
[13]Ervin V J,Roop J P.Variational formulation for the stationary fractional advection dis-persion equation[J].Numer Methods Partial Differ Equ,2005,(22):558-576.
[14]Ervin V J,Roop J P.Variational solution of fractional advection dispersion equations on bounded domains in Rd[J].Methods Partial Differential Equations,2007,(23):256-281.
[15]Ervin V J,Heuer N,Roop J P.Regularity of the solution to 1-D fractional order diffusion equations[J].Comp App Math,2018,87(313):2273-2294.
[16]Fix G J,Roop J P.Least squares finite-element solution of a fractional order two-Point boundary value problem[J].Comput Math Appl,2004,(48):1017-1033.
[17]Gorenflo R,Fabritiis G D.Discrete random walk models for symmetric Lévy-Feller diffusion processes[J].Phys A,1999,269(1):79-89.
[18]Gorenflo R,Luchko,Mainardi F.Wright functions as scale-invariant solutions of the diffusion-wave equation[J].Comp Appl Math,2000,(118):175-191.
[19]Gorenflo R,Mainardi F.Random walk models for space fractional diffusion processes[J].Fract Calc ApplAnal,1998,1(2):167-191.
[20]Gorenflo R,Mainardi F.Approximation of Levy-Feller diffusion by random walk[J].Anal Appl,1999,18(2):231-246.
[21]Gorenflo R,Mainardi F,Moretti D.Discrete random walk models for space-time fractional diffusion[J].Chem Phys.,2002,(284):521-541.
[22]Gru¨nward A K.Uber begrenzte derivationen und deren anwendung[J].Z angew Math und Phys,1867,(12):441-480.
[23]郭柏灵,蒲学科,黄凤辉.分数阶偏微分方程及其数值解[M].北京:科学出版社,2011:80-85.
[24]He J.Variational iteration method for autonomous ordinary differential system[J].Appl Math Comput,2000,(114):115-123.
[25]Huang J F,Nie N M,Tang F.A second order finite difference-spectral method for space fractional diffusion equations[J].Sci Chin Math,2014,(57):1303-1317.
[26]Jia J H,Wang H.A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh[J].Comput Phys,2015,299(15):842-862.
[27]Jia L L,Chen H,Wang H.Mixed-type Galerkin variational principle and numerical simulation for a generalized nonlocal elastic model[J].Sci Comput,2017,(71):660-681.
[28]Jin B T,Lazarov R,Lu X.A simple finite element method for boundary value problems with a Riemann-Liouville derivative[J].Comput Appl Math,2016,(293):94-111.
[29]Jin B,Lazarov R,Pasciak J.Variational formulation of problems involving fractional order differential operators[J].Math Comp,2015,(84):2665-2700.
[30]Jin B T,Lazarov R,Zhou Z.A Petrov-Galerkin finite element method for fractional convection-diffusion equations[J].SIAM J Numer Anal,2016,(54):481-503.
[31]Jin B T,Zhou Z.A finite element method with singularity reconstruction for fractional boundary valueproblems[J].ESAIM Math Model Numer Anal,2015,(49):1261-1283.
[32]Klafter J,Sokolov I M.Anomalous diffusion spreads its wings[J].Phys World,2005,18(8):29-32.
[33]Lacroix S F.Traite du Cazcuz Diff eˊRentiez et du Cazcuz Int eˊgraz[M].Paris:Mme Ve Courcier Tome Troisi,1819:409-410.
[34]Laurent H.Sur lecalcul des d eˊriv eˊes aˋindicies quelconques[J].Nouv Annales de Mathematiques,1884,(3):240-252.
[35]Letnikov A V.Theory of differentiation with an arbitrary index(Russian)[J].Mat sb,1868,(3):1-66.
[36]Letnikov A V.An explanation of the concepts of the theory of differentiation of arbitrary index(Russian)[J].Moscow Matem Sbornik,1872,(6):413-445.
[37]Li S,Zhou Z J.Legendre pseudo-spectral method for optimal control problem governed by time fractional diffusion equation[J].Comput Math,2018,95(6-7):1308-1325.
[38]Li X,Xu C.A space-time spectral method for the time fractional diffusion equation.[J].Numer Anal,2009,(47):2108-2131.
[39]Li S,Chen H,Wang H.A mixed-type Galerkin variational formulation and fast algorithms for variable-coefficient fractional diffusion equations[J].Math Method Appl Sci,2017,(40):5018-5034.
[40]Lin Y,Xu C.Finite difference spectral approximations for the time-fractional diffusion equation[J].Comput Phys,2007,(225):1533-1552.
[41]Liu F,Anh V,Turner I.Numerical solution of the space fractional Fokker-Planck equation[J].Comput Appl Math,2004,(166):209-219.
[42]Liu F,Anh V,Turner I.Numerical simulation for solute transport in fractal porous media[J].ANZIAM,2004,(45):C461-C473.
[43]刘青霞.空间分数阶对流一扩散方程的数值解及其应用[D].厦门:厦门大学,2008:30-35.
[44]Mao Z P,Chen S,Shen J.Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations[J].Appl Numer Math,2016,(106):165-181.
[45]Mandelbrot B B.The Fractal Geometry of Nature[M].New York:W H Freeman,1982:80-85.
[46]Meerschaert M M,Mortensen J,Wheatcraft S W.Fractional vector calculus for fractional advection-dispersion[J].Physica A,2006,(367):181-190.
[47]Meerschaert M M,Sikorskii A.Stochastic models for fractional calculus[J].Walter de Gruyter Gmb H and Co KG,2012,36(7):25-30.
[48]Meerschaert M M,Tadjeran C.Finite difference approximation for two-sided space-fractional partial differential equations[J].Appl Math,2006,(56):80-90.
[49]Meerschaert M M,Tadjeran C.Finite difference approximations for two-dimensional fractional dispersion equation[J].Comp Phy,2006,(211):249-261.
[50]Meerschaert M M,Tadjeran C.Finite difference approximations for fractional advection-dispersion flow equations[J].Comp Appl Math,2004,(172):65-77.
[51]Momani S.Analytic and approximate solutions of the space-and time-fractional telegraph equations[J].Appl Math Comput,2005,170(2):1126-1134.
[52]Nie N,Huang J,Wang W,Solving spatial-fractional partial differential diffusion equations by spectral method[J].Stat Comput Simul,2014,(84):1173-1189.
[53]ksendal B.Stochastic Differential Equations:An Introduction with Applications[M].Heidelberg:Springer,2010:105-110.
[54]Oldham K B,Spanier J.The Fractional Calculus[M].New York:Academic Press,1974:205-210.
[55]Pang H K,Sun H W.Multigrid method for fractional diffusion equations[J].Comput Phys,2012,(231):693-703.
[56]Podlubny I.Fractional Differential Equations[M].New York:Academic Press,1999:56-60.
[57]Pozrikidis C.The Fractional Laplacian[M].Massachusetts Amherst:CRC Press,2016:20-25.
[58]Rawashdeh E A.Numerical solution of fractional integro-differential equations by collocation method[J].Appl Math Comput,2006,(176):1-6.
[59]Samko S,Kilbas A,Marichev O.Fractional Integrals and Derivatives:Theory and Applications[M].London:Gordon and Breach,1993:102-104.
[60]Schey H M.Div Grad Curl and All That[M].New York:W W Norton,1992:70-76.
[61]Schumer R,Benson D A,Meerschaert M M.Eulerian derivation of the fractional advection-dispersion equation[J].Contaminant Hydrol,2001,(48):69-88.
[62]Sonin N Y.On differentiation with arbitrary index[J].Moscow Matem Sbornik,1869,6(1):1-38.
[63]Tadjeran C,Meerschaert M M,Scheffler H.A second-order accurate numerical approximation for the fractional diffusion equation[J].Comp Phy,2006,(213):205-213.
[64]Tsallis C,Lenzi E K.Anomalous diffusion:Nonlinear fractional Fokker-Plancker equation[J].Chem Phys,2002,(284):341-347.
[65]Wang F,Chen H,Wang H.Least-squares mixedgalerkin formulation for variable-coefficient fractional differential equations with mixed boundary conditions[J].Math Method Appl Sci,2017,(250):70-75.
[66]Wang F,Chen,Wang H.Finite element simulation and efficient algorithm for fractional cahn-hilliard equation[J].Comput Appl Math,2018,(152):560-570.
[67]Wang F,Chen H,Wang H.Numerical analysis of finite element simulation for fractional cahn-hilliard equation[J].Completed Math,2015,(78):65.
[68]Wang H,Basu T S.A fast finite difference method for two-dimensional space-fractional diffusion equations[J].SIAM J Sci Comput,2012,(34):A2444-A2458.
[69]Wang H,Du N.A fast finite difference method for three-dimensional time-dependent space-fractional diffusion equations and its efficientimplementation[J].Comput Phys,2013,(253):50-63.
[70]Wang H,Du N.A superfast-preconditioned iterative method for steady-state space-fractional diffusionequations[J].Comput Phys,2013,(240):49-57.
[71]Wang H,Du N.Fast solution methods for space-fractional diffusion equations[J].Comput Appl Math,2014,(255):376-383.
[72]Wang H,Wang K X.An O(Nlog2N)alternating-direction finite difference method for two-dimensional fractional diffusion equations[J].Comput Phys,2011,(230):7830-7839.
[73]Wang H,Wang K X,Sircar T.A direct O(N log2N)finite difference method for fractional diffusion equations[J].Comput Phys,2010,(229):8095-8104.
[74]Wang H,Yang D P.Wellposedness of variable-coefficient conservative fractional elliptic differential equations[J].SIAM J Numer Anal,2013,51(2):1088-1107.
[75]Wang H,Yang D P,Zhu S F.Inhomogeneous Dirichlet boundary-value problems of space-fractional diffusion equations and their finite element approximations[J].SIAM J Numer Anal,2014,(52):1292-1310.
[76]Wang H,Yang D P,Zhu S F.A Petrov-Galerkin finite element method for variable-coefficient fractional diffusion equations[J].Comput Methods Appl Mech Engrg,2015,(290):45-56.
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