分数阶偏微分方程交替方向有限元误差分析
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摘要
分数阶偏微分方程是一类很重要的微分方程,它源于许多科学领域,例如带记忆的热传导、多孔粘弹性介质的压缩、原子反应、动力学、生命科学、药理学、病理学、牛顿流体学等.在这些实际问题中,很多数学家以及应用领域中的研究者正在尝试着用分数阶微分方程来建模.由于分数阶导数是拟微分算子,它具有保记忆性(非局部性),因此在对实际问题进行优美刻画的同时,也给数值计算带来了相当大的困难,特别是对于高维问题的数值计算.
交替方向有限元方法是一类计算高维偏微分方程的有效数值方法,可以把多维问题降维,分解成连续解几个一维问题.因此它既具备了交替方向存储量少、计算量低又有有限元高精度的特点.本文旨在用交替方向有限元方法研究二维的分数阶偏微分方程的数值解,设计了这类方程稳定的有效的数值格式,建立了所给格式的误差估计.本文的工作主要包括以下三个部分：
第一部分利用交替方向隐式有限差分法研究二维的分数阶发展方程的数值解.时间方向采用向后欧拉方法,空间方向使用二阶差商,积分项用一阶卷积求积公式逼近,得到全离散格式.然后用离散的能量方法证明了全离散格式的稳定性和收敛性.最后用数值结果和数值模拟验证了理论分析的收敛阶和所给格式的有效性.该方法能够有效地减少由全局时间依赖性所引起的对存储量的要求,从而可以计算长时间的解.
第二部分讨论了利用交替方向隐式Galerkin有限元方法研究二维的分数阶发展方程的数值解,空间方向采用有限元方法,时间方向使用向后欧拉格式、Grank-Nicolson格式,积分项运用相应的卷积求积公式逼近,得到全离散格式.严格证明了全离散格式的稳定性和误差估计,并用数值例子检验理论分析的正确性.该方法在空间方向上能达到任意阶精度,也能够有效地减少由全局时间依赖性所引起的对存储量的要求.
第三部分利用交替方向隐式Galerkin有限元方法研究二维的时间分数阶扩散-波动方程的数值解.时间方向使用Grank-Nicolson格式,空间方向采用Galerkin有限元,得到全离散格式.证明了全离散格式的稳定性和收敛性,数值例子验证其结论的正确性.
The fractional partial differential equation is a class of very important differential equations, which originated from many science fields, such as, heat conduction with memory, compression of porous viscoelastic media, atomic reaction, dynamics, life sciences, pharmacology, pathology, Newton fluidics, etc. So many mathematicians and researchers in the application fields are trying to use fractional differential equations to model. Fractional derivative is pseudo-differential operator and has the character of memory (nonlocal). Although it can describe practical problems beautifully, it can also bring con-siderable difficulties in numerical computation, especially for high-dimensional problems.
Alternating direction implicit (ADI) finite element method is an effec-tive numerical methods to calculate high-dimensional partial differential equa-tions, which reduce a multidimensional problem to sets of independent one-dimensional problems. So ADI finite element method has less storage, low computation and the feature of high accuracy. The main objective of this thesis is to use ADI method to investigate the numerical solutions of the two-dimensional time fractional partial differential equations. Stable and efficient numerical schemes are proposed for these equations and the error estimates of our proposed numerical schemes are also established. The main work of this thesis contains the following three parts:
In the first part, we use ADI finite difference method to solve the nu-merical solutions of the two dimensional fractional evolution equation. We first use the second-order difference quotient for the spatial discretization and the backward Euler for the time stepping combined with order one convolu-tion quadrature approximating the integral term and obtain the full discrete scheme. Then, applying the method of discrete energy method, we prove that the full discrete scheme is unconditionally stable and convergent. Finally, some numerical results and numerical simulations are presented to confirm the rates of convergence and the robustness of the numerical schemes. This method can effectively reduce the storage capacity which is caused by overall time-dependent and can calculate the long time solution.
In the second part, ADI Galerkin schemes are formulated and analyzed for the two-dimensional time fractional evolution equation. We first use the Galerkin finite element for the spatial discretization and the backward Eu-ler, Grank-Nicolson for the time stepping combined with order one and two convolution quadrature and obtain the full discrete scheme. Then we rigorously prove the stability and error estimates of the full discrete scheme. Numerical examples is examined to the accuracy of the theoretical analysis. The method achieve arbitrary order accuracy in the spatial direction and effectively reduce the storage volume requirements caused by the overall time dependence.
In the third part, we apply ADI finite element method to investigate the numerical solutions of the two-dimensional fractional diffusion-wave equation. We use Galerkin finite element method in space and Crank-Nicolson method in time and obtain the full discrete scheme. We prove the stability and error estimates of the scheme. Numerical examples verify the correctness of the conclusions.
交替方向有限元方法是一类计算高维偏微分方程的有效数值方法,可以把多维问题降维,分解成连续解几个一维问题.因此它既具备了交替方向存储量少、计算量低又有有限元高精度的特点.本文旨在用交替方向有限元方法研究二维的分数阶偏微分方程的数值解,设计了这类方程稳定的有效的数值格式,建立了所给格式的误差估计.本文的工作主要包括以下三个部分：
第一部分利用交替方向隐式有限差分法研究二维的分数阶发展方程的数值解.时间方向采用向后欧拉方法,空间方向使用二阶差商,积分项用一阶卷积求积公式逼近,得到全离散格式.然后用离散的能量方法证明了全离散格式的稳定性和收敛性.最后用数值结果和数值模拟验证了理论分析的收敛阶和所给格式的有效性.该方法能够有效地减少由全局时间依赖性所引起的对存储量的要求,从而可以计算长时间的解.
第二部分讨论了利用交替方向隐式Galerkin有限元方法研究二维的分数阶发展方程的数值解,空间方向采用有限元方法,时间方向使用向后欧拉格式、Grank-Nicolson格式,积分项运用相应的卷积求积公式逼近,得到全离散格式.严格证明了全离散格式的稳定性和误差估计,并用数值例子检验理论分析的正确性.该方法在空间方向上能达到任意阶精度,也能够有效地减少由全局时间依赖性所引起的对存储量的要求.
第三部分利用交替方向隐式Galerkin有限元方法研究二维的时间分数阶扩散-波动方程的数值解.时间方向使用Grank-Nicolson格式,空间方向采用Galerkin有限元,得到全离散格式.证明了全离散格式的稳定性和收敛性,数值例子验证其结论的正确性.
The fractional partial differential equation is a class of very important differential equations, which originated from many science fields, such as, heat conduction with memory, compression of porous viscoelastic media, atomic reaction, dynamics, life sciences, pharmacology, pathology, Newton fluidics, etc. So many mathematicians and researchers in the application fields are trying to use fractional differential equations to model. Fractional derivative is pseudo-differential operator and has the character of memory (nonlocal). Although it can describe practical problems beautifully, it can also bring con-siderable difficulties in numerical computation, especially for high-dimensional problems.
Alternating direction implicit (ADI) finite element method is an effec-tive numerical methods to calculate high-dimensional partial differential equa-tions, which reduce a multidimensional problem to sets of independent one-dimensional problems. So ADI finite element method has less storage, low computation and the feature of high accuracy. The main objective of this thesis is to use ADI method to investigate the numerical solutions of the two-dimensional time fractional partial differential equations. Stable and efficient numerical schemes are proposed for these equations and the error estimates of our proposed numerical schemes are also established. The main work of this thesis contains the following three parts:
In the first part, we use ADI finite difference method to solve the nu-merical solutions of the two dimensional fractional evolution equation. We first use the second-order difference quotient for the spatial discretization and the backward Euler for the time stepping combined with order one convolu-tion quadrature approximating the integral term and obtain the full discrete scheme. Then, applying the method of discrete energy method, we prove that the full discrete scheme is unconditionally stable and convergent. Finally, some numerical results and numerical simulations are presented to confirm the rates of convergence and the robustness of the numerical schemes. This method can effectively reduce the storage capacity which is caused by overall time-dependent and can calculate the long time solution.
In the second part, ADI Galerkin schemes are formulated and analyzed for the two-dimensional time fractional evolution equation. We first use the Galerkin finite element for the spatial discretization and the backward Eu-ler, Grank-Nicolson for the time stepping combined with order one and two convolution quadrature and obtain the full discrete scheme. Then we rigorously prove the stability and error estimates of the full discrete scheme. Numerical examples is examined to the accuracy of the theoretical analysis. The method achieve arbitrary order accuracy in the spatial direction and effectively reduce the storage volume requirements caused by the overall time dependence.
In the third part, we apply ADI finite element method to investigate the numerical solutions of the two-dimensional fractional diffusion-wave equation. We use Galerkin finite element method in space and Crank-Nicolson method in time and obtain the full discrete scheme. We prove the stability and error estimates of the scheme. Numerical examples verify the correctness of the conclusions.
引文
[1]N.Y. Sonin, On differentiation with arbitrary index, Moscow Matem. Sbornik, 6(1869) 1-38.
[2]H. Laurent, Sur le calcul des derivees a indicies quelconques, Nouv. Annales de Mathematiques,3(1884) 240-252.
[3]A.K. Grunwald, Ueber'begrenzte'Derivationen und deren Anwendung, Z. angew. Math, und phys.,12(1867) 441-480.
[4]A. V. Letnikov, Theory of differentiation with arbitrary index, Moscow Matem. Sbornik(Russian),3(1868) 1-66.
[5]M. Caputo, Linear models of dissipation whose Q is almost frequency inde-pendent, Geophys. J. Royal astr. Soc.,13(1967) 529-539.
[6]P.G. Nutting, A new general law of deformation, Journal of the Franklin Institute,191(1921) 679-685.
[7]A. Gemant, A method of analyzing experimental Obtained from elasto viscous Bodies, J. Physics 7(1936) 311-317.
[8]A. Gemant, On fractional differentials, The Philosophical Magazine,25(1938) 540-549.
[9]G. Scott-Blair, J.E. Caffyn, An application of the theory of quasi-properties to the treatment of anomalous strain-stress relations, The Philosophical Mag-azine,40(1949) 80-94.
[10]Y. Rabotnov, Equilibrium of an elastic medium with after effect, Prikl. Matem. I Mekh,12(1948) 81-91.
[11]Y. Rabotnov, Elements of Hereditary solid Mechanics,1980.
[12]M. Gaputo, F. Mainardi, A new dissipation model based on memoral mecha-nism, Pure and Applied Geophysics 91(1971) 134-147.
[13]B.B. Mandelbrot, The Fractal Geometry of Nature,1983.
[14]R. Schumer, D. Benson, M. Meerschaert, et al, Eulerian dedrivation of the fractional advection-dispersion equation, J. Contam. Hydrol,48(2001) 69-88.
[15]B. Elli, CTRW pathways to the fractional diffusion equation, Chemical Physics,284(2002) 13-27.
[16]W. Wyss, The fractional diffusion equation, J. Math. Phys.27(1986) 2782-2785.
[17]W.R. Schneider, W. Wyss, Fractional diffusion and wave equations, J. Math. Phys.,30(1989) 34-144.
[18]R. Gorenflo, Y. Luchko, F. Mainardi, Wright function as scale-invariant solu-tions of the diffusion-wave equation, J. Comp. Appl. Math.,118(2000) 175-191.
[19]R. Gorenflo, F. Mainardi, D. Moretti, P. Paradisi, Time fractional diffusion: a discrete random walk approach, Nonlinear Dynam.,29(2002) 129-143.
[20]R. Gorenflo, F. Mainardi, and A. Vivoli, Continuous-time random walk and parametric subordination in fractional diffusion, Chaos Solitons Fractals, 34(2007) 87-103.
[21]F. Liu, V. Anh, I. Turner, P. Zhuang, Time fractional advection dispersion equation, J. Appl. Math. Comput.,13(2003) 233-245.
[22]F. Liu, S. Shen, V. Anh, I. Turner, Analysis of a discrete non-Markovian ran-dom walk approximation for the time fractional diffusion equation, ANZIAM J.,46(E)(2005) 488-504.
[23]F. Huang, F. Liu, The time fractional diffusion and advection-dispersion equa-tion, ANZIAM J.,46(2005) 1-14.
[24]O. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain,J. Nonlinear Dynamics,29(2002) 145-155.
[25]J. Chen, F. Liu, V. Anh, et al, Analytical solution for the time-fractional telegraph equation by the method of separating variables, J. Math. Anal. and Appl.,338(2008) 1364-1377.
[26]C. Lubich, Discretized fractional calculus, SIAM. J. Math. Anal.,17(1986) 704-719.
[27]K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Electr. Trans. Numer. Anal.,5(1997) 1-6.
[28]C.P. Li, A. Chen, J.J. Ye, Numerical approaches to fractional calculus and fractional ordinary differential equation, J. Comput. Phys.,230(2011) 3352-3368.
[29]H. Brunner, A survey of recent advances in the numerical treatment of Volterra integral and integral-differential equations, J. Comput. Appl. Math.,8(1982) 76-102.
[30]H. Brunner, A. Pedas & G. Vainikko, Piecewise polynomial collocation meth-ods for linear integro-differential equations with weakly singular kernels, SIAM J. Numer. Anal.,39(2001) 957-982.
[31]H. Brunner,Q.-Y. Hu, Q. Lin, Geometric meshes in collocation methods for Volterra integral equations with proportional delays, IMA J. Numer. Anal., 21(2001) 783-798.
[32]H. Brunner, D. Schotzau, hp-Discontinuous Galerkin time-stepping for Volterra integro-differential equations, SIAM J. Numer. Anal.,44(2006) 224-245.
[33]H. Brunner, The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays, Comm. Pure Appl. Anal., 5(2006) 261-276.
[34]H. Brunner, Q.-Y. Hu, Optimal superconvergence results for delay integro-differential equations of pantograph type, SIAM J. Numer. Anal.,45(2007) 986-1004.
[35]T. Tang, Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, J. Numer. Math.,61(1992) 373-382.
[36]T. Tang, A finite difference scheme for a partial integro-differential equation with a weakly singular kernel, Appl. Numer. Math.,2(1993) 309-319.
[37]T. Tang, X. Xu, J. Chen, On spectral methods for Volterra type integral equations and the convergence analysis, J. Comput. Math.,26(2008) 825-837.
[38]Z.Q. Xie, X.J. Li, T. Tang, Convergence analysis of spectral Galerkin methods for Volterra type integral equations, J. Sci. Comput.,52(2012) 414-434,
[39]D. Xu, Uniform L1 behaviour in a second-order difference-type method for a linear Volterra equation with comletely monotonic kernel Ⅰ:stability, IMA J. Numer. Anal.,31(3)(2011) 1154-1180.
[40]D. Xu, Stability of the difference type methods for linear Volterra equations in Hilbert spaces, Numer. Math.,109(2008) 571-595.
[41]D. Xu, Uniform Ll behavior for time discretization of a Volterra equation with completely monotonic kernel Ⅱ:convergence, SIAM J. Numer. Anal., 46(2008) 231-259.
[42]D. Xu, Uniform L1 error bounds for the semidiscrete solution of a Volterra equation with completely monotonic convolution kernel, Comput. Math. Appl.,43(2002) 1303-1318.
[43]D. Xu, The long-time global behaviour of time discretization for fractional order Volterra equation, CACOLO,35(1998) 93-116.
[44]D. Xu, The global behavior of time discretization for an abstract Volterra equation in Hilbert space, CALCOLO,34(1997) 71-104.
[45]D. Xu, Non-smooth initial data error estimates with the weight norms for the linear finite element method of parabolic partial differential equations, Appl. Math. Comput.,54 (1993) 1-24.
[46]D. Xu, On the discretization in time for a parabolic integro-differential equa-tion with a weakly singular kernel. Ⅰ:Smooth initial data, Appl. Math. Com-put.,58(1993)1-27
[47]D. Xu, On the discretization in time for a parabolic integro-differential equa-tion with a weakly singular kernel. Ⅱ:Non-smooth initial data, Appl. Math. Comput.,58(1993) 29-60.
[48]D. Xu, Finite element methods for the nonlinear integro-differential equations, Appl. Math. Comput.,58(1993) 241-273.
[49]W. Li, D. Xu, Finite central difference/finite element approximations for parabolic integro-differential equations, Computing,90(2010) 89-111.
[50]L. Li, D. Xu, Alternating direction implicit-Euler method for the two-dimensional fractional evolution equation, J. Comput. Phys.,236(2013) 157-168.
[51]T.A.M. Langlands, B.I. Henry, The accuracy and stability of an implicit solu-tion method for the fractional diffusion equation, J. Phys.,205(2005) 719-736.
[52]M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. APPl. Math.,172(2004) 65-77.
[53]Z. Sun, X. Wu, A fully discrete difference scheme for a diffusion-wave systym, Appl. Numer. Math.,2(2006) 193-209.
[54]R. Du, W. Cao, Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Model.,34(2010) 2998-3007.
[55]G. Gao, Z. Sun, A copmpact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys.,230(2011) 586-595.
[56]X. Zhao, Z. Sun, A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys.,230(2011) 6061-6074.
[57]Y. Zhang, Z. Sun, Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys.,230(2011) 8713-8728.
[58]Y. Zhang, Z. Sun, and X. Zhao, Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation, SIAM J. Numer. Anal.,50(2012) 1535-1555.
[59]F. Liu, V. Ahn, and I. Turner, Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math.,166 (2004) 209-219.
[60]P. Zhuang, F. Liu, Implicit difference appoximation for the time fractional diffusion equation, J. Appl. Math. Comput.,25(2006) 89-89.
[61]P. Zhuang, F. Liu, Implicit difference appoximation for the two-dimensional space-time fractional diffusion equation, J. Appl. Math. Comput.,25(2007) 269-282.
[62]P. Zhuang, F. Liu, V. Anh, et al, New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM J. Numer. Anal., (46)20081079-1095.
[63]S. Chen, F. Liu, ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation, J. Appl. Math. Comput.,26(2008) 295-311.
[64]P. Zhuang, F. Liu, V. Anh, and I. Turner, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal.,47(2009) 1760-1781.
[65]C. Chen, F. Liu, A numerical approximation method for solving a three-dimensional space Galilei invariant fractional advection-diffusion equation, J. Appl. Math. Comput.,30(2009) 219-236.
[66]Y. Gu, P. Zhuang, and F. Liu, An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation, CMES Comput. Model. Eng. Sci.,56(2010) 303-333.
[67]C. Chen, F. Liu, I. Turner, and V. Anh, Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation, Numer. Algor.,54(2010) 1-21.
[68]S. Shen, F. Liu V. Anh, Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation, Numer. Algor.,56(2011) 383-403
[69]S. Shen, F. Liu, J. Chen, I. Turner, V. Anh, Numerical techniques for the vari-able order time fractional diffusion equation, Appl. Math. Comput.,218(2012) 10861-10870.
[70]Y. Lin, C. Xu, Finite difference/spectral approxiations for the time-fractional diffusion equation, J. Comput. Phys.,225(2007) 1533-1552.
[71]X. Li, C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal.,47(2009) 2108-2131.
[72]X. Li, C. Xu, Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys.,8(2010) 1016-1051.
[73]Y. Lin, X. Li, C. Xu, Finite difference/spectral approximations for the frac-tional cable equation, Math. Comp.,80(2011) 1369-1396.
[74]W. Deng, Numerical algorithm for the time fractional Fokker-Planck equation, J. Comput. Phys.,227(2007) 1510-1522.
[75]W. Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal.,47(2008) 204-226.
[76]N. Zhang, W. Deng, Y. Wu, Finite difference/element method for a two-dimensional modified fractional diffusion equation, Adv. Appl. Math. Mech., 4(2012) 496-518.
[77]X. Ji, H. Tang, High-order accurate Runge-Kutta (Local) discontinuous Galerkin methods for one-and two-dimensional fractional diffusion equations, Numer. Math. Theor. Meth. Appl.,5(2012) 333-358.
[78]J.M. Sanz-serna, A numerical method for a partial integro-differential equa-tion, SIAM J. Numer. Anal.,25(1988) 319-327.
[79]J.C. Lopez-Marcos, A difference scheme for a nonlinear partial integro-differential equation, SIAM J. Numer. Anal.,27(1990) 20-31.
[80]W. Mclean, V. Thomee, Numerical solution of an evolution equation with a positive type memory term, J. Austral. Math. Soc. Ser.,35(1993) 23-70.
[81]W. Mclean, V. Thomee, L.B. Wahlbin, Discretization with variable time steps of an evolution equation with a positive-type memory term, J. Comput. Appl. Math.,69(1996) 49-69.
[82]C. Chen, V. Thomee, L. B. Wahlbin, Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel, Math. Comput.,58(1992) 587-602.
[83]W. Mclean and K. Mustapha, A second-order accurate numerical method for a fractional wave equation, Numer. Math.,105(2007) 481-510.
[84]C.H. Kim, U J. Choi, Spectral collocation methods for a partial integro-differential equations with a weakly singular kernel, J. Austral. Math. Soc. ser. B,39(1998) 408-430.
[85]K. Adolfsson, M. Enelund. S. Larsson, Adaptive discretization of an integro-differential equation with a weakly singular convolution kernel, Comput. Methods Appl. Mech. Engrg.,192(2003) 5285-5304.
[86]M. Rrenardy, W. Hrusa, J. Nohel, Mathematical problems in viscoelasticity, Pitman Monogr. Surveys Pure Appl. Math.35, Wiley, New York,1987.
[87]U. Jin Choi, R. C. MacCamy, Fractional order Volterra equations, in Volterra integrodifferential equations in Banach spaces and applications in Pitman Re-search Notes in mathematics, Volume 190 (Longman, Harlow, UK, and Wiley, New York,1989) 231-245.
[88]R. C. MacCamy, An integro-differential equation with application to heat flow, Quart. Appl. Math.,35(1977) 1-19.
[89]R. C. MacCamy, A model for one-dimensional nonlinear viscoelasticity, Quart. Appl. Math.,35(1977) 21-33.
[90]R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion:A fractional dynamics approach, Phys. Rep.,339(2000) 1-77.
[91]M. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys.,228(2009) 7792-7804.
[92]M. M. Meerschaert, C. Tadjeran. Finite difference approximations for frac-tional advection-dispersion flow equations, J. Comput. Appl. Math.,172(2004) 65-77.
[93]M. M. Meerschaert, C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56(2006) 80-90.
[94]Z.Q. Deng, V.P. Singh, F. Asce, L. Bengtsson, Numerical solution of fractional advection-dispersion equations, J. Hydraul. Eng.,130(2004) 422-431.
[95]J.T. Ma, J.Q. Liu, Z.Q. Zhou, Convergence analysis of moving finite element methods for space fractional differential equations, J. Comput. Appl. Math., 255(2014) 661-670.
[96]W.R. Schneider, W. Wyss, Fractional diffusion and wave equations, J. Math. Phys.30(1989) 134-144.
[97]J.E. Dendy, An analysis of some Galerkin schemes for the solution of nonlinear time-dependent problems, SIAM J. Numer. Anal.,12(1975) 541-565.
[98]J.E. Dendy, G. Fairweather, Alternating-direction Galerkin methods for parabolic and hyperbolic problems on rectangular polygons, SIAM J. Numer. Anal.,12(1975) 144-163.
[99]JR.J. Douglas, T. Dupont, Alternating direction Galerkin methods on rect-angles, Numerical Solution of Partial Differential Equations II (B. Hubbard ed.), New York:Academic Press, (1971) 133-214.
[100]Z. Zhang, D. Deng, A new alternating-direction finite element method for hyperbolic equation, Numer. Methods Partial Differ. Equ.,23(2007) 1530-1559.
[101]I. Podlubny, Fractional Differential Equations: An introduction to fractional derivatives, fractional differential equations, to The methods of their solution and some of their applications, Academic Press, San Diego,1999.
[102]C.P. Li, W.H. Deng, Remark on fractional derivatives, Appl. Math. Comput., 187(2007) 777-784.
[103]M. Moshrefi-Torbati, J.K. Hammond, Physical and geometrical interpretation of fractional operators, J. Franklin. Inst.,335B(1998) 1077-1086.
[104]R.R. Nigmatullin, Fractional integral and its physical interpretation, J. The-oret. Math. Phys.,90(1992) 242-251.
[105]S. Larsson, V. Thomee, LB Wahlbin, Numerical Solution of Parabolic Integro-differential equations by the discontinuous Galerkin methods, Math. Comput., 67(1998) 45-71.
[106]Y.Q. Huang, Time discretization scheme for an integro-differential equation of parabolic type, J. Comput. Math.,12(1994) 259-264.
[107]J. Bouchaud, A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep.,195(1990) 127-293.
[108]D.W. Peaceman, JR, H.H. Rachford, The numerical solution of elliptic and parabolic differential equations, J. Soc. Ind. Appl. Math.,3(1955) 28-41.
[109]H. Cheng, P. Lin, Q. Sheng, R.C.E. Tan, Solving degenerate reaction-diffusion equations via variable step Peaceman-Rachford splitting, SIAM J. Sci. Com-put.,25(2003) 1273-1292.
[110]S. Karaa, A high-order compact ADI method for parabolic problems with variable coefficients, Numer. Methods Partial Differ. Equ.,22(2006) 983-993.
[111]S. Karaa, A high-order ADI method for parabolic problems with variable coefficients, Int. J. Comput. Math.,86(2009) 109-120.
[112]J.E. Dendy, An analysis of some Galerkin schemes for the solution of nonlinear time dependent problems, SIAM J. Numer. Anal.,12(1975) 541-565.
[113]R.I. Fernandes, G. Fairweather, An alternating direction Galerkin method for a class of second-order hyperbolic equations in two space variables, SIAM J. Numer. Anal.,28(1991) 1265-1281.
[2]H. Laurent, Sur le calcul des derivees a indicies quelconques, Nouv. Annales de Mathematiques,3(1884) 240-252.
[3]A.K. Grunwald, Ueber'begrenzte'Derivationen und deren Anwendung, Z. angew. Math, und phys.,12(1867) 441-480.
[4]A. V. Letnikov, Theory of differentiation with arbitrary index, Moscow Matem. Sbornik(Russian),3(1868) 1-66.
[5]M. Caputo, Linear models of dissipation whose Q is almost frequency inde-pendent, Geophys. J. Royal astr. Soc.,13(1967) 529-539.
[6]P.G. Nutting, A new general law of deformation, Journal of the Franklin Institute,191(1921) 679-685.
[7]A. Gemant, A method of analyzing experimental Obtained from elasto viscous Bodies, J. Physics 7(1936) 311-317.
[8]A. Gemant, On fractional differentials, The Philosophical Magazine,25(1938) 540-549.
[9]G. Scott-Blair, J.E. Caffyn, An application of the theory of quasi-properties to the treatment of anomalous strain-stress relations, The Philosophical Mag-azine,40(1949) 80-94.
[10]Y. Rabotnov, Equilibrium of an elastic medium with after effect, Prikl. Matem. I Mekh,12(1948) 81-91.
[11]Y. Rabotnov, Elements of Hereditary solid Mechanics,1980.
[12]M. Gaputo, F. Mainardi, A new dissipation model based on memoral mecha-nism, Pure and Applied Geophysics 91(1971) 134-147.
[13]B.B. Mandelbrot, The Fractal Geometry of Nature,1983.
[14]R. Schumer, D. Benson, M. Meerschaert, et al, Eulerian dedrivation of the fractional advection-dispersion equation, J. Contam. Hydrol,48(2001) 69-88.
[15]B. Elli, CTRW pathways to the fractional diffusion equation, Chemical Physics,284(2002) 13-27.
[16]W. Wyss, The fractional diffusion equation, J. Math. Phys.27(1986) 2782-2785.
[17]W.R. Schneider, W. Wyss, Fractional diffusion and wave equations, J. Math. Phys.,30(1989) 34-144.
[18]R. Gorenflo, Y. Luchko, F. Mainardi, Wright function as scale-invariant solu-tions of the diffusion-wave equation, J. Comp. Appl. Math.,118(2000) 175-191.
[19]R. Gorenflo, F. Mainardi, D. Moretti, P. Paradisi, Time fractional diffusion: a discrete random walk approach, Nonlinear Dynam.,29(2002) 129-143.
[20]R. Gorenflo, F. Mainardi, and A. Vivoli, Continuous-time random walk and parametric subordination in fractional diffusion, Chaos Solitons Fractals, 34(2007) 87-103.
[21]F. Liu, V. Anh, I. Turner, P. Zhuang, Time fractional advection dispersion equation, J. Appl. Math. Comput.,13(2003) 233-245.
[22]F. Liu, S. Shen, V. Anh, I. Turner, Analysis of a discrete non-Markovian ran-dom walk approximation for the time fractional diffusion equation, ANZIAM J.,46(E)(2005) 488-504.
[23]F. Huang, F. Liu, The time fractional diffusion and advection-dispersion equa-tion, ANZIAM J.,46(2005) 1-14.
[24]O. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain,J. Nonlinear Dynamics,29(2002) 145-155.
[25]J. Chen, F. Liu, V. Anh, et al, Analytical solution for the time-fractional telegraph equation by the method of separating variables, J. Math. Anal. and Appl.,338(2008) 1364-1377.
[26]C. Lubich, Discretized fractional calculus, SIAM. J. Math. Anal.,17(1986) 704-719.
[27]K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Electr. Trans. Numer. Anal.,5(1997) 1-6.
[28]C.P. Li, A. Chen, J.J. Ye, Numerical approaches to fractional calculus and fractional ordinary differential equation, J. Comput. Phys.,230(2011) 3352-3368.
[29]H. Brunner, A survey of recent advances in the numerical treatment of Volterra integral and integral-differential equations, J. Comput. Appl. Math.,8(1982) 76-102.
[30]H. Brunner, A. Pedas & G. Vainikko, Piecewise polynomial collocation meth-ods for linear integro-differential equations with weakly singular kernels, SIAM J. Numer. Anal.,39(2001) 957-982.
[31]H. Brunner,Q.-Y. Hu, Q. Lin, Geometric meshes in collocation methods for Volterra integral equations with proportional delays, IMA J. Numer. Anal., 21(2001) 783-798.
[32]H. Brunner, D. Schotzau, hp-Discontinuous Galerkin time-stepping for Volterra integro-differential equations, SIAM J. Numer. Anal.,44(2006) 224-245.
[33]H. Brunner, The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays, Comm. Pure Appl. Anal., 5(2006) 261-276.
[34]H. Brunner, Q.-Y. Hu, Optimal superconvergence results for delay integro-differential equations of pantograph type, SIAM J. Numer. Anal.,45(2007) 986-1004.
[35]T. Tang, Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, J. Numer. Math.,61(1992) 373-382.
[36]T. Tang, A finite difference scheme for a partial integro-differential equation with a weakly singular kernel, Appl. Numer. Math.,2(1993) 309-319.
[37]T. Tang, X. Xu, J. Chen, On spectral methods for Volterra type integral equations and the convergence analysis, J. Comput. Math.,26(2008) 825-837.
[38]Z.Q. Xie, X.J. Li, T. Tang, Convergence analysis of spectral Galerkin methods for Volterra type integral equations, J. Sci. Comput.,52(2012) 414-434,
[39]D. Xu, Uniform L1 behaviour in a second-order difference-type method for a linear Volterra equation with comletely monotonic kernel Ⅰ:stability, IMA J. Numer. Anal.,31(3)(2011) 1154-1180.
[40]D. Xu, Stability of the difference type methods for linear Volterra equations in Hilbert spaces, Numer. Math.,109(2008) 571-595.
[41]D. Xu, Uniform Ll behavior for time discretization of a Volterra equation with completely monotonic kernel Ⅱ:convergence, SIAM J. Numer. Anal., 46(2008) 231-259.
[42]D. Xu, Uniform L1 error bounds for the semidiscrete solution of a Volterra equation with completely monotonic convolution kernel, Comput. Math. Appl.,43(2002) 1303-1318.
[43]D. Xu, The long-time global behaviour of time discretization for fractional order Volterra equation, CACOLO,35(1998) 93-116.
[44]D. Xu, The global behavior of time discretization for an abstract Volterra equation in Hilbert space, CALCOLO,34(1997) 71-104.
[45]D. Xu, Non-smooth initial data error estimates with the weight norms for the linear finite element method of parabolic partial differential equations, Appl. Math. Comput.,54 (1993) 1-24.
[46]D. Xu, On the discretization in time for a parabolic integro-differential equa-tion with a weakly singular kernel. Ⅰ:Smooth initial data, Appl. Math. Com-put.,58(1993)1-27
[47]D. Xu, On the discretization in time for a parabolic integro-differential equa-tion with a weakly singular kernel. Ⅱ:Non-smooth initial data, Appl. Math. Comput.,58(1993) 29-60.
[48]D. Xu, Finite element methods for the nonlinear integro-differential equations, Appl. Math. Comput.,58(1993) 241-273.
[49]W. Li, D. Xu, Finite central difference/finite element approximations for parabolic integro-differential equations, Computing,90(2010) 89-111.
[50]L. Li, D. Xu, Alternating direction implicit-Euler method for the two-dimensional fractional evolution equation, J. Comput. Phys.,236(2013) 157-168.
[51]T.A.M. Langlands, B.I. Henry, The accuracy and stability of an implicit solu-tion method for the fractional diffusion equation, J. Phys.,205(2005) 719-736.
[52]M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. APPl. Math.,172(2004) 65-77.
[53]Z. Sun, X. Wu, A fully discrete difference scheme for a diffusion-wave systym, Appl. Numer. Math.,2(2006) 193-209.
[54]R. Du, W. Cao, Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Model.,34(2010) 2998-3007.
[55]G. Gao, Z. Sun, A copmpact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys.,230(2011) 586-595.
[56]X. Zhao, Z. Sun, A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys.,230(2011) 6061-6074.
[57]Y. Zhang, Z. Sun, Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys.,230(2011) 8713-8728.
[58]Y. Zhang, Z. Sun, and X. Zhao, Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation, SIAM J. Numer. Anal.,50(2012) 1535-1555.
[59]F. Liu, V. Ahn, and I. Turner, Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math.,166 (2004) 209-219.
[60]P. Zhuang, F. Liu, Implicit difference appoximation for the time fractional diffusion equation, J. Appl. Math. Comput.,25(2006) 89-89.
[61]P. Zhuang, F. Liu, Implicit difference appoximation for the two-dimensional space-time fractional diffusion equation, J. Appl. Math. Comput.,25(2007) 269-282.
[62]P. Zhuang, F. Liu, V. Anh, et al, New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM J. Numer. Anal., (46)20081079-1095.
[63]S. Chen, F. Liu, ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation, J. Appl. Math. Comput.,26(2008) 295-311.
[64]P. Zhuang, F. Liu, V. Anh, and I. Turner, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal.,47(2009) 1760-1781.
[65]C. Chen, F. Liu, A numerical approximation method for solving a three-dimensional space Galilei invariant fractional advection-diffusion equation, J. Appl. Math. Comput.,30(2009) 219-236.
[66]Y. Gu, P. Zhuang, and F. Liu, An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation, CMES Comput. Model. Eng. Sci.,56(2010) 303-333.
[67]C. Chen, F. Liu, I. Turner, and V. Anh, Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation, Numer. Algor.,54(2010) 1-21.
[68]S. Shen, F. Liu V. Anh, Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation, Numer. Algor.,56(2011) 383-403
[69]S. Shen, F. Liu, J. Chen, I. Turner, V. Anh, Numerical techniques for the vari-able order time fractional diffusion equation, Appl. Math. Comput.,218(2012) 10861-10870.
[70]Y. Lin, C. Xu, Finite difference/spectral approxiations for the time-fractional diffusion equation, J. Comput. Phys.,225(2007) 1533-1552.
[71]X. Li, C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal.,47(2009) 2108-2131.
[72]X. Li, C. Xu, Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys.,8(2010) 1016-1051.
[73]Y. Lin, X. Li, C. Xu, Finite difference/spectral approximations for the frac-tional cable equation, Math. Comp.,80(2011) 1369-1396.
[74]W. Deng, Numerical algorithm for the time fractional Fokker-Planck equation, J. Comput. Phys.,227(2007) 1510-1522.
[75]W. Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal.,47(2008) 204-226.
[76]N. Zhang, W. Deng, Y. Wu, Finite difference/element method for a two-dimensional modified fractional diffusion equation, Adv. Appl. Math. Mech., 4(2012) 496-518.
[77]X. Ji, H. Tang, High-order accurate Runge-Kutta (Local) discontinuous Galerkin methods for one-and two-dimensional fractional diffusion equations, Numer. Math. Theor. Meth. Appl.,5(2012) 333-358.
[78]J.M. Sanz-serna, A numerical method for a partial integro-differential equa-tion, SIAM J. Numer. Anal.,25(1988) 319-327.
[79]J.C. Lopez-Marcos, A difference scheme for a nonlinear partial integro-differential equation, SIAM J. Numer. Anal.,27(1990) 20-31.
[80]W. Mclean, V. Thomee, Numerical solution of an evolution equation with a positive type memory term, J. Austral. Math. Soc. Ser.,35(1993) 23-70.
[81]W. Mclean, V. Thomee, L.B. Wahlbin, Discretization with variable time steps of an evolution equation with a positive-type memory term, J. Comput. Appl. Math.,69(1996) 49-69.
[82]C. Chen, V. Thomee, L. B. Wahlbin, Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel, Math. Comput.,58(1992) 587-602.
[83]W. Mclean and K. Mustapha, A second-order accurate numerical method for a fractional wave equation, Numer. Math.,105(2007) 481-510.
[84]C.H. Kim, U J. Choi, Spectral collocation methods for a partial integro-differential equations with a weakly singular kernel, J. Austral. Math. Soc. ser. B,39(1998) 408-430.
[85]K. Adolfsson, M. Enelund. S. Larsson, Adaptive discretization of an integro-differential equation with a weakly singular convolution kernel, Comput. Methods Appl. Mech. Engrg.,192(2003) 5285-5304.
[86]M. Rrenardy, W. Hrusa, J. Nohel, Mathematical problems in viscoelasticity, Pitman Monogr. Surveys Pure Appl. Math.35, Wiley, New York,1987.
[87]U. Jin Choi, R. C. MacCamy, Fractional order Volterra equations, in Volterra integrodifferential equations in Banach spaces and applications in Pitman Re-search Notes in mathematics, Volume 190 (Longman, Harlow, UK, and Wiley, New York,1989) 231-245.
[88]R. C. MacCamy, An integro-differential equation with application to heat flow, Quart. Appl. Math.,35(1977) 1-19.
[89]R. C. MacCamy, A model for one-dimensional nonlinear viscoelasticity, Quart. Appl. Math.,35(1977) 21-33.
[90]R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion:A fractional dynamics approach, Phys. Rep.,339(2000) 1-77.
[91]M. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys.,228(2009) 7792-7804.
[92]M. M. Meerschaert, C. Tadjeran. Finite difference approximations for frac-tional advection-dispersion flow equations, J. Comput. Appl. Math.,172(2004) 65-77.
[93]M. M. Meerschaert, C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56(2006) 80-90.
[94]Z.Q. Deng, V.P. Singh, F. Asce, L. Bengtsson, Numerical solution of fractional advection-dispersion equations, J. Hydraul. Eng.,130(2004) 422-431.
[95]J.T. Ma, J.Q. Liu, Z.Q. Zhou, Convergence analysis of moving finite element methods for space fractional differential equations, J. Comput. Appl. Math., 255(2014) 661-670.
[96]W.R. Schneider, W. Wyss, Fractional diffusion and wave equations, J. Math. Phys.30(1989) 134-144.
[97]J.E. Dendy, An analysis of some Galerkin schemes for the solution of nonlinear time-dependent problems, SIAM J. Numer. Anal.,12(1975) 541-565.
[98]J.E. Dendy, G. Fairweather, Alternating-direction Galerkin methods for parabolic and hyperbolic problems on rectangular polygons, SIAM J. Numer. Anal.,12(1975) 144-163.
[99]JR.J. Douglas, T. Dupont, Alternating direction Galerkin methods on rect-angles, Numerical Solution of Partial Differential Equations II (B. Hubbard ed.), New York:Academic Press, (1971) 133-214.
[100]Z. Zhang, D. Deng, A new alternating-direction finite element method for hyperbolic equation, Numer. Methods Partial Differ. Equ.,23(2007) 1530-1559.
[101]I. Podlubny, Fractional Differential Equations: An introduction to fractional derivatives, fractional differential equations, to The methods of their solution and some of their applications, Academic Press, San Diego,1999.
[102]C.P. Li, W.H. Deng, Remark on fractional derivatives, Appl. Math. Comput., 187(2007) 777-784.
[103]M. Moshrefi-Torbati, J.K. Hammond, Physical and geometrical interpretation of fractional operators, J. Franklin. Inst.,335B(1998) 1077-1086.
[104]R.R. Nigmatullin, Fractional integral and its physical interpretation, J. The-oret. Math. Phys.,90(1992) 242-251.
[105]S. Larsson, V. Thomee, LB Wahlbin, Numerical Solution of Parabolic Integro-differential equations by the discontinuous Galerkin methods, Math. Comput., 67(1998) 45-71.
[106]Y.Q. Huang, Time discretization scheme for an integro-differential equation of parabolic type, J. Comput. Math.,12(1994) 259-264.
[107]J. Bouchaud, A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep.,195(1990) 127-293.
[108]D.W. Peaceman, JR, H.H. Rachford, The numerical solution of elliptic and parabolic differential equations, J. Soc. Ind. Appl. Math.,3(1955) 28-41.
[109]H. Cheng, P. Lin, Q. Sheng, R.C.E. Tan, Solving degenerate reaction-diffusion equations via variable step Peaceman-Rachford splitting, SIAM J. Sci. Com-put.,25(2003) 1273-1292.
[110]S. Karaa, A high-order compact ADI method for parabolic problems with variable coefficients, Numer. Methods Partial Differ. Equ.,22(2006) 983-993.
[111]S. Karaa, A high-order ADI method for parabolic problems with variable coefficients, Int. J. Comput. Math.,86(2009) 109-120.
[112]J.E. Dendy, An analysis of some Galerkin schemes for the solution of nonlinear time dependent problems, SIAM J. Numer. Anal.,12(1975) 541-565.
[113]R.I. Fernandes, G. Fairweather, An alternating direction Galerkin method for a class of second-order hyperbolic equations in two space variables, SIAM J. Numer. Anal.,28(1991) 1265-1281.