[1] Podlubny, I. Fractional Differential Equations, Academic Press, New York (1999)
[2] Liu, F., Anh, V., Turner, I., and Zhang, P. Time fractional advection dispersion equation. Journal of Applied Mathematics and Computing, 13(1), 233–245 (2003)
[3] Yuste, S. B. and Acedo, L. An explicit finite difference method and a new von Numann-type stability analysis for fractional diffusion equation. SIAM Journal on Numerical Analysis, 42(5), 1862–1874 (2005)
[4] Lin, Y. M. and Xu, C. J. Finite difference/spectral approximations for the time-fractional diffusion equation. Journal of Computational Physics, 225(2), 1533–1552 (2007)
[5] Ervin, V. J. and Roop, J. P. Variational formulation for the stationary fractional advection dispersion equation. Numerical Methods for Partial Differential Equations, 22(3), 558–576 (2006)
[6] Zhang, H., Liu, F., and Anh, V. Galerkin finite element approximations of symmetric spacefractional partial differnetial equations. Applied Mathematics and Computation, 217(6), 2534– 2545 (2010)
[7] Li, C. P., Zhao, Z. G., and Chen, Y. Q. Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Computers and Mathematics with Applications, 62(3), 855–875 (2011)
[8] Ford, N. J., Xiao, J. Y., and Yan, Y. B. A finite element method for time fractional partial differential equations. Fractional Calculus and Applied Analysis, 14(3), 454–474 (2011)
[9] Zhang, X. D., Huang, P. Z., Feng, X. L., andWei, L. L. Finite element method for two-dimensional time-fractional Tricomi-type equations. Numerical Methods for Partial Differential Equations, 29(4), 1081–1096 (2013)
[10] Deng, W. H. Short memory principle and a predictor-corrector approach for fractional differential equations. Journal of Computational and Applied Mathematics, 206(1), 174–188 (2007)
[11] Ford, N. J. and Simpson, A. C. The numerical solution of fractional differential equations: speed versus accuracy. Numerical Algorithms, 26(4), 333–346 (2001)
[12] Holmes, P. J., Lumley, L., and Berkooz, G. Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge (1996)
[13] Jolliffe, I. T. Principal Component Analysis, Springer-Verlag, Berlin (2002)
[14] Fukunaga, K. Introduction to Statistical Pattern Recognition, Academic Press, New York (1990)
[15] Kunisch, K. and Volkwein, S. Galerkin proper orthogonal decomposition methods for parabolic problems. Numerische Mathematik, 90(1), 117–148 (2001)
[16] Kunisch, K. and Volkwein, S. Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM Journal on Numerical Analysis, 40(2), 492–515 (2002)
[17] Sun, P., Luo, Z. D., and Zhou, Y. J. Some reduced finite difference schemes based on a proper orthogonal decomposition technique for parabolic equations. Applied Numerical Mathematics, 60, 154–164 (2010)
[18] Luo, Z. D., Chen, J., Sun, P., and Yang, X. Z. Finite element formulation based on proper orthogonal decomposition for parabolic equations. Science in China Series A: Mathematics, 52(3), 585–596 (2009)
[19] Luo, Z. D., Li, H., Zhou, Y. J., and Huang, X. M. A reduced-order FVE formulation based on POD method and error analysis for two-dimensional viscoelastic problem. Journal of Mathematical Analysis and Applications, 385(1), 310–321 (2012)
[20] Luo, Z. D. A reduced-order SMFVE extrapolation algorithm based on POD technique and CN method for the non-stationary Navier-Stokes equations. Discrete and Continuous Dynamical Systems Series B, 20(4), 1189–1212 (2015)
[21] Liu, J. C., Li, H., Fang, Z. C., and Liu, Y. Application of low-dimensional finite element method to fractional diffusion equation. International Journal of Modeling, Simulation, and Scientific Computing, 5(4), 1450022 (2014)
[22] Adams, R. A. Sobolev Spaces, Academic Press, New York (1975)
[23] Thomée, V. Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin (1997) |