Numerical methods for solving the multi-term time-fractional wave-diffusion equation

Fractional Calculus and Applied Analysis - Tập 16 Số 1 - Trang 9-25 - 2013
Fawang Liu1, Mark M. Meerschaert2, Robert J. McGough2, Pinghui Zhuang3, Qingxia Liu3
1Queensland University of Technology.,
2Michigan State University
3[Xiamen University]

Tóm tắt

Từ khóa


Tài liệu tham khảo

B. Baeumer and M.M. Meerschaert, Stochastic solutions for fractional Cauchy problems. Fract. Calc. Appl. Anal. 4, No 4 (2001), 481–500.

B. Baeumer, S. Kurita and M.M. Meerschaert, Inhomogeneous fractional diffusion eqautions. Fract. Calc. Appl. Anal., 8, No 4 (2005), 371–376; at http://www.math.bas.bg/~fcaa.

D. Bolster, M.M. Meerschaert and A. Sikorskii, Product rule for vector fractional derivatives. Fract. Calc. Appl. Anal. 15, No 3 (2012), 463–478; DOI:10.2478/s13540-012-0033-0; at http://link.springer.com/article/10.2478/s13540-012-0033-0.

W. Chen, S. Holm, Modified Szabo’s wave equation models for lossy media obeying frequency power law. J. Acoust. Soc. Am. 114 (2003), 2570–2754.

W. Deng, C. Li, Q. Guo, Analysis of fractional differential equations with multi-orders. Fractals 15, No 2 (2007), 173–182.

W. Deng, C. Li, J. Lu, Stability analysis of linear fractional differential system with multiple time-delays. Nonlinear Dynamics 48, No 4 (2007), 409–416.

K. Diethelm, The Analysis of Fractional Differential Equations. Springer, Berlin etc. (2010).

F.A. Duck, Physical Properties of Tissue: A Comprehensive Reference Book. Academic Press (1990).

Y. Gu, P. Zhuang, F. Liu, An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation. Computer Modeling in Eng. & Sciences 56 (2010), 303–334.

M. Ilic, F. Liu, I. Turner, V. Anh, Numerical approximation of a fractional-in-space diffusion equation (I). Fract. Calc. Appl. Anal., 8, No 3 (2005), 323–341; at http://www.math.bas.bg/~fcaa.

M. Ilic, F. Liu, I. Turner, V. Anh, Numerical approximation of a fractional-in-space diffusion equation (II) — with nonhomogeneous boundary conditions. Fract. Calc. Appl. Anal., 9, No 4 (2006), 333–349; at http://www.math.bas.bg/~fcaa.

H. Jiang, F. Liu, I. Turner, K. Burrage, Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain. J. Math. Anal. Appl. 389 (2012), 1117–1127.

J.K. Kelly, R.J. McGough, M.M. Meerschaert, Analytical time-domain Green’s functions for power-law media. J. Acoust. Soc. Am. 124 (2008), 2861–2872.

C. Li, F. Zeng, F. Liu, Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15, No 3 (2012), 383–406; DOI:10.2478/s13540-012-0028-x; at http://link.springer.com/article/10.2478/s13540-012-0028-x

M. Liebler, S. Ginter, T. Dreyer, R.E. Riedlinger, Full wave modeling of therapeutic ultrasound: Efficient time-domain implementation of the frequency power-law attenuation. J. Acoust. Soc. Am. 116 (2004), 2742–2750.

F. Liu, V. Anh, I. Turner, Numerical solution of the space fractional Fokker-Planck equation. J. Comp. Appl. Math. 166 (2004), 209–219.

F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrag, Stability and convergence of the difference methods for the space-time fractional advectiondiffusion equation. J. Comp. Appl. Math. 191 (2007), 12–20.

F. Liu, C. Yang, K. Burrage, Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term. J. Comp. Appl. Math. 231 (2009), 160–176.

F. Liu, P. Zhuang, K. Burrage, Numerical methods and analysis for a class of fractional advection-dispersion models. Computers and Math. with Appl. 63 (2012), 1–22.

Q. Liu, F. Liu, I. Turner, V. Anh, Approximation of the Lévy- Feller advection-dispersion process by random walk and finite difference method. J. Comp. Phys. 222 (2007), 57–70.

Q. Liu, Y. Gu, P. Zhuang, F. Liu, Y. Nie, An implicit RBF meshless approach for time fractional diffusion equations. Comput. Mech. 48 (2011), 1–12.

Y. Luchko, Initial-boundary-value problems for the generalized multiterm time-fractional diffusion equation. J. Math. Anal. Appl. 374 (2011), 538–548.

M.M. Meerschaert and H.P. Scheffler, Semistable Lévy motion. Fract. Calc. Appl. Anal., 5, No 1 (2002), 27–54.

M.M. Meerschaert, J. Mortensen, H.P. Scheffler, Vector Grünwald formula for fractional derivatives. Fract. Calc. Appl. Anal. 7, No 1 (2004), 61–82.

M.M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations. J. Comp. Appl. Math. 172 (2004), 65–77.

M.M. Meerschaert, P. Straka, Y. Zhou, R.J. McGough, Stochastic solution to a time-fractional attenuated wave equation. Nonlinear Dynamics 70 (2012), 1273–1281.

R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1–77.

I. Podlubny, Fractional Differential Equations. Academic Press, New York (1999).

J.P. Roop, Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R2. J. Comp. Appl. Math. 193 (2006), 243–268.

R. Schumer, D.A. Benson, M.M Meerschaert, B. Baeumer, Fractal mobile/immobile solute transport. Water Resources Researces 39 (2003), 1296–1307.

S. Shen, F. Liu, V. Anh, Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation. Numerical Algorithm 56 (2011), 383–404.

M. Stojanovic, Numerical method for solving diffusion-wave phenomena. J. Comp. Appl. Math. 235 (2011), 3121–3137.

P. Straka, M.M. Meerschaert, R.J. McGough, and Y. Zhou, Fractional wave equations with attenuation. Fract. Calc. Appl. Anal. 16, No 1 (2013), 262–272 (same issue); DOI:10.2478/s13540-013-0016-9; at http://link.springer.com/journal/13540.

T.L. Szabo, Time domain wave equations for lossy media obeying a frequency power law. J. Acoust. Soc. Am. 96 (1994), 491–500.

C. Yang, F. Liu, A computationally effective predictor-corrector method for simulating fractional order dynamical control system. ANZIAM J. 47 (2006), 168–184.

Y. Zhang, D.A. Benson, D.M. Reeves, Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications. Advances in Water Resources 32 (2009), 561–581.

F. Zhang, C. Li, Stability analysis of fractional differential systems with order lying in (1,2). Advances in Difference Equations (2011), ID 213485.

P. Zhuang, F. Liu, V. Anh, 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.