A High Order Finite Difference/Spectral Approximations to the Time Fractional Diffusion Equations


Article Preview

In this paper, we consider the numerical solution of a time-fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order α, with 03-α+N-m) , where Δt,N and m are the time step size, the polynomial degree and the regularity of the exact solution, respectively.



Advanced Materials Research (Volumes 875-877)

Edited by:

Duanling Li, Dawei Zheng and Jun Shi




J. Y. Cao et al., "A High Order Finite Difference/Spectral Approximations to the Time Fractional Diffusion Equations", Advanced Materials Research, Vols. 875-877, pp. 781-785, 2014

Online since:

February 2014




[1] H. Sun, A. Abdelwahab, and B. Onaral. Linear approximation of transfer function with a pole of fractional power. Automatic Control, IEEE Transactions on 1984; 29(5): 441–444.

DOI: https://doi.org/10.1109/tac.1984.1103551

[2] R.L. Bagley and R.A. Calico. Fractional order state equations for the control of viscoelastically damped structures. Journal of Guidance, Control, and Dynamics 1991; 14(2): 304–311.

DOI: https://doi.org/10.2514/3.20641

[3] M. Ichise, Y. Nagayanagi, and T. Kojima. An analog simulation of non-integer order trans fer functions for analysis of electrode processes. Journal of Electroanalytical Chemistry and Interfacial Electrochemistry 1971; 33(2): 253–265.

DOI: https://doi.org/10.1016/s0022-0728(71)80115-8

[4] J.P. Bouchaud and A. Georges. Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Phys. Rep. 1990; 195(4-5): 127–293.

DOI: https://doi.org/10.1016/0370-1573(90)90099-n

[5] R.J. Marks and M.W. Hall. Differintegral interpolation from a bandlimited signal's samples. Acoustics, Speech and Signal Processing, IEEE Transactions on 1981; 29(4): 872–877.

DOI: https://doi.org/10.1109/tassp.1981.1163636

[6] D.A. Benson, S.W. Wheatcraft, and M.M. Meerschaert. Application of a fractional advection dispersion equation. Water Resour. Res. 2000; 36(6): 1403–1412.

DOI: https://doi.org/10.1029/2000wr900031

[7] B. Mandelbrot. Some noises with 1/f spectrum, a bridge between direct current and white noise. Information Theory, IEEE Transactions on 1967; 13(2): 289–298.

DOI: https://doi.org/10.1109/tit.1967.1053992

[8] G.M. Zaslavsky. Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 2002; 371(6): 461–580.

[9] S. B. Yuste, L. Acedo, and K. Lindenberg. Reaction front in an A+ B → C reaction subdiffusion process. Phys. Rev. E 2004; 69(3): 36–126.

DOI: https://doi.org/10.1103/physreve.69.036126

[10] R. Schumer, D.A. Benson, M.M. Meerschaert, and B. Baeumer. Multiscaling fractional advection-dispersion equations and their solutions. Water Resour. Res 2003; 39(1): 1022–1032.

DOI: https://doi.org/10.1029/2001wr001229

[11] M. Raberto, E. Scalas, and F. Mainardi. Waiting-times and returns in high-frequency financial data: an empirical study. Phys. A 2002; 314(1-4): 749–755.

DOI: https://doi.org/10.1016/s0378-4371(02)01048-8

[12] J. Sabatier, O.P. Agrawal, and J.A.T. Machado. Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering . Dordrecht: Springer; (2007).

[13] R.P. Agarwal, V. Lakshmikantham, and J.J. Nieto. On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. 2010; 72(6): 2859–2862.

DOI: https://doi.org/10.1016/j.na.2009.11.029

[14] A. Ashyralyev. A note on fractional derivatives and fractional powers of operators. J. Math. Anal. Appl. 2009; 357(1): 232–236.

[15] Y. Lin and C. Xu. Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 2007; 225(2): 1533–1552.

[16] I. Podlubny. Fractional differential equations. New York: Acad. Press; (1999).

Fetching data from Crossref.
This may take some time to load.