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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 139530, 6 pages
A Spectral Deferred Correction Method for Fractional Differential Equations
1College of Mathematics, Qingdao University, Qingdao 266071, China
2National Laboratory for Scientific Computing, MCTI, Avenida Getulio Vargas 333, 25651-075 Petropolis, RJ, Brazil
Received 31 May 2013; Revised 7 August 2013; Accepted 19 September 2013
Academic Editor: Dumitru Baleanu
Copyright © 2013 Jia Xin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
- D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, racrional Calculus Models and Numerical Methods, World Scientific, Singapore, 2012.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, The Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Amsterdam, The Netherlands, 1993.
- I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, San Diego, Calif, USA, 1999.
- K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, Germany, 2010.
- R. L. Bagley and R. A. Calico, “Fractional order state equations for the control of viscoelastically damped structures,” Journal of Guidance, Control, and Dynamics, vol. 14, no. 2, pp. 304–311, 1991.
- J. Bai and X. C. Feng, “Fractional-order anisotropic diffusion for image denoising,” IEEE Transactions on Image Processing, vol. 16, no. 10, pp. 2492–2502, 2007.
- V. E. Tarasov, “Fractional vector calculus and fractional Maxwell's equations,” Annals of Physics, vol. 323, no. 11, pp. 2756–2778, 2008.
- D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “Application of a fractional advection-dispersion equation,” Water Resources Research, vol. 36, no. 6, pp. 1403–1412, 2000.
- R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, pp. 1–77, 2000.
- B. Baeumer, M. M. Meerschaert, D. A. Benson, and S. W. Wheatcraft, “Subordinated advection-dispersion equation for contaminant transport,” Water Resources Research, vol. 37, no. 6, pp. 1543–1550, 2001.
- R. Schumer, D. A. Benson, M. M. Meerschaert, and B. Baeumer, “Multiscaling fractional advection-dispersion equations and their solutions,” Water Resources Research, vol. 39, no. 1, pp. 1022–1032, 2003.
- S. B. Yuste and L. Acedo, “Some exact results for the trapping of subdiffusive particles in one dimension,” Physica A, vol. 336, no. 3-4, pp. 334–346, 2004.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
- I. Podlubny, “Matrix approach to discrete fractional calculus,” Fractional Calculus & Applied Analysis, vol. 3, no. 4, pp. 359–386, 2000.
- K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 3–22, 2002.
- K. Diethelm, N. J. Ford, and A. D. Freed, “Detailed error analysis for a fractional Adams method,” Numerical Algorithms, vol. 36, no. 1, pp. 31–52, 2004.
- R. Lin and F. Liu, “Fractional high order methods for the nonlinear fractional ordinary differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 4, pp. 856–869, 2007.
- Ch. Lubich, “Discretized fractional calculus,” SIAM Journal on Mathematical Analysis, vol. 17, no. 3, pp. 704–719, 1986.
- P. Kumar and O. P. Agrawal, “An approximate method for numerical solution of fractional differential equations,” Signal Processing, vol. 86, no. 10, pp. 2602–2610, 2006.
- J. F. Huang, Y. F. Tang, and L. Vázquez, “Convergence analysis of a block-by-block method for fractional differential equations,” Numerical Mathematics: Theory, Methods and Applications, vol. 5, no. 2, pp. 229–241, 2012.
- Y. L. Li, “Solving a nonlinear fractional differential equation using Chebyshev wavelets,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 9, pp. 2284–2292, 2010.
- R. Scherer, S. L. Kalla, Y. F. Tang, and J. F. Huang, “The Grünwald-Letnikov method for fractional differential equations,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 902–917, 2011.
- Y. Ben Nakhi and S. L. Kalla, “Some boundary value problems of temperature fields in oil strata,” Applied Mathematics and Computation, vol. 146, no. 1, pp. 105–119, 2003.
- V. J. Ervin, N. Heuer, and J. P. Roop, “Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation,” SIAM Journal on Numerical Analysis, vol. 45, no. 2, pp. 572–591, 2007.
- F. Liu, P. Zhuang, and K. Burrage, “Numerical methods and analysis for a class of fractional advection-dispersion models,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 2990–3007, 2012.
- C. Tadjeran, M. M. Meerschaert, and H. P. Scheffler, “A second-order accurate numerical approximation for the fractional diffusion equation,” Journal of Computational Physics, vol. 213, no. 1, pp. 205–213, 2006.
- T. S. Basu and H. Wang, “A fast second-order finite difference method for space-fractional diffusion equations,” International Journal of Numerical Analysis and Modeling, vol. 9, no. 3, pp. 658–666, 2012.
- S. B. Yuste and J. Quintana-Murillo, “A finite difference method with non-uniform timesteps for fractional diffusion equations,” Computer Physics Communications, vol. 183, no. 12, pp. 2594–2600, 2012.
- A. Dutt, L. Greengard, and V. Rokhlin, “Spectral deferred correction methods for ordinary differential equations,” BIT Numerical Mathematics, vol. 40, no. 2, pp. 241–266, 2000.
- J. Huang, J. Jia, and M. Minion, “Accelerating the convergence of spectral deferred correction methods,” Journal of Computational Physics, vol. 214, no. 2, pp. 633–656, 2006.
- K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229–248, 2002.