Abstract and Applied Analysis

High-Order Algorithms for Riesz Derivative and Their Applications ( I )

Hengfei Ding, Changpin Li, and YangQuan Chen

Full-text: Open access

Abstract

We firstly develop the high-order numerical algorithms for the left and right Riemann-Liouville derivatives. Using these derived schemes, we can get high-order algorithms for the Riesz fractional derivative. Based on the approximate algorithm, we construct the numerical scheme for the space Riesz fractional diffusion equation, where a fourth-order scheme is proposed for the spacial Riesz derivative, and where a compact difference scheme is applied to approximating the first-order time derivative. It is shown that the difference scheme is unconditionally stable and convergent. Finally, numerical examples are provided which are in line with the theoretical analysis.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 653797, 17 pages.

Dates
First available in Project Euclid: 6 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412606725

Digital Object Identifier
doi:10.1155/2014/653797

Mathematical Reviews number (MathSciNet)
MR3214445

Zentralblatt MATH identifier
07022832

Citation

Ding, Hengfei; Li, Changpin; Chen, YangQuan. High-Order Algorithms for Riesz Derivative and Their Applications $(I)$. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 653797, 17 pages. doi:10.1155/2014/653797. https://projecteuclid.org/euclid.aaa/1412606725


Export citation

References

  • B. L. Guo, X. K. Pu, and F. H. Huang, Fractional Partial Differential Equations and Their Numerical Solutions, Science Press, Beijing, China, 2011 (Chinese).
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.
  • K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, NY, USA, 1974.
  • J. Chen, F. Liu, and V. Anh, “Analytical solution for the time-fractional telegraph equation by the method of separating variables,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1364–1377, 2008.
  • F. Huang and B. Guo, “General solutions to a class of time fractional partial differential equations,” Applied Mathematics and Mechanics, vol. 31, no. 7, pp. 815–826, 2010.
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  • N. T. Shawagfeh, “Analytical approximate solutions for nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 131, no. 2-3, pp. 517–529, 2002.
  • Ch. Lubich, “Discretized fractional calculus,” SIAM Journal on Mathematical Analysis, vol. 17, no. 3, pp. 704–719, 1986.
  • C. Li, A. Chen, and J. Ye, “Numerical approaches to fractional calculus and fractional ordinary differential equation,” Journal of Computational Physics, vol. 230, no. 9, pp. 3352–3368, 2011.
  • E. Hanert, “On the numerical solution of space-time fractional diffusion models,” Computers & Fluids, vol. 46, pp. 33–39, 2011.
  • W. McLean and K. Mustapha, “A second-order accurate numerical method for a fractional wave equation,” Numerische Mathematik, vol. 105, no. 3, pp. 481–510, 2007.
  • K. Mustapha and W. McLean, “Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations,” SIAM Journal on Numerical Analysis, vol. 51, no. 1, pp. 491–515, 2013.
  • C. Piret and E. Hanert, “A radial basis functions method for fractional diffusion equations,” Journal of Computational Physics, vol. 238, pp. 71–81, 2013.
  • A. I. Saichev and G. M. Zaslavsky, “Fractional kinetic equations: solutions and applications,” Chaos, vol. 7, no. 4, pp. 753–764, 1997.
  • G. M. Zaslavsky, “Chaos, fractional kinetics, and anomalous transport,” Physics Reports, vol. 371, no. 6, pp. 461–580, 2002.
  • H. Zhang and F. Liu, “The fundamental solutions of the space, space-time Riesz fractional partial differential equations with periodic conditions,” Numerical Mathematics: A Journal of Chinese Universities, English Series, vol. 16, no. 2, pp. 181–192, 2007.
  • H. Zhang, F. Liu, and V. Anh, “Galerkin finite element approximation of symmetric space-fractional partial differential equations,” Applied Mathematics and Computation, vol. 217, no. 6, pp. 2534–2545, 2010.
  • J. Chen, F. Liu, I. Turner, and V. Anh, “The fundamental and numerical solutions of the Riesz space-fractional reaction-dispersion equation,” The ANZIAM Journal, vol. 50, no. 1, pp. 45–57, 2008.
  • S. Shen, F. Liu, and V. Anh, “Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation,” Numerical Algorithms, vol. 56, no. 3, pp. 383–403, 2011.
  • S. Shen, F. Liu, V. Anh, and I. Turner, “A novel numerical approximation for the space fractional advection-dispersion equation,” IMA Journal of Applied Mathematics, 2012.
  • N. Özdemir, D. Avc\i, and B. B. \.Iskender, “The numerical solutions of a two-dimensional space-time Riesz-Caputo fractional diffusion equation,” International Journal of Optimization and Control: Theories & Applications, vol. 1, no. 1, pp. 17–26, 2011.
  • Q. Yang, F. Liu, and I. Turner, “Numerical methods for fractional partial differential equations with Riesz space fractional derivatives,” Applied Mathematical Modelling, vol. 34, no. 1, pp. 200–218, 2010.
  • C. Çelik and M. Duman, “Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative,” Journal of Computational Physics, vol. 231, no. 4, pp. 1743–1750, 2012.
  • D. Wang, A. Xiao, and W. Yang, “Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative,” Journal of Computational Physics, vol. 242, pp. 670–681, 2013.
  • M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for fractional advection-dispersion flow equations,” Journal of Computational and Applied Mathematics, vol. 172, no. 1, pp. 65–77, 2004.
  • E. Sousa, “A second order explicit finite difference method for the fractional advection diffusion equation,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3141–3152, 2012.
  • M. D. Ortigueira, “Riesz potential operators and inverses via fractional centred derivatives,” International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 48391, 12 pages, 2006.
  • W. Y. Tian, H. Zhou, and W. H. Deng, “A class of second čommentComment on ref. [27?]: Please update the information of this reference, if possible.order difference approximation for solving space fractional diffusion equations,” http://arxiv.org/abs/1201.5949.
  • V. J. Ervin and J. P. Roop, “Variational formulation for the stationary fractional advection dispersion equation,” Numerical Methods for Partial Differential Equations, vol. 22, no. 3, pp. 558–576, 2006.
  • V. K. Tuan and R. Gorenflo, “Extrapolation to the limit for numerical fractional differentiation,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 75, no. 8, pp. 646–648, 1995.
  • W. Zhang, Finite Difference Methods for Partial Differential Equations in Science Computation, Higher Education Press, Beijing, China, 2006 (Chinese).
  • R. H.-F. Chan and X.-Q. Jin, An Introduction to Iterative Toeplitz Solvers, vol. 5 of Fundamentals of Algorithms, SIAM, Philadelphia, Pa, USA, 2007.
  • R. H. Chan, “Toeplitz preconditioners for Toeplitz systems with nonnegative generating functions,” IMA Journal of Numerical Analysis, vol. 11, no. 3, pp. 333–345, 1991. \endinput