Abstract and Applied Analysis

Fourth-Order Compact Difference Schemes for the Riemann-Liouville and Riesz Derivatives

Yuxin Zhang, Hengfei Ding, and Jincai Luo

Full-text: Open access

Abstract

We propose two new compact difference schemes for numerical approximation of the Riemann-Liouville and Riesz derivatives, respectively. It is shown that these formulas have fourth-order convergence order by means of the Fourier transform method. Finally, some numerical examples are implemented to testify the efficiency of the numerical schemes and confirm the convergence orders.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 540692, 4 pages.

Dates
First available in Project Euclid: 6 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412607138

Digital Object Identifier
doi:10.1155/2014/540692

Mathematical Reviews number (MathSciNet)
MR3206799

Zentralblatt MATH identifier
07022586

Citation

Zhang, Yuxin; Ding, Hengfei; Luo, Jincai. Fourth-Order Compact Difference Schemes for the Riemann-Liouville and Riesz Derivatives. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 540692, 4 pages. doi:10.1155/2014/540692. https://projecteuclid.org/euclid.aaa/1412607138


Export citation

References

  • B. L. Guo, X. K. Pu, and F. H. Huang, Fractional Partial Differential Equations and Their Numerical Solutions, Science Press, Beijin, China, 2011.
  • 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.
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  • M. H. Chen and W. H. Deng, “Fourth order accurate scheme for the space čommentComment on ref. [1?]: Please update the information of these references [2,16?], if possible.fractional diffusion equations,” SIAM Journal on Numerical Analysis. In press.
  • M. H. Chen and W. H. Deng, “Fourth order difference approximations for space Riemann-Liouville derivatives based on weighted and shifted Lubich difference operators,” Communications in Computational Physics. In press.
  • K. Diethelm, N. J. Ford, A. D. Freed, and Yu. Luchko, “Algorithms for the fractional calculus: a selection of numerical methods,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 6–8, pp. 743–773, 2005.
  • 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.
  • D. A. Murio, “On the stable numerical evaluation of Caputo fractional derivatives,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1539–1550, 2006.
  • T. Miyakoda, “Discretized fractional calculus with a series of Chebyshev polynomial,” Electronic Notes in Theoretical Computer Science, vol. 225, pp. 239–244, 2009.
  • Z. Odibat, “Approximations of fractional integrals and Caputo fractional derivatives,” Applied Mathematics and Computation, vol. 178, no. 2, pp. 527–533, 2006.
  • I. Podlubny, A. Chechkin, T. Skovranek, Y. Chen, and B. M. V. Jara, “Matrix approach to discrete fractional calculus. II. Partial fractional differential equations,” Journal of Computational Physics, vol. 228, no. 8, pp. 3137–3153, 2009.
  • E. Sousa, “How to approximate the fractional derivative of order $1<\alpha \leq 2$,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 22, no. 4, Article ID 1250075, 13 pages, 2012.
  • W. Y. Tian, H. Zhou, and W. H. Deng, “A class of second order difference approximation for solving space fractional diffusion equations,” http://arxiv.org/abs/1201.5949.
  • R. F. Wu, H. F. Ding, and C. P. Li, “Determination of coefficients of high-order schemes for Riemann-Liouville derivative,” The Scientific World Journal, vol. 2014, Article ID 402373, 21 pages, 2014.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
  • 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. \endinput