## Abstract and Applied Analysis

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

#### 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

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

#### 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