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High-Order Numerical Algorithms for Riesz Derivatives via Constructing New Generating Functions

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Abstract

A class of high-order numerical algorithms for Riesz derivatives are established through constructing new generating functions. Such new high-order formulas can be regarded as the modification of the classical (or shifted) Lubich’s difference ones, which greatly improve the convergence orders and stability for time-dependent problems with Riesz derivatives. In rapid sequence, we apply the 2nd-order formula to one-dimension Riesz spatial fractional partial differential equations to establish an unconditionally stable finite difference scheme with convergent order \({\mathcal {O}}(\tau ^2+h^2)\), where \(\tau \) and h are the temporal and spatial stepsizes, respectively. Finally, some numerical experiments are performed to confirm the theoretical results and testify the effectiveness of the derived numerical algorithms.

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References

  1. Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bailey, W.N.: Generalized Hypergeometric Series. Stechert-Hafner Inc, New York (1964)

    MATH  Google Scholar 

  3. Çelik, C., Duman, M.: Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J. Comput. Phys. 231, 1743–1750 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chan, R.H.: Toeplitz preconditioners for Toeplitz systems with nonnegative generating functions. IMA J. Numer. Anal. 11, 333–345 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chan, R.H., Jin, X.Q.: An Introduction to Iterative Toeplitz Solvers. SIAM, Philadelphia (2007)

    Book  MATH  Google Scholar 

  6. Chen, M.H., Deng, W.H.: WSLD operators: a class of fourth order difference approximations for space Riemann–Liouville derivative. SIAM J. Numer. Anal. 52, 1418–1438 (2014)

    Article  MathSciNet  Google Scholar 

  7. Chen, S., Shen, J., Wang, L.L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 79, 807–827 (2016)

  8. Chen, S., Jiang, X., Liu, F., Turner, I.: High order unconditionally stable difference schemes for the Riesz space-fractional telegraph equation. J. Comput. Appl. Math. 278, 119–129 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ding, H.F., Li, C.P., Chen, Y.Q.: High-order algorithms for Riesz derivative and their applications (II). J. Comput. Phys. 293, 218–237 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22, 558–576 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Feller, W.: On a generalization of Marcel Riesz’ potentials and the semi-groups generated by them. Commun. Sém. Math. Univ. Lund. 1, 73–81 (1952)

  12. Garrappaa, R., Moretb, I., Popolizioc, M.: Solving the time-fractional Schrödinger equation by Krylov projection methods. J. Comput. Phys. 293, 115–134 (2015)

    Article  MathSciNet  Google Scholar 

  13. Huang, J., Tang, Y., Vázquez, L., Yang, J.: Two finite difference schemes for time fractional diffusion-wave equation. Numer. Algorithms 64, 707–720 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  15. Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704–719 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, C.P., Ding, H.F.: Higher order finite difference method for the reaction and anomalous-diffusion equation. Appl. Math. Model. 38, 3802–3821 (2014)

    Article  MathSciNet  Google Scholar 

  17. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  18. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection–dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Metzler, R., Klafter, J.: The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nörlund, N.E.: Vorlesungen über Differenzenrechnung. Springer, Berlin (1924)

    Book  MATH  Google Scholar 

  21. Ortigueira, M.D.: Riesz potential operators and inverses via fractional centred derivatives. Int. J. Math. Math. Sci. 2006, Article ID 48391, 1–2 (2006)

  22. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  23. Podlubny, I.: Fractional Differential Equations. Academin Press, San Diego (1999)

    MATH  Google Scholar 

  24. Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, 2nd edn. Springer, New York (2007)

    Book  MATH  Google Scholar 

  25. Rall, L.B.: Perspectives on Automatic Differentiation: Past, Present, and Future? Automatic Differentiation: Applications, Theory, and Implementations. Springer, Berlin (2006)

    MATH  Google Scholar 

  26. Srivastava, H.M.: An explicit formula for the generalized Bernoulli polynomials. J. Math. Anal. Appl. 130, 509–513 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  27. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, London (1993)

    MATH  Google Scholar 

  28. Saichev, A.I., Zaslavsky, G.M.: Fractional kinetic equations: solutions and applications. Chaos 7, 753–764 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tricomi, F.G., Erdélyi, A.: The asymptotic expansion of a ratio of Gamma functions. Pac. J. Math. 1, 133–142 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tian, W.Y., Zhou, H., Deng, W.H.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84, 1703–1727 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang, H., Du, N.: Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations. J. Comput. Phys. 258, 305–318 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, Z., Vong, S.: Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation. J. Comput. Phys. 277, 1–15 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Xu, Q., Hesthaven, J.S., Chen, F.: A parareal method for time-fractional differential equations. J. Comput. Phys. 293, 173–183 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Yuste, S.B., Acedo, L.: An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42, 1862–1874 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36, A40–A62 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zeng, F.H., Li, C.P., Liu, F.W., Turner, I.: The use of finite difference/element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35, A2976–A3000 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhao, X., Sun, Z.Z.: Compact Crank–Nicolson schemes for a class of fractional Cattaneo equation in inhomogeneous medium. J. Sci. Comput. 62, 747–771 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhao, X., Sun, Z.Z., Hao, Z.P.: A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrodinger equation. SIAM J. Sci. Comput. 36, A2865–A2886 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11372170, 11561060 and 11671251), the Scientific Research Program for Young Teachers of Tianshui Normal University (Grant Nos. TSA1405) and Tianshui Normal University Key Construction Subject Project (Big data processing in dynamic image).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Tianshui Normal University, Tianshui, 741001, China

    Hengfei Ding

  2. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Changpin Li

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  1. Hengfei Ding
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  2. Changpin Li
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Correspondence to Hengfei Ding.

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Ding, H., Li, C. High-Order Numerical Algorithms for Riesz Derivatives via Constructing New Generating Functions. J Sci Comput 71, 759–784 (2017). https://doi.org/10.1007/s10915-016-0317-3

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  • DOI: https://doi.org/10.1007/s10915-016-0317-3

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