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Finite Difference Method for Two-Sided Space-Fractional Partial Differential Equations

  • Kamal Pal
  • Fang Liu
  • Yubin YanEmail author
  • Graham Roberts
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9045)

Abstract

Finite difference methods for solving two-sided space-fractional partial differential equations are studied. The space-fractional derivatives are the left-handed and right-handed Riemann-Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. The Hadamard finite-part integrals are approximated by using piecewise quadratic interpolation polynomials and a numerical approximation scheme of the space-fractional derivative with convergence order \(O(\varDelta x^{3- \alpha }), \, 1<\alpha <2\) is obtained. A shifted implicit finite difference method is introduced for solving two-sided space-fractional partial differential equation and we prove that the order of convergence of the finite difference method is \(O (\varDelta t + \varDelta x^{\min (3- \alpha , \beta )}), 1< \alpha <2, \, \beta >0\), where \(\varDelta t, \varDelta x\) denote the time and space step sizes, respectively. Numerical examples are presented and compared with the exact analytical solution for its order of convergence.

Notes

Acknowledgements

We wish to express our sincere gratitude to Professor Neville. J. Ford for his encouragement, discussions and valuable criticism during the research of this work.

References

  1. 1.
    Choi, H.W., Chung, S.K., Lee, Y.J.: Numerical solutions for space fractional dispersion equations with nonlinear source terms. Bull. Korean Math. Soc. 47, 1225–1234 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order. Electron. Trans. Numer. Anal. 5, 1–6 (1997)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Ford, N.J., Rodrigues, M.M., Xiao, J., Yan, Y.: Numerical analysis of a two-parameter fractional telegraph equation. J. Comput. Appl. Math. 249, 95–106 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Ford, N.J., Xiao, J., Yan, Y.: A finite element method for time fractional partial differential equations. Fract. Calc. Appl. Anal. 14, 454–474 (2011)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Ford, N.J., Xiao, J., Yan, Y.: Stability of a numerical method for space-time-fractional telegraph equation. Comput. Methods Appl. Math. 12, 1–16 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56, 80–90 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Yan, Y., Pal, K., Ford, N.J.: Higher order numerical methods for solving frational differential equations. BIT Numer. Math. 54(2), 555–584 (2014). doi: 10.1007/s10543-013-0443-3 zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Dimitrov, Y.: Numerical approximations for fractional differential equations. J. Frac. Calc. Appl. 5(3S), 1–45 (2014)MathSciNetGoogle Scholar
  9. 9.
    Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22, 558–576 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Ervin, V.J., Roop, J.P.: Variational solution of fractional advection dispersion equations on bounded domains in \(\mathbb{R}^{d}\). Numer. Methods Partial Differ. Equ. 23, 256–281 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Ervin, V.J., Heuer, N., Roop, J.P.: Numerical approximation of a time dependent nonlinear, space-fractional diffusion equation. SIAM J. Numer. Anal. 45, 572–591 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Li, C.P., Zeng, F.: Finite difference methods for fractional differential equations. Int. J. Bifurc. Chaos 22, 1230014 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Liu, F., Anh, V., Turner, I.: Numerical solution of space fractional Fokker-Planck equation. J. Comp. Appl. Math. 166, 209–219 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Podlubny, I.: Matrix approach to discrete fractional calculus. Fract. Calc. Appl. Anal. 3, 359–386 (2000)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Roop, J.P.: Computational aspects of FEM approximation of fractional advection dispersion equations on bouinded domains in \(\mathbb{R}^{2}\). J. Comput. Appl. Math. 193, 243–268 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Shen, S., Liu, F., Anh, V., Turner, I.: The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation. IMA J. Appl. Math. 73, 850–872 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Sousa, E.: Finite difference approximations for a fractional advection diffusion problem. J. Comput. Phys. 228, 4038–4054 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Tadjeran, C., Meerschaert, M.M., Scheffler, H.: A second-order acurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213, 205–213 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Yang, Q., Liu, F., Turner, I.: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 34, 200–218 (2010)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Kamal Pal
    • 1
  • Fang Liu
    • 2
  • Yubin Yan
    • 1
    Email author
  • Graham Roberts
    • 1
  1. 1.Department of MathematicsUniversity of ChesterChesterUK
  2. 2.Department of MathematicsLuliang UniversityLishiPR China

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