Acta Mechanica Sinica

, Volume 34, Issue 3, pp 515–527 | Cite as

Enriched reproducing kernel particle method for fractional advection–diffusion equation

  • Yuping Ying
  • Yanping Lian
  • Shaoqiang Tang
  • Wing Kam Liu
Research Paper


The reproducing kernel particle method (RKPM) has been efficiently applied to problems with large deformations, high gradients and high modal density. In this paper, it is extended to solve a nonlocal problem modeled by a fractional advection–diffusion equation (FADE), which exhibits a boundary layer with low regularity. We formulate this method on a moving least-square approach. Via the enrichment of fractional-order power functions to the traditional integer-order basis for RKPM, leading terms of the solution to the FADE can be exactly reproduced, which guarantees a good approximation to the boundary layer. Numerical tests are performed to verify the proposed approach.


Meshfree method Fractional calulus Enriched reproducing kernel Advection–diffusion equation Fractional-order basis 



The project was supported partly by the National Natural Science Foundation of China (Grant 11521202). Ying is grateful for the support from the Chinese Scholarship Council. Lian and Liu are partially support by an Army Research Office (Grant W911NF-15-1-0569).


  1. 1.
    Chen, J.S., Pan, C., Roque, C.M.O.L., et al.: A Lagrangian reproducing kernel particle method for metal forming analysis. Comput. Mech. 22, 289–307 (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, J.S., Pan, C., Wu, C., et al.: Reproducing kernel particle methods for large deformation analysis of non-linear structures. Comput. Methods Appl. Mech. 139, 195–227 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, J.S., Pan, C., Wu, C.: Large deformation analysis of rubber based on a reproducing kernel particle method. Comput. Mech. 19, 211–227 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Lian, Y., Zhang, X., Liu, Y.: An adaptive finite element material point method and its application in extreme deformation problems. Comput. Methods Appl. Mech. 241, 275–285 (2012)CrossRefzbMATHGoogle Scholar
  5. 5.
    Belytschko, T., Lu, Y., Gu, L.: Crack propagation by element-free Galerkin methods. Eng. Fract. Mech. 51, 295–315 (1995)CrossRefGoogle Scholar
  6. 6.
    Belytschko, T., Tabbara, M.: Dynamic fracture using element-free Galerkin methods. Int. J. Numer. Mech. Eng. 39, 923–938 (1996)CrossRefzbMATHGoogle Scholar
  7. 7.
    Guan, P.C., Chi, S.W., Chen, J.S., et al.: Semi-Lagrangian reproducing kernel particle method for fragment-impact problems. Int. J. Impact Eng. 38, 1033–1047 (2011)CrossRefGoogle Scholar
  8. 8.
    Chi, S., Lee, C., Chen, J.S., et al.: A level set enhanced natural kernel contact algorithm for impact and penetration modeling. Int. J. Numer. Mech. Eng. 102, 839–866 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Liu, W.K., Chen, Y.: Wavelet and multiple scale reproducing kernel methods. Int. J. Numer. Methods Fluids 21, 901–931 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Li, S., Liu, W.K.: Moving least-square reproducing kernel method part II: Fourier analysis. Comput. Methods Appl. Mech. 139, 159–193 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Liu, W.K., Jun, S., Li, S., et al.: Reproducing kernel particle methods for structural dynamics. Int. J. Numer. Mech. Eng. 38, 1655–1679 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liu, W.K., Jun, S., Zhang, Y.: Reproducing kernel particle methods. Int. J. Numer. Methods Fluids 20, 1081–1106 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Liu, W.K., Chen, Y., Jun, S., et al.: Overview and applications of the reproducing kernel particle methods. Arch. Comput. Methods Eng. 3, 3–80 (1996)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bessa, M.A., Foster, J.T., Belytschko, T., et al.: A meshfree unification: reproducing kernel peridynamics. Comput. Mech. 53, 1251–1264 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Carpinteri, A., Mainardi, F.: Fractals and Fractional Calculus in Continuum Mechanics. Springer, Vienna (1997)CrossRefzbMATHGoogle Scholar
  16. 16.
    Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection–dispersion equation. Water Resour. Res. 36, 1403–1412 (2000)CrossRefGoogle Scholar
  18. 18.
    Chen, W., Sun, H., Zhang, X., et al.: Anomalous diffusion modeling by fractal and fractional derivatives. Comput. Math. Appl. 59, 1754–1758 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    West, B.J.: Colloquium: fractional calculus view of complexity: a tutorial. Rev. Mod. Phys. 86, 1169 (2014)CrossRefGoogle Scholar
  20. 20.
    Chen, W., Liang, Y., Hu, S.: Fractional derivative anomalous diffusion equation modeling prime number distribution. Fract. Calc. Appl. Anal. 18, 789–798 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lei, D., Liang, Y., Xiao, R.: A fractional model with parallel fractional Maxwell elements for amorphous thermoplastics. Physica A 450, 465–475 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Xiao, R., Sun, H., Chen, W.: A finite deformation fractional viscoplastic model for the glass transition behavior of amorphous polymers. Int. J. Nonlinear Mech. 93, 7–14 (2017)CrossRefGoogle Scholar
  23. 23.
    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic press, New York (1998)zbMATHGoogle Scholar
  24. 24.
    Li, C., Zeng, F.: Numerical Methods for Fractional Calculus. CRC Press, Boca Raton (2015)zbMATHGoogle Scholar
  25. 25.
    Chen, W., Ye, L., Sun, H.: Fractional diffusion equations by the Kansa method. Comput. Math. Appl. 59, 1614–1620 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pang, G., Chen, W., Sze, K.Y.: A comparative study of finite element and finite difference methods for two-dimensional space-fractional advection–dispersion equation. Adv. Appl. Math. Mech. 8, 166–186 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ding, H., Li, C., Chen, Y.: High-order algorithms for Riesz derivative and their applications (II). J. Comput. Phys. 293, 218–237 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ying, Y., Lian, Y., Tang, S., et al.: High-order central difference scheme for Caputo fractional derivative. Comput. Methods Appl. Mech. 317, 42–54 (2017)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Ervin, V.J., Roop, J.P.: Variational solution of fractional advection dispersion equations on bounded domains in \({\mathbb{R}}^{d}\). Numer. Methods Parial Differ. Equ. 23, 256–281 (2007)CrossRefzbMATHGoogle Scholar
  30. 30.
    Fix, G.J., Roof, J.P.: Least squares finite-element solution of a fractional order two-point boundary value problem. Comput. Math. Appl. 48, 1017–1033 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22, 558–576 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Lian, Y., Ying, Y., Tang, S., et al.: A Petrov–Galerkin finite element method for the fractional advection–diffusion equation. Comput. Methods Appl. Mech. 309, 388–410 (2016)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Luan, S., Lian, Y., Ying, Y., et al.: An enriched finite element method to fractional advection–diffusion equation. Comput. Mech. 60, 181–201 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Zayernouri, M., Karniadakis, G.E.: Fractional Sturm–Liouville eigen-problems: theory and numerical approximation. J. Comput. Phys. 252, 495–517 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Zayernouri, M., Karniadakis, G.E.: Exponentially accurate spectral and spectral element methods for fractional ODEs. J. Comput. Phys. 257, 460–480 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation methods for linear and nonlinear variable order FPDEs. J. Comput. Phys. 293, 312–338 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Kharazmi, E., Zayernouri, M., Karniadakis, G.E.: Petrov–Galerkin and spectral collocation methods for distributed order differential equations. SIAM J. Sci. Comput. 39, A1003–A1037 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Fleming, M., Chu, Y.A., Moran, B., et al.: Enriched element-free Galerkin methods for crack tip fields. Int. J. Numer. Mech. Eng. 40, 1483–1504 (1997)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Liu, W.K., Li, S., Belytschko, T.: Moving least-square reproducing kernel methods (I): methodology and convergence. Comput. Methods Appl. Mech. 143, 113–154 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratory of Computational PhysicsInstitute of Applied Physics and Computational MathematicsBeijingChina
  2. 2.College of EngineeringPeking UniversityBeijingChina
  3. 3.Department of Mechanical EngineeringNorthwestern UniversityEvanstonUSA
  4. 4.HEDPS, CAPT, and LTCSPeking UniversityBeijingChina

Personalised recommendations