Journal of Scientific Computing

, Volume 79, Issue 2, pp 700–717 | Cite as

Finite Element Method for Two-Sided Fractional Differential Equations with Variable Coefficients: Galerkin Approach

  • Zhaopeng Hao
  • Moongyu ParkEmail author
  • Guang Lin
  • Zhiqiang Cai


This paper develops a Galerkin approach for two-sided fractional differential equations with variable coefficients. By the product rule, we transform the problem into an equivalent formulation which additionally introduces the fractional low-order term. We prove the existence and uniqueness of the solutions of the Dirichlet problems of the equations with certain diffusion coefficients. We adopt the Galerkin formulation, and prove its error estimates. Finally, several numerical examples are provided to illustrate the fidelity and accuracy of the proposed theoretical results.


Fractional diffusion equation Two-sided fractional derivative Galerkin methods Error estimate 

Mathematics Subject Classification

26A33 65M06 65M12 65M55 65T50 



Hao would like to acknowledge the support by National Natural Science Foundation of China (No. 11671083), China Scholarship Council (No. 201506090065). Lin gratefully acknowledges the support from National Science Foundation (DMS-1555072, DMS-1736364, and DMS-1821233). Cai would like to acknowledge the support by the NSF Grant DMS-1522707.


  1. 1.
    Adams, R.A., Fournier, J.F.: Sobolev Spaces. Academic Press, New York (2003)zbMATHGoogle Scholar
  2. 2.
    Benson, D.A., Tadjeran, C., Meerschaert, M.M., Farnham, I., Pohll, G.: Radial fractional-order dispersion through fractured rock. Water Resour. Res. 40, 1–9 (2004)CrossRefGoogle Scholar
  3. 3.
    Benson, D.A., Schumer, R., Meerschaert, M.M., Wheatcraft, S.W.: Fractional dispersion, Levy motion, and the made tracer tests. Transp. Porous Media 42, 211–240 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brenner, S., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)CrossRefzbMATHGoogle Scholar
  5. 5.
    Carreras, B.A., Lynch, V.E., Zaslavsky, G.M.: Anomalous diffusion and exit time distribution of particle travers in plasma turbulence models. Phys. Plasma 8, 5096–5103 (2001)CrossRefGoogle Scholar
  6. 6.
    Chen, M.H., Deng, W.H.: Fourth order accurate scheme for the space fractional diffusion equatioins. SIAM J. Numer. Anal. 52(3), 1418–1438 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    De Nápoli, P.L., Drelichman, I.: Elementary proofs of embedding theorems for potential spaces of radial functions. In: Ruzhansky, M., Tikhonov, S. (eds.) Methods of Fourier Analysis and Approximation Theory. Applied and Numerical Harmonic Analysis, pp. 115–138. Birkhauser, Cham (2016)CrossRefGoogle Scholar
  8. 8.
    Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22(3), 558–576 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ervin, V.J., Heuer, N., Roop, J.P.: Regularity of the solution to 1-D fractional order diffusion equations. Math. Comput. 87, 2273–2294 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hao, Z., Sun, Z., Cao, W.: A fourth-order approximation of fractional derivatives with its applications. J. Comput. Phys. 281, 787–805 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hilfer, R.: Applications of Fractional Calculus in Physics. Word Scientific, Singapore (2000)CrossRefzbMATHGoogle Scholar
  13. 13.
    Kirchner, J.W., Feng, X., Neal, C.: Fractal stream chemistry and its implications for containant transport in catchments. Nature 403, 524–526 (2000)CrossRefGoogle Scholar
  14. 14.
    Li, X., Xu, C.: The existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Commun. Comput. Phys. 8, 1016–1051 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker–Planck equation. J. Comput. Appl. Math. 166, 209–219 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publishers, Danbury (2006)Google Scholar
  17. 17.
    Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection–dispersion flow equations. J. Comput. Phys. 172, 65–77 (2004)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56, 80–90 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)zbMATHGoogle Scholar
  20. 20.
    Sabatelli, L., Keating, S., Dudley, J., Richmond, P.: Waiting time distributions in financial markets. Eur. Phys. J. B 27, 273–27 (2002)MathSciNetGoogle Scholar
  21. 21.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Yverdon (1993)zbMATHGoogle Scholar
  22. 22.
    Schumer, R., Benson, D.A., Meerschaert, M.M., Wheatcraft, S.W.: Eulerian derivation of the fractional advection–dispersion equation. J. Contam. Hydrol. 48(1), 69–88 (2001)CrossRefGoogle Scholar
  23. 23.
    Shlesinger, M.F., West, B.J., Klafter, J.: Levy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58, 1100–1103 (1987)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Tadjeran, C., Meerschaert, M.M., Scheffler, H.-P.: A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213(1), 205–213 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wang, H., Yang, D.: Wellposedness of variable-coefficient conservative fractional elliptic differential equations. SIAM J. Numer. Anal. 51(2), 1088–1107 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wang, H., Basu, T.S.: A fast finite difference method for two-dimensional space-fractional diffusion equations. SIAM J. Sci. Comput. 34, A2444–A2458 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wang, H., Wang, K.: An \(O(N \log ^2 N)\) alternating-direction finite difference method for two-dimensional fractional diffusion equations. J. Comput. Phys. 230, 7830–7839 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wang, H., Yang, D., Zhu, S.: A Petrov–Galerkin finite element method for variable-coefficient fractional diffusion equations. Comput. Methods Appl. Mech. Eng. 290, 45–56 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wang, H., Yang, D., Zhu, S.: Accuracy of finite element methods for boundary-value problems of steady-state fractional diffusion equations. J. Sci. Comput. 70, 429–449 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Xu, Q., Hesthaven, J.S.: Discontinuous Galerkin method for fractional convection diffusion equations. SIAM J. Numer. Anal. 52, 405–423 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Yang, D.P., Wang, H.: Wellposedness and regularity of steady-state two-sided variable-coefficient conservative space-fractional diffusion equations (2016). arXiv:1606.04912 [math.NA]
  32. 32.
    Zaslavsky, G.M., Stevens, D., Weitzner, H.: Self-similar transport in incomplete chaos. Phys. Rev. E 48, 1683–1694 (1993)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsSoutheast UniversityNanjingPeople’s Republic of China
  2. 2.Department of Mathematics and StatisticsOakland UniversityRochesterUSA
  3. 3.Department of MathematicsPurdue UniversityWest LafayetteUSA
  4. 4.School of Mechanical EngineeringPurdue UniversityWest LafayetteUSA

Personalised recommendations