Journal of Scientific Computing

, Volume 70, Issue 1, pp 386–406 | Cite as

A Galerkin Finite Element Method for a Class of Time–Space Fractional Differential Equation with Nonsmooth Data

  • Zhengang ZhaoEmail author
  • Yunying Zheng
  • Peng Guo


In this article, a Galerkin finite element approximation for a class of time–space fractional differential equation is studied, under the assumption that \(u_{tt}, u_{ttt}, u_{2\alpha ,tt}\) are continuous for \(\varOmega \times (0,T]\), but discontinuous at time \(t=0\). In spatial direction, the Galerkin finite element method is presented. And in time direction, a Crank–Nicolson time-stepping is used to approximate the fractional differential term, and the product trapezoidal method is employed to treat the temporal fractional integral term. By using the properties of the fractional Ritz projection and the fractional Ritz–Volterra projection, the convergence analyses of semi-discretization scheme and full discretization scheme are derived separately. Due to the lack of smoothness of the exact solution, the numerical accuracy does not achieve second order convergence in time, which is \(O(k^{3-\beta }+k^{3}t_{n+1}^{-\beta }+k^{3}t_{n+1}^{-\beta -1})\), \(n=0,1,\ldots ,N-1\). But the convergence order in time is shown to be greater than one. Numerical examples are also included to demonstrate the effectiveness of the proposed method.


Time–space fractional differential equation Riesz fractional derivative Crank–Nicolson scheme Product trapezoidal method Fractional Ritz–Volterra projection Galerkin finite element method 



This work was supported by the National Natural Science Foundation of China under Grant Number 11301333; Innovation Program of Shanghai Municipal Education Commission under Grant Number 14YZ165; Funding Scheme for Training Young Teachers in Shanghai Colleges under Grant Number zzhg12001; Natural Science Foundation of Anhui provence under Grant Number 1408085MA1; and Funding Scheme for Training Young Teachers in Shanghai Colleges under Grant Number 14AZ17.


  1. 1.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  2. 2.
    Benson, D.A., Wheatcraft, S.W., Meerschaeert, M.M.: The fractional order governing equations of Lévy motion. Water Resour. Res. 36, 1413–1423 (2000)CrossRefGoogle Scholar
  3. 3.
    Shlesinger, M.F., West, B.J., Klafter, J.: Lévy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58, 1100–1103 (1987)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gurtin, M.E., Pipkin, A.C.: A general theory of heat conduction with finite wave speed. Arch. Ration. Mech. Anal. 31, 113–126 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Miller, R.K.: An integro-differential equation for grid heat conductions with memory. J. Math. Anal. Appl. 66, 313–332 (1978)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Baleanu, D., Güvenc, Z.B., Tenreiro Machado, J.A.: New Trends in Nanotechnology and Fractional Calculus Applications. Springer, Dordrecht (2009)zbMATHGoogle Scholar
  7. 7.
    Christensen, R.M.: Theory of Viscolasticity. Academic Press, New York (1971)Google Scholar
  8. 8.
    Renardy, M.: Mathematical analysis of viscoelastic flows. Ann. Rev. Fluid Mech. 21, 21–36 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput. 191, 12–20 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Yang, Q., Liu, F., Turner, I.: Stability and convergence of an effective numerical method for the time–space fractional Fokker–Planck equation with a nonlinear source term. Int. J. Diff. Eq. (2010). doi: 10.1155/2010/464321
  11. 11.
    Liu, F., Zhuang, P., Burrage, K.: Numerical methods and analysis for a class of fractional advection-dispersion models. Comput. Math. Appl. 64, 2990–3007 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Zhao, Z.G., Li, C.P.: Fractional difference/finite element approximations for the time–space fractional telegraph equation. Appl. Math. Comput. 219, 2975–2988 (2012)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Yu, Q., Liu, F., Turner, I., Burrage, K.: Numerical investigation of three types of space and time fractional Bloch–Torrey equations in 2D. Cent. Eur. J. Phys. 11, 646–665 (2013)Google Scholar
  14. 14.
    Yu, Q., Liu, F., Turner, I., Burrage, K.: Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation. Phil. Trans. R. Soc. (2013). doi: 10.1098/rsta.2012.0150
  15. 15.
    Song, J., Yu, Q., Liu, F., Turner, I.: A spatially second-order accurate implicit numerical method for the space and time fractional Bloch–Torrey equation. Numer. Algo. 66, 911–932 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zhang, H., Liu, F., Zhuang, P., Turner, I., Anh, V.: Numerical analysis of a new space-time variable fractional order advection-dispersion equation. Appl. Math. Comput. 242, 541–550 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Zheng, M., Liu, F., Turner, I., Anh, V.: A novel high order space-time spectral method for the time-fractional Fokker–Planck equation. SIAM J. Sci. Comput. 37, A701–A724 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zeng, F.H., Liu, F., Li, C.P., Burrage, K., Turner, I., Anh, V.: A Crank–Nicolson ADI spectral method for a two-dimensional Riesz fractional nonlinear reaction-diffusion equation. SIAM J. Numer. Anal. 52, 2599–2622 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Li, C.P., Zeng, F.H.: Numerical Methods for Fractional Calculus. Chapman and Hall/CRC, Boca Raton (2015)zbMATHGoogle Scholar
  21. 21.
    Sanz-Serna, J.M.: A numerical method for a partial integro-differential equation. SIAM J. Numer. Anal. 25, 319–327 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lopez-Marcos, J.C.: A difference scheme for a nonlinear partial integro-differential equation. SIAM J. Numer. Anal. 27, 20–31 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Tang, T.: A finite difference scheme for partial integro-differential equation with a weakly singualr kernel. Appl. Numer. Math. 11, 309–319 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Meth. Partial Diff. Equ. 22, 558–576 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    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
  26. 26.
    Deng, W.H.: Finite element method for the space and time fractional Fokker–Planck equation. SIAM J. Numer. Anal. 47, 204–226 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Li, C.P., Zhao, Z.G., Chen, Y.Q.: Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Math. Appl. 62, 855–875 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Bu, W.P., Tang, Y.F., Yang, J.Y.: Galerkin finite element method for two dimensional Riesz space fractional diffusion equations. J. Comput. Phys. 276, 26–38 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zhao, Z.G., Zheng, Y.Y., Guo, P.: A Galerkin finite element scheme for time–space fractional diffusion quation. Int. J. Comput. Math. (2015). doi: 10.1080/00207160.2015.1044986
  30. 30.
    Samko, S.C., Kilbas, A.A., Maxitchev, O.I.: Integrals and Derivatives of the Fractional Order and Some of Their Applications. Nauka i Tekhnika, Minsk (1987). (in Russian) Google Scholar
  31. 31.
    Chen, C.M., Shih, T.: Finite Element Methods for Integrodifferential Equations. Word Scientific, Singapore (1998)CrossRefzbMATHGoogle Scholar
  32. 32.
    Zheng, Y.Y., Li, C.P., Zhao, Z.G.: A note on the finite element method for the space-fractional advection diffusion equation. Comput. Math. Appl. 59, 1718–1726 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York, Berlin (1994)CrossRefzbMATHGoogle Scholar
  34. 34.
    Cannon, J.R., Lin, Y.P.: A priori \(L^{2}\) error estimates for Galerkin methods for nonlinear parbolic integro-differential equations. SIAM J. Numer. Anal. 21, 595–602 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Larsson, S., Thomé, V., Wahlbin, L.B.: Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method. Math. Comput. 67, 45–71 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Ma, J.T.: Finite element method for partial Volterra integro-diffeential equations on two-dimensions unbounded spatial domains. Appl. Math. Comput. 186, 598–609 (2007)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Zeng, F.H., Cao, J.X., Li, C.P.: Gronwall inequalities, In: Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis. World Scientific, Singapore, pp. 49–68 (2013)Google Scholar

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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Fundamental CoursesShanghai Customs CollegeShanghaiChina
  2. 2.School of Mathematical SciencesHuaibei Normal UniversityHuaibeiChina
  3. 3.Department of Mathematics and PhysicsShanghai DianJi UniversityShanghaiChina

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