Variational formulation for fractional inhomogeneous boundary value problems


The steady state fractional convection diffusion equation with inhomogeneous Dirichlet boundary is considered. By utilizing standard boundary shifting trick, a homogeneous boundary problem is derived with a singular source term which does not belong to \(L^2\) anymore. The variational formulation of such problem is established, based on which the finite element approximation scheme is developed. Inf-sup conditions for both continuous case and discrete case are demonstrated thus the corresponding well-posedness is verified. Furthermore, rigorous regularity analysis for the solutions of both original equation and dual problem is carried out, based on which the error estimates for the finite element approximation are derived. Numerical results are presented to illustrate the theoretical analysis.

This is a preview of subscription content, log in to check access.


  1. 1.

    del-Castillo-Negrete, D.: Fractional diffusion models of nonlocal transport. Phys. Plasmas 13(8), 082308 (2006)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Baeumer, B., Kovács, M., Meerschaert, M.M., Sankaranarayanan, H.: Boundary conditions for fractinal diffusion. J. Comput. Appl. Math. 336, 408–424 (2018)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Gracia, J.L., O’Riordan, E., Stynes, M.: Convergence analysis of a finite difference scheme for a two-point boundary value problem with a Riemann–Liouville–Caputo fractional derivative. Bit Numer. Math. (2019).

    Article  MATH  Google Scholar 

  4. 4.

    Jia, L., Chen, H., Ervin, V.J.: Existence and regularity of solutions to 1-D fractional order diffusion equations. Electron. J. Differ. Equ. 2019(93), 1–21 (2019)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Part Differ. Equ. 22(3), 558–576 (2006)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Ervin, V.J., Roop, J.P.: Variational solution of fractional advection dispersion equations on bounded domains in \(R^{d}\). Numer. Methods Part Differ. Equ. 23(2), 256–281 (2007)

    Article  Google Scholar 

  7. 7.

    Wang, H., Yang, D., Zhu, S.: Inhomogeneous Dirichlet boundary-value problems of space-fractional diffusion equations and their finite element approximations. SIAM J. Numer. Anal. 51(3), 1292–1310 (2014)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Jin, B., Lazarov, R., Pasciak, J., Rundell, W.: Variational formulation of problems involving fractional order differential operators. Math. Comput. 84(296), 2665–2700 (2015)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Jin, B., Lazarov, R., Zhou, Z.: A Petrov–Galerkin finite element method for fractional convection–diffusion equations. SIAM J. Numer. Anal. 54(1), 481–503 (2016)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Zayernouri, M., Karniadakis, G.E.: Fractional Sturm–Liouville eigen-problems. J. Comput. Phys. 252, 495–517 (2013)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Zayernouri, M., Karniadakis, G.E.: Exponentially accurate spectral and spectral element methods for fractional ODEs. J. Comput. Phys. 257, 460–480 (2014)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Chen, S., Shen, J., Wang, L.-L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85(300), 1603–1638 (2015)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Mao, Z., Chen, S., Shen, J.: Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations. Appl. Numer. Math. 106, 165–181 (2016)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Mao, Z., Shen, J.: Spectral element method with geometric mesh for two-sided fractional differential equations. Adv. Comput. Math. 44(3), 745–771 (2018)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Sheng, C., Shen, J.: A hybrid spectral element method for fractional two-point boundary value problems. Numer. Math. Theory Methods Appl. 10(2), 437–464 (2017)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Hou, D., Xu, C.: A fractional spectral method with applications to some singular problems. Adv. Comput. Math. 43(5), 911–944 (2017)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Hou, D., Hasan, M.T., Xu, C.: Müntz spectral methods for the time-fractional diffusion equation. Comput. Methods Appl. Math. 18(1), 43–62 (2018)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Academic Press, New York (1998)

    Google Scholar 

  19. 19.

    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Elsevier, Amsterdam (2003)

    Google Scholar 

  20. 20.

    Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements, Volume 159 of Applied Mathematical Sciences. Springer, New York (2004)

    Google Scholar 

Download references


We would like to thank the anonymous referees for many constructive comments and suggestions which led to an improved presentation of this paper.

Author information



Corresponding author

Correspondence to Zhoushun Zheng.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the National Key Research and Development Program of China (Grant Nos. 2017YFB0701700, 2017YFB0305601).

Communicated by Mihaly Kovacs.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fu, T., Zheng, Z. & Duan, B. Variational formulation for fractional inhomogeneous boundary value problems. Bit Numer Math (2020).

Download citation


  • Fractional calculus
  • Variational formulation
  • Inhomogeneous boundary value
  • Well-posedness

Mathematics Subject Classification

  • 26A33
  • 65N12
  • 65N30