Journal of Scientific Computing

, Volume 76, Issue 1, pp 166–188 | Cite as

A Spectral Collocation Method for Nonlinear Fractional Boundary Value Problems with a Caputo Derivative

  • Chuanli Wang
  • Zhongqing WangEmail author
  • Lilian Wang


In this paper, we consider the nonlinear boundary value problems involving the Caputo fractional derivatives of order \(\alpha \in (1,2)\) on the interval (0, T). We present a Legendre spectral collocation method for the Caputo fractional boundary value problems. We derive the error bounds of the Legendre collocation method under the \(L^2\)- and \(L^\infty \)-norms. Numerical experiments are included to illustrate the theoretical results.


Spectral collocation method Caputo fractional derivative Fredholm integral equations Convergence analysis 

Mathematics Subject Classification

65N35 45D05 41A05 41A10 41A25 


  1. 1.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)zbMATHGoogle Scholar
  2. 2.
    Chen, S., Shen, J., Wang, L.L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85, 1603–1638 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    del-Castillo-Negrete, D., Carreras, B.A., Lynch, V.E.: Front dynamics in reaction–diffusion systems with Levy fights: a fractional diffusion approach. Phys. Rev. Lett. 91, 018302 (2003)CrossRefGoogle Scholar
  4. 4.
    Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, vol. 2004. Springer, Berlin (2010)zbMATHGoogle Scholar
  5. 5.
    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
  6. 6.
    Esmaeili, S., Shamsi, M.: A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 16, 3646–3654 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ito, K., Jin, B., Takeuchi, T.: On a Legendre tau method for fractional boundary value problems with a Caputo derivative. Fract. Calc. Appl. Anal. 19, 357–378 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jin, B., Lazarov, R., Pasciak, J., Rundell, W.: Variational formulation of problems involving fractional order differential operators. Math. Comput. 84, 2665–2700 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jin, B., Lazarov, R., Pasciak, J., Zhou, Z.: Error analysis of a finite element method for the space-fractional parabolic equation. SIAM J. Numer. Anal. 52, 2272–2294 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  11. 11.
    Kopteva, N., Stynes, M.: An efficient collocation method for a Caputo two-point boundary value problem. BIT 55, 1105–1123 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Li, C., Zeng, F., Liu, F.: Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15, 383–406 (2012)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47, 2108–2131 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Machado, J.T., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16, 1140–1153 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London (2010)CrossRefzbMATHGoogle Scholar
  16. 16.
    Mastroianni, G., Occorsto, D.: Optimal systems of nodes for Lagrange interpolation on bounded intervals: a survey. J. Comput. Appl. Math. 134, 325–341 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mokhtary, P., Ghoreishi, F.: The \(L^2\)-convergence of the Legendre spectral tau matrix formulation for nonlinear fractional integro-differential equations. Numer. Algor. 58, 475–496 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pedas, A., Tamme, E.: Piecewise polynomial collocation for linear boundary value problems of fractional differential equations. J. Comput. Appl. Math. 236, 3349–3359 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, vol. 41. Springer, Berlin (2011)Google Scholar
  20. 20.
    Tadjeran, C., Meerschaert, M.M., Scheffler, H.P.: A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213, 205–213 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    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. 52, 1292–1310 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wang, Z.Q., Guo, Y.L., Yi, L.J.: An \(hp\)-version Legendre–Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels. Math. Comput. 86, 2285–2324 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Wang, Z.Q., Sheng, C.T.: An \(hp\)-spectral collocation method for nonlinear Volterra integral equations with vanishing variable delays. Math. Comput. 85, 635–666 (2016)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.College of ScienceUniversity of Shanghai for Science and TechnologyShanghaiChina
  2. 2.School of Mathematics and StatisticsHenan University of Science and TechnologyLuo YangChina
  3. 3.School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore

Personalised recommendations