Skip to main content
SpringerLink
Log in

On a Legendre Tau Method for Fractional Boundary Value Problems with a Caputo Derivative

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

In this paper, we revisit a Legendre-tau method for two-point boundary value problems with a Caputo fractional derivative in the leading term, and establish an L2 error estimate for smooth solutions. Further, we apply the method to the Sturm-Liouville problem. Numerical experiments indicate that for the source problem, it converges steadily at an algebraic rate even for nonsmooth data, and the convergence rate enhances with problem data regularity, whereas for the Sturm-Liouville problem, it always yields excellent convergence for eigenvalue approximations.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D.A. Benson, S.W. Wheatcraft, M.M. Meerschaert, The fractional-order governing equation of Lévy motion. Water Resour. Res. 36, No 6 (2000), 1413–1424; DOI: 10.1029/2000WR900032.

    Article  Google Scholar 

  2. C. Bernardi, Y. Maday, Spectral methods. Handbook of Numerical Analysis Ser. Handb. Numer. Anal. V North-Holland, Amsterdam, (1997), 209–485; DOI: 10.1016/S1570-8659(97)80003-8.

    Article  MathSciNet  Google Scholar 

  3. A.H. Bhrawy, M.M. Al-Shomrani, A shifted Legendre spectral method for fractional-order multi-point boundary value problems. Adv. Diff. Eq. 2012, No 1 (2012), 1–19; DOI: 10.1186/1687-1847-2012-8.

    Article  MathSciNet  Google Scholar 

  4. C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics. Springer-Verlag, New York, (1988).

    Book  Google Scholar 

  5. C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods. Springer-Verlag, Berlin, (2006).

    Book  Google Scholar 

  6. D. del-Castillo-Negrete, B.A. Carreras, V.E. Lynch, Front dynamics in reaction-diffusion systems with Levy flights. Phys. Rev. Lett. 91, No 1 (2003), 018302–4; DOI: 10.1103/PhysRevLett.91.018302.

    Article  Google Scholar 

  7. V.J. Ervin, J.P. Roop, Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Diff. Eq. 22, No 3 (2006), 558–576; DOI: 10.1002/num.20112.

    Article  MathSciNet  Google Scholar 

  8. N.J. Ford, M.L. Morgado, M. Rebelo, Nonpolynomial collocation approximation of solutions to fractional differential equations. Fract. Calc. Appl. Anal. 16, No 4 (2013), 874–891; DOI: 10.2478/s13540-013-0054-3; http://www.degruyter.com/view/j/fca.2013.16.issue-4/issue-files/fca.2013.16.issue-4.xml.

    Article  MathSciNet  Google Scholar 

  9. I.M. Gel’fand, G.E. Shilov, Generalized Functions. I Academic Press, New York, (1964).

  10. R. Gorenflo, Y. Luchko, M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev spaces. Fract. Calc. Appl. Anal. 18, No 3 (2015), 799–820; DOI: 10.1515/fca-2015-0048; http://www.degruyter.com/view/j/fca.2015.18.issue-3/issue-files/fca.2015.18.issue-3.xml.

    Article  MathSciNet  Google Scholar 

  11. J.L. Gracia, M. Stynes, Formal consistency versus actual convergence rates of difference schemes for fractional-derivative boundary value problems. Fract. Calc. Appl. Anal. 18, No 2 (2015), 419–436; DOI: 10.1515/fca-2015-0027; http://www.degruyter.com/view/j/fca.2015.18.issue-2/issue-files/fca.2015.18.issue-2.xml.

    Article  MathSciNet  Google Scholar 

  12. E. Hanert, On the numerical solution of space-time fractional diffusion models. Comput. Fluids. 46 (2011), 33–39; DOI: 10.1016/j.compfluid.2010.08.010.

    Article  MathSciNet  Google Scholar 

  13. J. Henderson, N. Kosmatov, Eigenvalue comparison for fractional boundary value problems with the Caputo derivative. Fract. Calc. Appl. Anal. 17, No 3 (2014), 872–880; DOI: 10.2478/s13540-014-0202-4; http://www.degruyter.com/view/j/fca.2014.17.issue-3/issue-files/fca.2014.17.issue-3.xml.

    Article  MathSciNet  Google Scholar 

  14. K. Ito, B. Jin, T. Takeuchi, On the sectorial property of the Caputo derivative operator. Appl. Math. Lett. 47 (2015), 43–46; DOI: 10.1016/j.aml.2015.03.001.

    Article  MathSciNet  Google Scholar 

  15. K. Ito, R. Teglas, Legendre-tau approximations for functional-differential equations. SIAM J. Control Optim. 24, No 4 (1986), 737–759; DOI: 10.1137/0324046.

    Article  MathSciNet  Google Scholar 

  16. B. Jin, R. Lazarov, J. Pasciak, W. Rundell, Variational formulation of problems involving fractional order differential operators. Math. Comp. 84, No 296 (2015), 2665–2700; DOI: 10.1090/mcom/2960.

    Article  MathSciNet  Google Scholar 

  17. B. Jin, R. Lazarov, J. Pasciak, Z. Zhou, Error analysis of a finite element method for the space-fractional parabolic equation. SIAM J. Numer. Anal. 52, No 5 (2014), 2272–2294; DOI: 10.1137/13093933X.

    Article  MathSciNet  Google Scholar 

  18. B. Jin, W. Rundell, An inverse Sturm-Liouville problem with a fractional derivative. J. Comput. Phys. 231, No 14 (2012), 4954–4966; DOI: 10.1016/j.jcp.2012.04.005.

    Article  MathSciNet  Google Scholar 

  19. B. Jin, Z. Zhou, A singularity reconstructed finite element method for fractional boundary value problems. ESAIM Math. Model. Numer. Anal. 49, No 5 (2015), 1261–1283; DOI: 10.1051/m2an/2015010.

    Article  MathSciNet  Google Scholar 

  20. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, (2006).

    MATH  Google Scholar 

  21. C. Li, F. Zeng, F. Liu, Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15, No 3 (2012), 383–406; DOI: 10.2478/s13540-012-0028-x; http://www.degruyter.com/view/j/fca.2012.15.issue-3/issue-files/fca.2012.15.issue-3.xml.

    Article  MathSciNet  Google Scholar 

  22. X. Li, C. Xu, A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47, No 3 (2009), 2108–2131; DOI: 10.1137/080718942.

    Article  MathSciNet  Google Scholar 

  23. X. Li, C. Xu, Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Commun. Comput. Phys. 8, No 5 (2010), 1016–1051.

    Article  MathSciNet  Google Scholar 

  24. J.-L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications. I Springer-Verlag, New York, (1972).

  25. P. Mokhtary, F. Ghoreishi, The L2-convergence of the Legendre spectral tau matrix formulation for nonlinear fractional integro differential equations. Numer. Algor. 58 (2011), 475–496; DOI: 10.1007/s11075-011-9465-6.

    Article  Google Scholar 

  26. A. Pedas, E. Tamme, Piecewise polynomial collocation for linear boundary value problems of fractional differential equations. J. Comput. Appl. Math. 236, No 13 (2012), 3349–3359; DOI: 10.1016/j.cam.2012.03.002.

    Article  MathSciNet  Google Scholar 

  27. C. Shen, M.S. Phanikumar, An efficient space-fractional dispersion approximation for stream solute transport modeling. Adv. Water Res. 32, No 10 (2009), 1482–1494; DOI: 10.1016/j.advwatres.2009.07.001.

    Article  Google Scholar 

  28. J. Shen, A spectral-tau approximation for the Stokes and Navier-Stokes equations. RAIRO Modél. Math. Anal. Numér. 22, No 4 (1988), 677–693.

    Article  MathSciNet  Google Scholar 

  29. E. Sousa, How to approximate the fractional derivative of order 1 < α ≤ 2. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22, No 4 (2012), 1250075, pp13; DOI: 10.1142/S0218127412500757.

    Article  MathSciNet  Google Scholar 

  30. M. Stynes, J.L. Gracia, A finite difference method for a two-point boundary value problem with a Caputo fractional derivative. IMA J. Numer. Anal. 35, No 2 (2015), 698–721; DOI: 10.1093/imanum/dru011.

    Article  MathSciNet  Google Scholar 

  31. C. Tadjeran, M.M. Meerschaert, H.-P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213, No 1 (2006), 205–213; DOI: 10.1016/j.jcp.2005.08.008.

    Article  MathSciNet  Google Scholar 

  32. W.Y. Tian, W. Deng, Y. Wu, Polynomial spectral collocation method for space fractional advection-diffusion equation. Numer. Methods Partial Diff. Eq. 30, No 2 (2014), 514–535; DOI: 10.1002/num.21822.

    Article  MathSciNet  Google Scholar 

  33. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam, (1978).

    MATH  Google Scholar 

  34. H. Wang, D. Yang, S. Zhu, Inhomogeneous Dirichlet boundary-value problems of space-fractional diffusion equations and their finite element approximations. SIAM J. Numer. Anal. 52, No 3 (2014), 1292–1310; DOI: 10.1137/130932776.

    Article  MathSciNet  Google Scholar 

  35. X. Zhang, M. Lv, J.W. Crawford, I.M. Young, The impact of boundary on the fractional advection-dispersion equation for solute transport in soil: Defining the fractional dispersive flux with the Caputo derivatives. Adv. Water Res. 30, No 5 (2007), 1205–1217; DOI: 10.1016/j.advwatres.2006.11.002.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Kazufumi Ito

  2. Department of Computer Science, University College London Gower Street, London, WC1E 6BT, UK

    Bangti Jin

  3. Institute of Industrial Science, The University of Tokyo, Tokyo, Japan

    Tomoya Takeuchi

Authors
  1. Kazufumi Ito
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bangti Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tomoya Takeuchi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kazufumi Ito.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ito, K., Jin, B. & Takeuchi, T. On a Legendre Tau Method for Fractional Boundary Value Problems with a Caputo Derivative. FCAA 19, 357–378 (2016). https://doi.org/10.1515/fca-2016-0019

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1515/fca-2016-0019

MSC 2010

Key Words and Phrases

Search

Navigation