Article
Licensed
Unlicensed Requires Authentication

On a Legendre Tau method for fractional boundary value problems with a Caputo derivative

  • EMAIL logo , and
Published/Copyright: May 4, 2016
Become an author with De Gruyter Brill
Author Information Explore this Subject
Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 19 Issue 2

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 indicatethat 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.

Key Words and Phrases: Caputo fractional derivative; Legendre tau method; error estimates; eigenvalue approximation
MSC 2010: Primary 65N35; Secondary 65N15, 34B10

Acknowledgements

The work of the second author (B. Jin) is partially supported by NSFDMS 1319052.

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: 10.1029/2000WR900032.10.1029/2000WR900032Search in Google Scholar

[2] C. Bernardi, Y. Maday, Spectral methods. In:Handbook of Numerical Analysis. Ser. Handb. Numer. Anal., Vol.V, North-Holland, Amsterdam (1997), 209–485; DOI: 10.1016/S1570-8659(97)80003-8.10.1016/S1570-8659(97)80003-8Search in 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: 10.1186/1687-1847-2012-8.10.1186/1687-1847-2012-8Search in Google Scholar

[4] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics. Springer-Verlag, New York (1988).10.1007/978-3-642-84108-8Search in Google Scholar

[5] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods. Springer-Verlag, Berlin (2006).10.1007/978-3-540-30726-6Search in 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 pp; DOI: 10.1103/PhysRevLett.91.018302.10.1103/PhysRevLett.91.018302Search in Google Scholar PubMed

[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.20112.10.1002/num.20112Search in 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.10.2478/s13540-013-0054-3;Search in Google Scholar

[9] I.M. Gel’fand, G.E. Shilov, Generalized Functions, Vol.I. Academic Press, New York (1964).Search in Google Scholar

[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.10.1515/fca-2015-0048;Search in 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.10.1515/fca-2015-0027;Search in Google Scholar

[12] E. Hanert, On the numerical solution of space-time fractional diffusion models. Comput. Fluids46, (2011), 33–39; DOI: 10.1016/j.compfluid.2010.08.010.10.1016/j.compfluid.2010.08.010Search in 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.xmlfca.2014.17.issue-3.xml.10.2478/s13540-014-0202-4;Search in 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.2015.03.001.10.1016/j.aml.2015.03.001Search in 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: 10.1137/0324046.10.1137/0324046Search in 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/2960.10.1090/mcom/2960Search in 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.10.1137/13093933XSearch in 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.2012.04.005.10.1016/j.jcp.2012.04.005Search in 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/2015010.10.1051/m2an/2015010Search in Google Scholar

[20] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).Search in 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.10.2478/s13540-012-0028-x;Search in 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: 10.1137/080718942.10.1137/080718942Search in 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), 1016–1051.10.4208/cicp.020709.221209aSearch in Google Scholar

[24] J.-L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol.I. Springer-Verlag, New York (1972).10.1007/978-3-642-65161-8Search in Google Scholar

[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/s11075-011-9465-6.10.1007/s11075-011-9465-6Search in 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.2012.03.002.10.1016/j.cam.2012.03.002Search in 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.2009.07.001.10.1016/j.advwatres.2009.07.001Search in 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), 677–693.10.1051/m2an/1988220406771Search in 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, 13pp; DOI: 10.1142/S0218127412500757.10.1142/S0218127412500757Search in 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/dru011.10.1093/imanum/dru011Search in 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.2005.08.008.10.1016/j.jcp.2005.08.008Search in 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.21822.10.1002/num.21822Search in Google Scholar

[33] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978).Search in 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: 10.1137/130932776.10.1137/130932776Search in 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.2006.11.002.10.1016/j.advwatres.2006.11.002Search in Google Scholar

Received: 2015-2-13
Revised: 2015-9-1
Published Online: 2016-5-4
Published in Print: 2016-4-1

© 2016 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA-Volume 19-2-2016)
  4. Survey Paper
  5. A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations
  6. Survey Paper
  7. A free fractional viscous oscillator as a forced standard damped vibration
  8. Research Paper
  9. On a Legendre Tau method for fractional boundary value problems with a Caputo derivative
  10. Research Paper
  11. Nonlinear Dirichlet problem with non local regional diffusion
  12. Research Paper
  13. Generalized fraction evolution equations with fractional Gross Laplacian
  14. Research Paper
  15. A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation
  16. Research Paper
  17. Convolutional approach to fractional calculus for distributions of several variables
  18. Research Paper
  19. Existence theorems for semi-linear Caputo fractional differential equations with nonlocal discrete and integral boundary conditions
  20. Research Paper
  21. Nonlinear Riemann-Liouville fractional differential equations with nonlocal Erdelyi-Kober fractional integral conditions
  22. Research Paper
  23. Spatial dispersion of elastic waves in a bar characterized by tempered nonlocal elasticity
  24. Research Paper
  25. Applications of the fractional Sturm-Liouville problem to the space-time fractional diffusion in a finite domain
  26. Short Paper
  27. Measurement of para-xylene diffusivity in zeolites and analyzing desorption curves using the Mittag-Leffler function
  28. Short Paper
  29. Monotonicity of functions and sign changes of their Caputo derivatives
  30. Short Note
  31. On the Volterra μ-functions and the M-Wright functions as kernels and eigenfunctions of Volterra type integral operators
https://doi.org/10.1515/fca-2016-0019
Keywords for this article
Caputo fractional derivative; Legendre tau method; error estimates; eigenvalue approximation

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA-Volume 19-2-2016)
  4. Survey Paper
  5. A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations
  6. Survey Paper
  7. A free fractional viscous oscillator as a forced standard damped vibration
  8. Research Paper
  9. On a Legendre Tau method for fractional boundary value problems with a Caputo derivative
  10. Research Paper
  11. Nonlinear Dirichlet problem with non local regional diffusion
  12. Research Paper
  13. Generalized fraction evolution equations with fractional Gross Laplacian
  14. Research Paper
  15. A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation
  16. Research Paper
  17. Convolutional approach to fractional calculus for distributions of several variables
  18. Research Paper
  19. Existence theorems for semi-linear Caputo fractional differential equations with nonlocal discrete and integral boundary conditions
  20. Research Paper
  21. Nonlinear Riemann-Liouville fractional differential equations with nonlocal Erdelyi-Kober fractional integral conditions
  22. Research Paper
  23. Spatial dispersion of elastic waves in a bar characterized by tempered nonlocal elasticity
  24. Research Paper
  25. Applications of the fractional Sturm-Liouville problem to the space-time fractional diffusion in a finite domain
  26. Short Paper
  27. Measurement of para-xylene diffusivity in zeolites and analyzing desorption curves using the Mittag-Leffler function
  28. Short Paper
  29. Monotonicity of functions and sign changes of their Caputo derivatives
  30. Short Note
  31. On the Volterra μ-functions and the M-Wright functions as kernels and eigenfunctions of Volterra type integral operators
Downloaded on 2.4.2026 from https://www.degruyterbrill.com/document/doi/10.1515/fca-2016-0019/html
Scroll to top button