Skip to main content
SpringerLink
Log in

A fractional spectral method with applications to some singular problems

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

In this paper we propose and analyze fractional spectral methods for a class of integro-differential equations and fractional differential equations. The proposed methods make new use of the classical fractional polynomials, also known as Müntz polynomials. We first develop a kind of fractional Jacobi polynomials as the approximating space, and derive basic approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces. We then construct efficient fractional spectral methods for some integro-differential equations which can achieve spectral accuracy for solutions with limited regularity. The main novelty of the proposed methods is that the exponential convergence can be attained for any solution u(x) with u(x 1/λ) being smooth, where λ is a real number between 0 and 1 and it is supposed that the problem is defined in the interval (0,1). This covers a large number of problems, including integro-differential equations with weakly singular kernels, fractional differential equations, and so on. A detailed convergence analysis is carried out, and several error estimates are established. Finally a series of numerical examples are provided to verify the efficiency of the methods.

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. Agrawal, O.P.: Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368–379 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 191–206 (2004)

    Article  MathSciNet  Google Scholar 

  3. Bardeh, A.K., Eslahchi, M., Dehghan, M.: A method for obtaining the operational matrix of fractional Jacobi functions and applications. J. Vib. Control 20(5), 736–748 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  4. Benson, D.A., Schumer, R., Meerschaert, M.M., Wheatcraft, S.W.: Fractional dispersion, Lévy motion, and the MADE tracer tests. Transp. Por. Media 42(1/2), 211–240 (2001)

    Article  Google Scholar 

  5. Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36(6), 1403–1412 (2000)

    Article  Google Scholar 

  6. Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: The fractional-order governing equation of Lévy motion. Water Resour. Res. 36, 1413–1423 (2006)

    Article  Google Scholar 

  7. Bernardi, C., Maday, Y.: Spectral methods. Handbook of Numerical Analysis 5, 209–485 (1997)

    MathSciNet  Google Scholar 

  8. Bologna, M.: Asymptotic solution for first and second order linear Volterra integro-differential equations with convolution kernels. J. Phys. A. Math. Theor. 43, 1–13 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Borwein, P., Erdélyi, T., Zhang, J.: Müntz System and Orthogonal Müntz-Legendre Polynomials. Trans. Amer. Math. Soc. 342(3), 523–5421 (1994)

    MATH  MathSciNet  Google Scholar 

  10. Brunner, H.: Polynomial spline collocation methods for Volterra integro-differential equations with weakly singular kernels. IMA J. Numer. Anal. 8, 221–239 (1986)

    Article  MATH  Google Scholar 

  11. Brunner, H.: Collocation methods for Volterra integral and related functional differential equations. Cambridge University Press (2004)

  12. Brunner, H., Pedas, A., Vainikko, G.: Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels. SIAM J. Numer. Anal. 39, 957–982 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chen, S., Shen, J., Wang, L.L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85(300), 1603–1638 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chen, Y.P., Tang, T.: Spectral methods for weakly singular Volterra Integral equations with smooth solutions. J. Comput. Appl. Math. 233, 938–950 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Chen, Y.P., Tang, T.: Convergence analysis of the Jacobi spectral-collocation methods for Volterra Integral equations with a weakly singular kernel. Math. Comput. 79, 147–167 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. Diethelm, K.: The Analysis of Fractional Differential Equations. Springer-Verlag, Berlin (2010)

  17. Diethelm, K., Ford, N.J., Freed, A.D.: Detailed error analysis for a fractional Adams method. Numerical algorithms 36(1), 31–52 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Elliott, W.C.: Introduction to Approximation Theory, 2nd edn. McGraw-Hill Book Co., New York-Toronto (1982)

  19. Esmaeili, S., Shamsi, M., Luchko, Y.: Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials. J. Comput. Math. Appl. 62, 918–929 (2011)

    Article  MATH  Google Scholar 

  20. Gafiychuk, V., Datsko, B., Meleshko, V.: Mathematical modeling of time fractional reactiondiffusion systems. J. Math. Anal. Appl. 220(1-2), 215–225 (2008)

    MATH  Google Scholar 

  21. Gautschi, W.: On Generating Orthogonal Polynomials. SIAM J. Sci. Stat. Comput. 3(3), 289–317 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  22. Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Oxford University Press, New York (2004)

  23. Gorenflof, R., Mainardi, F., Scalas, E., Raberto, M.: Fractional calculus and continuous-time finance iii : The limit, diffusion. Trends Math. 287(3), 171–180 (2000)

    MATH  Google Scholar 

  24. Guo, B.Y., Shen, J., Wang, L.L.: Generalized Jacobi polynomials/functions and their applications. Appl. Numer. Math. 59(5), 1011–1028 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Hardy, G., Littlewood, J., Pólya, G.: Inequalities. Cambridge University Press (1934)

  26. Kazem, S., Abbasbandy, S., Kumar, S.: Fractional-order Legendre functions for solving fractional-order differential equations. J. Appl. Math. Model. 37(7), 5498–5510 (2013)

    Article  MathSciNet  Google Scholar 

  27. Koeller, R.C.: Applcation of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51, 229–307 (1984)

    Article  MathSciNet  Google Scholar 

  28. Kusnezov, D., Bulgac, A., Dang, G.D.: Quantum levy processes and fractional kinetics. Phys. Rev. Lett. 82, 1136–1139 (1999)

    Article  Google Scholar 

  29. Li, C.P., Zeng, F.H., Liu, F.W.: Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15(3), 383–406 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  30. Li, X.J., Xu, C.J.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47(3), 2108–2131 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  31. Li, X.J., Xu, C.J.: Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method, approximation. Comm. Comput. Phy. 8(5), 1016–1051 (2010)

    MATH  MathSciNet  Google Scholar 

  32. Mccarthy, P.C., Sayre, J.E., Shawyer, B.L.R.: Generalized Legendre Polynomials. J. Math. Anal. Appl. 177(2), 530–537 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  33. Meerschaert, M.M., Scalas, E.: Coupled continuous time random walks in finance. Phys. A 390, 114–118 (2006)

    Article  MathSciNet  Google Scholar 

  34. Ortiz, E.L., Pinkus, A., Müntz, H.: A Mathematician’s Odyssey. Math. Intell. 27(1), 22–31 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  35. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

  36. Raberto, M., Scalas, E., Mainardi, F.: Waiting-times and returns in high-frequency finanical data: An empirical study. Phys. A 314, 749–755 (2002)

    Article  MATH  Google Scholar 

  37. Rawashdeh, E., Mcdowell, A., Rakesh, L.: Polynomial spline collocation methods for second-order Volterra equations, integro-differential. IJMMS 56, 3011–3022 (2004)

    MATH  Google Scholar 

  38. Shen, J., Tang, T., Wang, L.L.: Spectral methods, Algorithms, Analysis and Applications. Springer, Berlin (2010)

  39. Shen, J., Wang, Y.W.: Muntz-Galerkin methods and applications to mixed Dirichlet-Neumann boundary value problems. SIAM J. Sci. Comput. 38:A2357–A2381 (2016)

  40. Song, F.Y., Xu, C.J.: Spectral direction splitting methods for two-dimensional space fractional diffusion equations. J. Comput. Phys. 299:196–214 (2015)

  41. Szegö, G.: Orthogonal Polynomials, 4th edition, vol. 23 (1975). AMS Coll. Publ.

  42. Tang, T.: Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations. Numer. Math. 61:373–382 (1992)

  43. Tang, T.: A finite difference scheme for partial integro-differential equations with a weakly singular kernel. Appl. Numer. Math. 11(4), 309–319 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  44. Tang, T., Xu, X., Cheng, J.: On spectral methods for Volterra integral equations and the convergence analysis. J. Comput. Math. 26(6), 825–837 (2008)

    MATH  MathSciNet  Google Scholar 

  45. Tarang, M.: Stability of the spline collocation method for second order Volterra integro-differential equations. Math. Model. Anal. 9, 79–90 (2004)

    MATH  MathSciNet  Google Scholar 

  46. Wang, L.L., Shen, J.: Error Analysis for Mapped Jacobi Spectral Methods. J. Sci. Comput. 24(2), 183–218 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  47. Wei, Y.X., Chen, Y.P.: Convergence analysis of the Legendre spectral collocation methods for second order Volterra equations, integro-differential. Numer. Math. Theor. Meth. Appl. 4, 419–438 (2011)

    MATH  MathSciNet  Google Scholar 

  48. Wei, Y.X., Chen, Y.P.: Convergence Analysis of the Spectral Methods for Weakly Singular Volterra Integro-Differential Solutions, Equations with Smooth. Adv. Appl. Math. Mech. 4(1), 1–20 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  49. Zarebnia, M., Nikpour, Z.: Solution of linear Volterra integro-differential equations via Sinc functions. Int. J. Appl. Math. Comput. 2, 1–10 (2010)

    Google Scholar 

  50. Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36(1), A41—A62 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  51. Zayernouri, M., Karniadakis, G.E.: Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation. J. Comput. Phys. 252, 495–517 (2014)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Xiamen, 361005, China

    Dianming Hou & Chuanju Xu

Authors
  1. Dianming Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chuanju Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chuanju Xu.

Additional information

Communicated by: Martin Stynes

This research is partially supported by NSF of China (Grant numbers 11471274, 11421110001, and 51661135011).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hou, D., Xu, C. A fractional spectral method with applications to some singular problems. Adv Comput Math 43, 911–944 (2017). https://doi.org/10.1007/s10444-016-9511-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-016-9511-y

Keywords

Mathematics Subject Classification (2010)

Search

Navigation