Skip to main content
Log in

A second-order accurate numerical method for a fractional wave equation

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The scheme uses a non-uniform grid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k 2 + h 2, where k and h are the parameters for the time and space meshes, respectively.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Brunner H., Pedas A., Vainikko G. (2001) Piecewise polynomial collocation methods for Volterra integro-differential equations. SIAM J. Numer. Anal. 39, 957–982

    Article  MathSciNet  MATH  Google Scholar 

  2. Chandler G.A., Graham I.G. (1988) Product integration-collocation methods for noncompact integral operator equations. Math. Comp. 50, 125–138

    Article  MathSciNet  MATH  Google Scholar 

  3. Cuesta E., Lubich C., Palencia C. (2006) Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comp. 75, 673–696

    Article  MathSciNet  MATH  Google Scholar 

  4. Fairweather G. (1994) Spline collocation methods for a class of hyperbolic partial integro-differential equations. SIAM J. Numer. Anal. 31, 444–460

    Article  MathSciNet  MATH  Google Scholar 

  5. Fujita Y.: Integral equation that interpolates the heat equation and the wave equation, I, II. Osaka J. Math. 27, 309–321, 797–804 (1990)

    Google Scholar 

  6. Kiryakova V.S. (2000) Multiple (multiindex) Mittag–Leffler functions and relation to generalized fractional calculus. J. Comput. Appl. Math. 118, 241–259

    Article  MathSciNet  MATH  Google Scholar 

  7. López-Fernández M., Palencia C. (2004) On the numerical inversion of the Laplace transform of certain holomorphic mappings. Appl. Numer. Math. 51, 289–303

    Article  MathSciNet  MATH  Google Scholar 

  8. López-Fernández M., Palencia C. (2006) A spectral order method for inverting sectorial Laplace transforms. SIAM J. Numer. Anal. 44, 1332–1350

    Article  MathSciNet  MATH  Google Scholar 

  9. López Marcos J.-C. (1990) A difference scheme for a nonlinear partial integrodifferential equation. SIAM J. Numer. Anal. 27, 20–31

    Article  MathSciNet  MATH  Google Scholar 

  10. Lubich Ch. (1986) Discretized fractional calculus. SIAM J. Math. Anal. 17, 704–719

    Article  MathSciNet  MATH  Google Scholar 

  11. Lubich C. (1988) Convolution quadrature and discretized operational calculus, I, II. Numer. Math. 53, 129–145, 413–425

    Article  MathSciNet  Google Scholar 

  12. Lubich C.H., Sloan I.H., Thomée V. (1996) Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term. Math. Comp. 65, 1–17

    Article  MathSciNet  MATH  Google Scholar 

  13. McLean W., Mustapha K.: A second-order accurate numerical method for a fractional wave equation. Preprint AMR05/37, School of Mathematics, The University of New South Wales

  14. McLean W., Thomée V. (1993) Numerical solution of an evolution equation with a positive-type memory term. J. Austral. Math. Soc. Ser. B 35, 23–70

    Article  MathSciNet  MATH  Google Scholar 

  15. McLean W., Thomée V. (2004) Time discretization of an evolution equation via Laplace transform. IMA. J. Numer. Anal. 24, 439–463

    Article  MathSciNet  MATH  Google Scholar 

  16. McLean W., Thomée V., Wahlbin L.B. (1996) Discretization with variable time steps of an evolution equation with a positive-type memory term. J. Comput. Appl. Math. 69, 49–69

    Article  MathSciNet  MATH  Google Scholar 

  17. Metzler R., Klafter J. (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Reports 339, 1–77

    Article  MathSciNet  MATH  Google Scholar 

  18. Mittag–Leffler G. (1904) Sur la représentation analytique d’une branche uniforme d’une fonction monogéne. Acta Math. 29, 101–181

    Article  MathSciNet  Google Scholar 

  19. Pani A., Fairweather G. (2002) An H 1-Galerkin mixed finite element method for an evolution equation with a positive type memory term. SIAM J. Numer. Anal. 40: 1475–1490

    Article  MathSciNet  MATH  Google Scholar 

  20. te Riele H.J.J. (1982) Collocation methods for weakly singular second-kind Volterra integral equations with nonsmooth solution. IMA J. Numer. Anal. 2, 437–449

    MathSciNet  MATH  Google Scholar 

  21. Schneider W.R., Wyss W. (1989) Fractional diffusion and wave equations. J. Math. Phys. 30, 134–144

    Article  MathSciNet  MATH  Google Scholar 

  22. Sanz-Serna M.J. (1988) A numerical method for a partial integro-differential equation. SIAM J. Numer. Anal. 25, 319–327

    Article  MathSciNet  MATH  Google Scholar 

  23. Thomée V.: Galerkin finite element methods for parabolic problems, Lecture Notes in Mathematics, vol. 1054. Springer, Berlin Heidelberg Newyork (1984)

  24. Yan Y., Fairweather G. (1992) Orthogonal spline collocation methods for some partial integrodifferential equations. SIAM J. Numer. Anal. 29, 755–768

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to William McLean.

Rights and permissions

Reprints and permissions

About this article

Cite this article

McLean, W., Mustapha, K. A second-order accurate numerical method for a fractional wave equation. Numer. Math. 105, 481–510 (2007). https://doi.org/10.1007/s00211-006-0045-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-006-0045-y

Mathematics Subject Classification (2000)

Navigation