Skip to main content
Springer Nature Link
Log in

A Compact Difference Scheme for Time Fractional Diffusion Equation with Neumann Boundary Conditions

  • Conference paper
AsiaSim 2012 (AsiaSim 2012)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 323))

Included in the following conference series:

  • 2026 Accesses

Abstract

This paper is devoted to the numerical treatment of time fractional diffusion equation with Neumann boundary conditions. A compact difference scheme is derived for solving this problem, by combining the classic finite difference method for Caputo derivative in time, the second order central difference method in space and the compact difference treatment for Neumann boundary conditions. The solvability, stability and convergence of this scheme are rigorously discussed. We prove that the convergence order of this proposed scheme is O(τ2 − α + h2), where τ, α and h are the time step size, the index of fractional derivative and space step size respectively. Numerical experiments are carried out to demonstrate the theoretical analysis.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36, 1403–1412 (2006)

    Article  Google Scholar 

  2. 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 

  3. Gorenflo, R., Mainardi, F., Scalas, E., Raberto, M.: Fractional calculus and continuous-time finace. III, The diffusion limit. In: Mathematical Finance. Trends in Math., pp. 171–180. Birkhäuser, Basel (2001)

    Chapter  Google Scholar 

  4. Gorenflo, R., Mainardi, F., Moretti, D., Paradisi, P.: Time fractional diffusion: a discrete random walk approach. Nonlinear Dynam. 29, 129–143 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Langlands, T.A.M., Henry, B.I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205, 719–736 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Lin, Y.M., Xu, C.J.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Liu, F., Shen, S., Anh, V., Turner, I.: Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation. ANZIAM J. 46(E), 488–504 (2005)

    MathSciNet  Google Scholar 

  10. Mainardi, F.: Fractional calculus: Some basicproblems in continuum and statisticalmechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics. Springer, Wien (1997)

    Google Scholar 

  11. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Meerschaert, M.M., Scalas, E.: Coupled continuous time random walks in finance. Phys. A 370, 114–118 (2006)

    Article  MathSciNet  Google Scholar 

  13. Podlubny, I.: Fractional differential equations. Academic Press, New York (1999)

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  15. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional integrals and derivative: theory and applications. Gordon and Breach, New York (1993)

    MATH  Google Scholar 

  16. Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Sun, Z.Z.: Compact difference schemes for heat equation with Neumann boundary conditions. Numer. Meth. Part. D. E. 25, 1320–1341 (2009)

    Article  MATH  Google Scholar 

  18. Yuste, S.B.: Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216, 264–274 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Zhao, J., Dai, W.Z., Niu, T.C.: Fourth-order compact schemes of a heat conduction problem with Neumann boundary conditions. Numer. Meth. Part. D. E. 23, 949–959 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Zhao, X., Sun, Z.Z.: A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 230, 6061–6074 (2011)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Jianfei Huang, Yifa Tang & Jiye Yang

  2. Yuanpei College, Peking University, Beijing, 100871, China

    Wenjia Wang

Authors
  1. Jianfei Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yifa Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wenjia Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jiye Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. National CIMS Engineering Research Center, Department of Automation, Tsinghua University, 100084, Beijing, China

    Tianyuan Xiao

  2. School of Automation Science and Electrical Engineering, Beihang University, 100191, Beijing, China

    Lin Zhang

  3. School of Mechatronics Engineering and Automation, Shanghai University, 200072, Shanghai, China

    Minrui Fei

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Huang, J., Tang, Y., Wang, W., Yang, J. (2012). A Compact Difference Scheme for Time Fractional Diffusion Equation with Neumann Boundary Conditions. In: Xiao, T., Zhang, L., Fei, M. (eds) AsiaSim 2012. AsiaSim 2012. Communications in Computer and Information Science, vol 323. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34384-1_33

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-34384-1_33

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-34383-4

  • Online ISBN: 978-3-642-34384-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics