Advertisement

Numerical Algorithms

, Volume 57, Issue 3, pp 313–327 | Cite as

On the spline collocation method for the single layer equation related to time-fractional diffusion

  • Jukka T. Kemppainen
  • Keijo Matti Ruotsalainen
Original Paper

Abstract

The boundary element spline collocation method is studied for the time-fractional diffusion equation in a bounded two-dimensional domain. We represent the solution as the single layer potential which leads to a Volterra integral equation of the first kind. We discretize the boundary integral equation with the spline collocation method on uniform meshes both in spatial and time variables. In the stability analysis we utilize the Fourier analysis technique developed for anisotropic pseudodifferential equations. We prove that the collocation solution is quasi-optimal under some stability condition for the mesh parameters. We have to assume that the mesh parameter in time satisfies \((h_t=c h^{\frac{2}{\alpha}})\), where (h) is the spatial mesh parameter.

Keywords

Boundary element method Single layer potential Time-fractional diffusion Spline collocation Fourier analysis Anisotropic pseudodifferential operators 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnold, D., Saranen, J.: On the asymptotic convergence of spline collocation methods for partial differential equations. SIAM J. Numer. Anal. 21, 459–472 (1984)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Arnold, D.N., Wendland, W.L.: On the asymptotic convergence of collocation methods. Math. Comput. 41, 349–381 (1983)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Costabel, M.: Boundary integral operators for the heat equation. Int. Eq. Oper. Th. 13(4), 498–552 (1990)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Costabel, M., Saranen, J.: Spline collocation for convolutional parabolic boundary integral equations. Numer. Math. 84, 417–449 (2000)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Costabel, M., Saranen, J.: Parabolic boundary integral operators, symbolic representations and basic properties. Int. Eq. Oper. Th. 40, 185–211 (2001)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Hsiao, G., Saranen, J.: Coercivity of the Single Layer Heat Operator, Report 89-2. Center for Mathematics and Waves. Newark, Delaware (1989)Google Scholar
  7. 7.
    Hsiao, G.C., Saranen, J.: Boundary integral solution of the two-dimensional heat equation. Math. Methods Appl. Sci. 16, 87–114 (1993)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Hämäläinen, J.: Spline collocation for the single layer heat equation. Ann. Acad. Sci. Fenn. Math. Diss. 113, 67 pp. (1998)Google Scholar
  9. 9.
    Hörmander, L.: The analysis of linear partial differential operators I. Distribution Theory and Fourier Analysis. Springer-Verlag, Berlin (2003)MATHGoogle Scholar
  10. 10.
    Kemppainen, J., Ruotsalainen, K.: Boundary integral solution of the time-fractional diffusion equation. Int. Eq. Oper. Th. 64, 239–249 (2009)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Kemppainen, J., Ruotsalainen, K.: Boundary element collocation method for time-fractional diffusion equation. In: Proc. of the 10th International Conference on Integral Methods in Science and Engineering. Birkhäuser, 7–10 July 2008 (2010)Google Scholar
  12. 12.
    Kilbas, A.A., Saigo, M.: H-transforms: Theory and Applications. CRC, LLC (2004)MATHCrossRefGoogle Scholar
  13. 13.
    Lions, J.L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications I. Springer, Berlin (1972)MATHGoogle Scholar
  14. 14.
    Lions, J.L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications II. Springer, Berlin (1972)MATHGoogle Scholar
  15. 15.
    Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series, vol. 3. More special functions. Overseas Publishers Association, Amsterdam (1990)Google Scholar
  16. 16.
    Saranen, J., Vainikko, G.: Periodic Integral and Pseudodifferential Equations with Numerical Approximations. Springer-Verlag, Berlin (2002)Google Scholar
  17. 17.
    Saranen, J., Wendland, W.L.: On the asymptotic convergence of collocation methods with spline functions of even degree. Math. Comput. 45, 91–108 (1985)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  • Jukka T. Kemppainen
    • 1
  • Keijo Matti Ruotsalainen
    • 1
  1. 1.University of OuluOuluFinland

Personalised recommendations