Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Fundamental solution of a multidimensional difference equation with periodical and matrix coefficients

  • 52 Accesses

  • 4 Citations

Summary

The multidimensional (partial) difference equation with periodical coefficients is transformed into an equation for a vector sequence. Integral formulae for the vector fundamental solution are developed and some results about its asymptotic properties are explained. As an example, the results are used for a simple difference equation on a hexagonal grid.

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

References

  1. [1]

    de Boor, C., Höllig, K. andRiemenschneider, S.,Fundamental solutions of multivariate difference equations. J. Amer. Math. Soc.111 (1989), 403–415.

  2. [2]

    Bosák, M.,On the bibo-stability of multidimensional difference equation with periodical coefficients. (Unpublished report.)

  3. [3]

    Duffin, R. andShaffer, D.,Asymptotic expansions of double Fourier transforms. Duke Math. J.27 (1960), 581–596.

  4. [4]

    Lynch, R.,Fundamental solutions of nine-point discrete Laplacians. Appl. Numer. Math.10 (1992), 325–334.

  5. [5]

    Veit, J.,Fundamentallösung der zweidimensionalen ‘singulären’ Differenzengleichung. Z. Angew. Math. Mech.74 (1994), 362–364.

  6. [6]

    Ermilov, A. N., Kabanovich andKurbatov, A. M.,An analytic representation of the Green function for a finite-difference Laplace equation (Russian). Zh. Vychisl. Mat. i Mat. Fiz.28 (1988), 1258–1260.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Veit, J. Fundamental solution of a multidimensional difference equation with periodical and matrix coefficients. Aeq. Math. 49, 47–56 (1995). https://doi.org/10.1007/BF01827928

Download citation

AMS (1991) subject classification

  • Primary: 39A10, 39A11, 40B05
  • Secondary: 35J05, 65N06, 65R10