Skip to main content
SpringerLink
Log in

A Discrete Theorem of Goursat

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

In order to describe stress and displacement fields in the neightborhood of singularities in fracture mechanics the so-called theory of complex stress function based on Kolosov and Muskhelishvili can be used. The relation between linear elasticity and complex function theory is based on the Theorem of Goursat. In this paper a discrete version of the Theorem is proved.

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. G.W. Kolosov, Über eine Anwendung der Theorie der Funktion einer komplexen Veränderlichen auf das ebene Problem der mathematischen Elastizitätstheorie (Russian). Ph. D. Thesis.Yuriew (Dorpat) 1909.

  2. Kolosov G.W.: Über einige Eigenschaften des ebenen Problems der Elastizitätstheorie. Zeitschrift fü r Mathematik und Physik 62, 383–409 (1914)

    Google Scholar 

  3. Muskhelishvili N.I.: Einige Grundaufgaben der mathematischen Elastizitätstheorie. VEB Fachbuchverlag, Leipzig (1971)

    Google Scholar 

  4. Meleshko V.V.: Selected topics in the history of the two-dimensional biharmonic problem. Journal of Applied Mechanics 56((1), 33–85 (2003)

    Article  Google Scholar 

  5. E. Goursat, Sur l’équation Δ Δu = 0. . Bulletin de la Société Mathématique de France 26 (1898), 236.

  6. S. Bock, K. Gü rlebeck, On a Spatial Generalization of the Kolosov-Muskhelishvili Formulae. Math. Meth. Appl. Sciences 32 (2) (2009), 223-240

  7. A.A. Samarskij, Theorie der Differenzenverfahren. Akademische Verlagsgesellschaft Geest und Portig K.-G. Leipzig 1984.

  8. Duffin R.J.: Discrete potential theory. Duke Math. J. 20, 233–251 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  9. Duffin R.J.: Basic properties of discrete analytic functions. Duke Math. J. 23, 335–363 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  10. J. Ferrand, Fonctions preharmonique et fonctions preholomorphes. Bulletin des Sciences Mathematique, sec. series.

  11. N. Faustino, U. Kähler, F. Sommen, Dirac Operators in Clifford Analysis. Adv. appl. Clifford alg. Birkhäuser Verlag Basel, doi:10.1007/s00006-007-0041-z

  12. H. Ridder, Schepper H, Kähler U, Sommen F. Discrete function theory based on skew Weyl relations. Proceedings of the American Mathematical Society, 2009

  13. A. Hommel, Fundamentallösungen partieller Differenzenoperatoren und die Lösung diskreter Randwertprobleme mit Hilfe von Differenzenpotentialen. Dissertation, Bauhaus Universität Weimar 1998.

  14. F Brackx, H. Schepper, F. Sommen, L.V. de Voorde, Discrete Clifford analysis: an overview. Cubo 11 (2009), 55-71.

Download references

Author information

Authors and Affiliations

  1. University of Applied Sciences Zwickau, Faculty of Economics, POB 201037, 08012, Zwickau, Germany

    Angela Hommel

Authors
  1. Angela Hommel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Angela Hommel.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hommel, A. A Discrete Theorem of Goursat. Adv. Appl. Clifford Algebras 24, 1039–1045 (2014). https://doi.org/10.1007/s00006-014-0500-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00006-014-0500-2

Keywords

Search

Navigation