Approximate Solution of the Dirichlet Problem for Elliptic PDE and Its Error Estimate

  • Serguey Zemskov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3718)


The proposed in [7] uniform error estimate allows to control the accuracy of the symbolic approximate solution of the Dirichlet problem for elliptic PDE in the whole domain of the problem considered. The present paper demonstrates the techniques of finding such an approximate solution with Mathematica and the use of the uniform error estimate for a concrete example.


Approximate Solution Dirichlet Problem Helmholtz Equation Linear Algebraic System Orthonormal Eigenfunctions 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Wolfram, S.: The Mathematica Book, 4th edn. Wolfram Media, Champain (1999)zbMATHGoogle Scholar
  2. 2.
    Dzyadyk, V.K.: Approximated methods for solving differential and integral equations. Nauk. dumka, Kiev (1988)Google Scholar
  3. 3.
    Hantzschmann, K.: Zur Lösung von Randwertaufgaben bei Systemen gewönlichen Deifferentialgleichungen mit dem Ritz-Galerkin-Verfahren. Habilitationsschrift Technische Universitet Dresden (1983)Google Scholar
  4. 4.
    Lehmann, N.J.: Fehlerschranken für Näherungslösungen bei Differentialgleichungen. Numerische Mathematik 10, 261–288 (1967)Google Scholar
  5. 5.
    Becken, O., Jung, A.: Error estimation in the case of linear ordinary differential equations. Rostoker Informatik-Berichte 22 (1998)Google Scholar
  6. 6.
    Rösler, T.: Adaptiv-iteratives Verfahren zur Lsung von Differenzialgleichungen. Rostoker Informatik-Berichte 28, 89–108 (2003)Google Scholar
  7. 7.
    Zemskov, S.: The Error Estimate of the Approximate Solution of the Dirichlet Problem for Elliptical Partial Differential Equation. Computer Algebra in Scientific Computing. In: Proceedings of the Seventh International Workshop, pp. 479–484. Technische Universität München (2004)Google Scholar
  8. 8.
    Mihajlov, V.P.: Partial differential equations. Nauka, Moskva, (1976) (in Russian)Google Scholar
  9. 9.
    Polianin, A.D.: Handbook of Linear Partial Differential Equations for Engineers and Scientists. Chapman & Hill/CRC, Boca Raton (2002)Google Scholar
  10. 10.
    Koshliakov, N.S., Gliner, E.B., Smirnov, M.M.: Differential equations of mathematical physics. Moskva, State Publishing House for physical and mathematical literature (1962) (in Russian)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Serguey Zemskov
    • 1
  1. 1.University of Rostock 

Personalised recommendations