Advertisement

Two-Operator Boundary-Domain Integral Equations for Variable Coefficient Dirichlet Problem in 2D

  • Tsegaye G. Ayele
  • Solomon T. Bekele
Chapter

Abstract

The Dirichlet problem for the second order elliptic PDE with variable coefficient is considered in two-dimensional bounded domain. Using an appropriate parametrix (Levi function) and applying the two-operator approach, this problem is reduced to two systems of boundary-domain integral equations (BDIEs). Although the theory of BDIEs in 3D is well developed, the BDIEs in 2D need a special consideration due to their different equivalence properties. As a result, we need to set conditions on the domain or on the associated Sobolev spaces to ensure the invertibility of corresponding parametrix-based integral layer potentials and hence the unique solvability of BDIEs. The properties of corresponding potential operators are investigated. The equivalence of the original BVP and the obtained BDIEs is analysed and the invertibility of the BDIE operators is proved in appropriate Sobolev-Slobodecki (Bessel potential) spaces.

Notes

Acknowledgements

The work on this paper of the first author was supported by EMS-Simmons for Africa and ISP Sweden and of the second author was supported by ISP, Sweden. They also thank the Department of Mathematics at Brunel University for hosting their research visit.

References

  1. [AyMi11]
    Tsegaye G. Ayele, Sergey E. Mikhailov (2011), Analysis of two-operator boundary-domain integral equations for a variable-coefficient BVP, in: Eurasian Math. J., 2:3 (2011), 20–41. Google Scholar
  2. [ChEtAl09b]
    O. Chakuda, S.E. Mikhailov, D. Natroshvili, Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient. I: Equivalence and Invertibility. J. Integral Equat. and Appl. 21 (2009), 499–543.MathSciNetCrossRefGoogle Scholar
  3. [Co00]
    Constanda, C.: Direct and Indirect Boundary Integral Equation Methods. Chapman & Hall/CRC (2000).Google Scholar
  4. [Mi15]
    Mikhailov, S.E.: Analysis of Segregated Boundary-Domain Integral Equations for variable coefficient Dirichlet and Neumann problems with general date. ArXiv:1509.03501, 1–32 (2015).Google Scholar
  5. [Co88]
    M. Costabel, Boundary integral operators on Lipchitz domains: elementary results. SIAM journal on Mathematical Analysis 19 (1988), 613–626.MathSciNetCrossRefGoogle Scholar
  6. [DuMi15]
    Dufera, T.T. and Mikhailov, S.E.: Analysis of Boundary-Domain Integral Equations for Variable-Coefficient Dirichlet BVP in 2D. In: Integral Methods in Science and Engineering: Computational and Analytic Aspects, C. Constanda and A. Kirsch (eds.),Birkhäuser, Boston (2015), pp. 163–175.CrossRefGoogle Scholar
  7. [Mc00]
    W. McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge, 2000.zbMATHGoogle Scholar
  8. [Mi05a]
    Mikhailov S.E.Analysis of boundary-domain integral and integro-differential equations for a Dirichlet problem with variable coefficient. In: Integral Methods in Science and Engineering: Theoretical and Practical Aspects (Edited by C.Constanda, Z.Nashed, D.Rolins), Boston-Basel-Berlin: Birkhäuser, ISBN 0-8176-4377-X, 161–176.Google Scholar
  9. [Mi02]
    S.E. Mikhailov, Localized boundary-domain integral formulations for problems with variable coefficients, Int. J. Engineering Analysis with Boundary Elements 26 (2002), 681–690.CrossRefGoogle Scholar
  10. [Mi05b]
    S.E. Mikhailov, Localized formulations for scalar nonlinear BVPs with variable coefficients. J. Eng. Math., 51 (2005), 283–302.CrossRefGoogle Scholar
  11. [Mi11]
    S.E. Mikhailov, Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains. J. Math. Analysis and Appl. 378 (2011), 324–342.MathSciNetCrossRefGoogle Scholar
  12. [Se08]
    Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements, Springer New York (2008).Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Tsegaye G. Ayele
    • 1
  • Solomon T. Bekele
    • 1
  1. 1.Addis Ababa UniversityAddis AbabaEthiopia

Personalised recommendations