Skip to main content
SpringerLink
Log in
Book cover

Integral Methods in Science and Engineering pp 91–108Cite as

Analysis of Some Localized Boundary–Domain Integral Equations for Transmission Problems with Variable Coefficients

  • Chapter

Abstract

Some segregated systems of direct localized boundary-domain integral equations (LBDIEs) associated with several transmission problems for scalar PDEs with variable coefficients are formulated and analyzed for a bounded domain composed of two subdomains with a coefficient jump over the interface. The main results established in the paper are the LBDIE equivalence to the original transmission problems and the invertibility of the corresponding localized boundary-domain integral operators in corresponding Sobolev spaces function spaces.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Buchukuri, T., Chkadua, O., Natroshvili, D.: Mixed boundary value problems of thermopiezoelectricity for solids with interior cracks. Integral Equations and Operator Theory, 64, n. 4, 495–537 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  2. Chkadua, O., Mikhailov, S., Natroshvili, D.: Analysis of direct boundary–domain integral equations for a mixed BVP with variable coefficient. Part I. Equivalence and invertibility. J. Integral Equations Appl., 21, 499–542 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  3. Chkadua, O., Mikhailov, S., Natroshvili, D.: Analysis of some localized boundary–domain integral equations. J. Integral Equations Appl., 21, 407–447 (2009).

    Article  MATH  Google Scholar 

  4. Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of segregated boundary-domain integral equations for variable-coefficient problems with cracks. Numerical Methods for PDEs, 27, n. 1, 121–140 (2011).

    MathSciNet  MATH  Google Scholar 

  5. Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal., 19, 613–626 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  6. Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 4, Integral Equations and Numerical Methods, Springer-Verlag, Berlin (1990).

    Google Scholar 

  7. Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations, Applied Mathematical Sciences, Springer-Verlag, Berlin–Heidelberg (2008).

    Book  Google Scholar 

  8. Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. 1, Springer-Verlag, New York–Heidelberg (1972).

    Google Scholar 

  9. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, UK (2000).

    MATH  Google Scholar 

  10. Mikhailov, S.E.: Localized boundary–domain integral formulation for problems with variable coefficients. Int. J. Engineering Analysis with Boundary Elements, 26, 681–690 (2002).

    Article  MATH  Google Scholar 

  11. Mikhailov, S.E., Nakhova, I.S.: Mesh-based numerical implementation of the localized boundary–domain integral equation method to a variable-coefficient Neumann problem. J. Engineering Math., 51, 251–259 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  12. Miranda, C.: Partial Differential Equations of Elliptic Type, Second revised edition, Springer-Verlag, New York–Berlin (1970).

    MATH  Google Scholar 

  13. Sladek, J., Sladek, V., Atluri, S.N.: Local boundary integral equation (LBIE) method for solving problems of elasticity with nonhomogeneous material properties. Comput. Mech., 24, n. 6, 456–462 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  14. Mikhailov, S.E.: Analysis of united boundary-domain integro-differential and integral equations for a mixed BVP with variable coefficient. Math. Methods in Applied Sciences, 29, 715–739 (2006).

    Article  MATH  Google Scholar 

  15. Zhu, T., Zhang, J.-D., Atluri, S.N.: A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach. Comput. Mech., 21, n. 3, 223–235 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhu, T., Zhang, J.-D., Atluri, S.N.: A meshless numerical method based on the local boundary integral equation (LBIE) to solve linear and non-linear boundary value problems. Eng. Anal. Bound. Elem., 23, n. 5–6, 375–389 (1999).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. A.Razmadze Mathematical Institute, Tbilisi, Georgia

    O. Chkadua

  2. Brunel University West London, Uxbridge, UK

    S. E. Mikhailov

  3. Georgian Technical University, Tbilisi, Georgia

    D. Natroshvili

Authors
  1. O. Chkadua
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. E. Mikhailov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. Natroshvili
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. E. Mikhailov .

Editor information

Editors and Affiliations

  1. , Department of Mathematical and Computer, The University of Tulsa, 800 South Tucker Drive, Tulsa, 74104-9700, USA

    Christian Constanda

  2. School of Computing, Engineering, and Ma, University of Brighton, Lewes Road, Brighton, BN2 4GJ, United Kingdom

    Paul J. Harris

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Chkadua, O., Mikhailov, S.E., Natroshvili, D. (2011). Analysis of Some Localized Boundary–Domain Integral Equations for Transmission Problems with Variable Coefficients. In: Constanda, C., Harris, P. (eds) Integral Methods in Science and Engineering. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8238-5_10

Download citation

Publish with us

Policies and ethics

Search

Navigation