Skip to main content
SpringerLink
Log in

Line element method of solving singular integral equations

  • Original Research
  • Published:
Indian Journal of Pure and Applied Mathematics Aims and scope Submit manuscript

Abstract

The work in this paper is concerned with application of a very simple numerical technique called line element method to solve singular integral equations with weakly singular kernel and hypersingular kernel. Of the integral equations considered here are first kind Abel integral equation and integral equation with log kernel and hypersingular integral equations of first and second kind. In this method, the range of integration as well as the interval of definition of the integral equation are discretised into finite number of small subintervals and the unknown function satisfying the integral equation is assumed to be constant in each small subinterval. This reduces the integral equations to a system of algebraic equations which is then solved to obtain the unknown function in each subinterval. The method is illustrated with examples. It is observed that a very accurate result is obtained by applying this method. The error analysis for this method is also given.

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.

Similar content being viewed by others

References

  1. Mandal, B.N., Chakrabarti, A. 2011. Applied singular integral equations, CRC Press.

  2. Mandal, B.N., Chakrabarti, A. 2000. Water wave scattering by barriers, WIT Press.

  3. Banerjea S., Dutta, B. 2008. On a weakly singular integral equation and its application, Appl. Math. Lett. 21, 729-734.

    Article  MathSciNet  Google Scholar 

  4. Banerjea, S., Chakraborty, R., and  Samanta, A., 2019. Boundary element approach of solving fredholm and volterra integral equations, Int. J. Math. Model. Numer. Optim. 9(1), 1–11.

    Google Scholar 

  5. Chakrabarti, A., Mandal, B.N., Banerjea, S. and Sahoo, T. 1995. Solution of a class of mixed boundary value problems for Laplace’s equation arising in water wave scattering, J Indian Instt. Sci., 75, 577-585.

    MathSciNet  MATH  Google Scholar 

  6. Pozrikidis, C., 2002. A practical guide to boundary element methods with the software library BEMLIB. CRC Press, London.

    Book  Google Scholar 

  7. Stephen Kirkup, 2007. The Boundary Element Method in Acoustics.

  8. Katsikadelis, John T. 2002, Boundary Elements Theory and Applications, Amsterdam: Elsevier.

    Google Scholar 

  9. Beer, G. and J.O. Watson. 1992. Introduction to Finite and Boundary Element Methods for Engineers. Wiley: New York.

    MATH  Google Scholar 

  10. Beer, G., Smith, I. and Duenser, C. 2008, The boundary Element Method with Programming. Springer.

  11. Becker, A.A. 1992. The boundary element method for engineering, A complete course. McGraw Hill Book Company.

    Google Scholar 

  12. Luminita G.,2007. A Direct Boundary Element Approach with Linear Boundary Elements for the Compressible Fluid Flow Around Obstacles,The XVI Conference on Applied and Industrial Mathematics CAIM.

  13. Brebbia, C.A. Dominguez,J. 1994. Boundary Elements: An Introductory Course, WIT Press.

  14. Goldberg, Chen, 1994. Boundary Elements: An Introductory Course, WIT Press.

  15. A.C. Kaya, F. Erdogom, 1987. On the solution of integral equation with strongly singular kernels, Quart. Appl. Math. 55 (1) 105-122.

    Article  MathSciNet  Google Scholar 

  16. Y. Chan, A.C. Fannjiang, G.H. Paulino, 2003. Integral equations with hyper singular kernels-theory and applications to fracture mechanics, J. Eng. Sci. 41 683-720.

    Article  Google Scholar 

  17. P.A. Martin, 1992 Exact solution of a simple hyper singular integral equation, J. Integral Equation Appl. 4 197-204.

    Article  Google Scholar 

  18. N.F. Parsons and P.A. Martin, 1992 Scattering of water waves by submerged plates using hypersingular integral equations. Applied Ocean Research 14 313-21.

    Article  Google Scholar 

  19. A. Chakrabarti, B.N. Mandal, U. Basu and S. Banerjea, 1997 Solution of a hypersingular integral equation of second kind, ZAMM 77 319-320.

    Article  MathSciNet  Google Scholar 

  20. M. Abramowitz and I.A. Stegun, 1965, Handbook of Mathematical functions. Dover. New York.

    MATH  Google Scholar 

Download references

Acknowledgements

The present work is supported by DST (SERB) through Matric Project No. MTR/2017/000363 through SB and RUSA 2.0 through Jadavpur University. AS thankfully acknowledge Department of Science and Technology(DST),Government of India, for awarding Inspire fellowship \((No.- IF170841)\).

Author information

Authors and Affiliations

  1. Department of Mathematics, Jadavpur University, Kolkata, 700032, India

    Anushree Samanta & Sudeshna Banerjea

  2. Department of Mathematics, Diamond Harbour Women’s University, South 24 Parganas, 743368, India

    Rumpa Chakraborty

Authors
  1. Anushree Samanta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rumpa Chakraborty
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sudeshna Banerjea
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sudeshna Banerjea.

Additional information

Communicated by V D Sharma.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Samanta, A., Chakraborty, R. & Banerjea, S. Line element method of solving singular integral equations. Indian J Pure Appl Math 53, 528–541 (2022). https://doi.org/10.1007/s13226-021-00115-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13226-021-00115-7

Keywords

Search

Navigation