10.1186/2251-7456-6-15

Methods of solution of singular integral equations

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  • Aloknath Chakrabarti1,
  • Subash Chandra Martha*2,
  1. Department of Mathematics, Indian Institute of Science, Bangalore, 560012, IN
  2. Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, Punjab, 140001, IN

Published in Issue 2012-07-20

How to Cite

Chakrabarti, A., & Martha, S. C. (2012). Methods of solution of singular integral equations. Mathematical Sciences, 6(1). https://doi.org/10.1186/2251-7456-6-15
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Abstract

Abstract Purpose This paper investigates a different method to evaluate different real improper integrals and also to obtain the solutions of various types of Cauchy-type singular integral equations of the first kind. Methods Methods using the analysis of functions of real variables only are reviewed and utilized for the above purpose. These methods clearly demonstrate that details of complex function theory which are normally employed in handling such integral equations for their solutions can be avoided altogether. Also, some approximate methods of solution of such integral equations are developed. Results The solutions of real singular integral equations over different intervals such as (−1,1); ( a , b ); (0, a ) ∪ ( b , c ); (−1, k ) ∪ ( k , 1); (− ∞ , b ); ( a , + ∞ ); (− ∞ , + ∞ ); infinite intervals with a gap are obtained by using the proposed methods. Conclusion The proposed methods are new and each has its own structure.

Keywords

  • Real singular integral equations,
  • Cauchy-type kernels,
  • Real variable method,
  • MSC,
  • 45E05
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References

  1. Muskhelishvili (1953) Noordhoff
  2. Gakhov (1966) Pergamon
  3. Chakrabarti et al. (1997) Solution of a hypersingular integral equation of the second kind (pp. 319-320) https://doi.org/10.1002/zamm.19970770424
  4. Chakrabarti and Mandal (1998) Derivation of the solution of a simple hypersingular integral equation 29(1) (pp. 47-53) https://doi.org/10.1080/0020739980290105
  5. Chakrabarti (1999) Numerical solution of a singular integro-differential equation 79(4) (pp. 233-241) https://doi.org/10.1002/(SICI)1521-4001(199904)79:4<233::AID-ZAMM233>3.0.CO;2-6
  6. Chakrabarti and Berghe Vanden (2004) Approximate solution of singular integral equations (pp. 553-559) https://doi.org/10.1016/S0893-9659(04)90125-5
  7. Chakrabarti (2007) Solution of a simple hyper-singular integral equation 19(4) (pp. 465-471) https://doi.org/10.1216/jiea/1192628619
  8. Frankel (1995) A Galerkin solution to a regularized Cauchy singular integro-differential equation (pp. 245-258)
  9. Mandal and Bera (2007) Approximate solution for a class of singular integral equations of second kind 206(1) (pp. 189-195) https://doi.org/10.1016/j.cam.2006.06.011
  10. Martin (1992) Exact solution of a simple hyper-singular integral equation (pp. 197-204) https://doi.org/10.1216/jiea/1181075681
  11. Ahlfors (1979) McGraw-Hill Book Company
  12. Ursell (1947) The effect of a fixed vertical barrier on surface waves in deep water (pp. 374-382) https://doi.org/10.1017/S0305004100023604
  13. Tricomi (1951) The airfoil equation for a double interval https://doi.org/10.1007/BF02579701
  14. Tricomi (1957) Interscience Publishers, Inc.