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On Collocation Methods for Nonlinear Cauchy Singular Integral Equations

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Toeplitz Matrices and Singular Integral Equations

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 135))

Abstract

Collocation methods with respect to the Chebyshev nodes of first and of second kind together with both the modified and the classical Newton iteration method for a class of nonlinear Cauchy singular integral equations are investigated. The proof of the convergence of the Newton methods is based on the stability of the respective collocation methods applied to linear Cauchy singular integral equations, which is in turn proved by using Banach algebra techniques. The effective numerical realization of the methods and their applicability to a nonlinear Cauchy singular integral equation arising from a free surface seepage problem are discussed.

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References

  1. M.R. Capobianco, P. Junghanns, U. Luther and G. Mastroianni, Weighted uniform convergence of the quadrature method for Cauchy singular integral wquations, in: Singular Integral Operators and Related Topics, ed. by A. Böttcher and I. Gohberg, Operator Theory, Advances and Applications, Vol. 90, Birkhäuser Verlag, 1996, 153-181.

    Google Scholar 

  2. I. Gohberg and N. Krupnik, One-dimensional Linear Singular Integral Equations, Birkhäuser Verlag, Basel, Boston, Berlin, 1992.

    Google Scholar 

  3. G. Heinig and K. Rost, Algebraic Methods for Toeplitz-like Matrices and Operators, Akademie Verlag, Berlin, and Birkhäuser Verlag, Boston, 1984.

    Google Scholar 

  4. G. Heinig and K. Rost, Representations of Cauchy matrices with Chebyshev nodes using trigonometric transforms, in: Structured Matrices, Recent Advances and Applications, ed. by D.A. Bini, E. Tyrtyshnikov, P. Yalamov, Advances in the Theory of Computational Mathematics, Vol. 4, Nova Science Publishers, 2001, 135-147.

    Google Scholar 

  5. T. Huckle, Cauchy matrices and iterative methods for Toeplitz matrices, Proceedings of SPIE-95, 2563 (1995), 281–292.

    Article  Google Scholar 

  6. P. Junghanns, Numerical analysis of Newton projection methods for nonlinear singular integral equations, J. Comp. Appl. Math., 55 (1994), 145–163.

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Junghanns, On the numerical solution of nonlinear singular integral equations, ZAMM, 76 (1996), 152–166.

    Google Scholar 

  8. P. Junghanns, Numerical solution of a free surface seepage problem from nonlinear channel, Applicable Analysis, 63 (1996), 87–110.

    Article  MathSciNet  Google Scholar 

  9. P. Junghanns and U. Luther, Cauchy singular integral equations in spaces of continuous functions and methods for their numerical solution, J. Comp. Appl. Math., 77 (1997), 201–237.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Junghanns and K. Müller, A collocation method for nonlinear Cauchy singular integral equations, J. Comp. Appl. Math., 115 (2000), 283–300.

    Article  MATH  Google Scholar 

  11. P. Junghanns, S. Roch and B. Silbermann, Collocation methods for systems of Cauchy singular integral equations on an interval, Comp. Techn., 6 (2001), 88–124.

    MathSciNet  MATH  Google Scholar 

  12. P. Junghanns, G. Semmler, U. Weber and E. Wegert, Non-linear singular integral equations on a finite interval, Math. Meth. Appl. Sci., 24 (2001).

    Google Scholar 

  13. P. Junghanns and U. Weber, On the solvability of nonlinear singular integral equations, Zeitschr. Anal. Anw., 12 (1993), 683–698.

    MathSciNet  MATH  Google Scholar 

  14. P. Junghanns and U. Weber, Local theory of projection methods for Cauchy singular integral equations on an interval, in: Boundary Integral Methods — Numerical and Mathematical Aspects, Comp. Mech. Publ., Southampton, 1999, 219–256.

    Google Scholar 

  15. U. Luther, Generalized Besov Spaces and Cauchy Singular Integral Equations, PhD-Thesis, TU Chemnitz, 1998.

    Google Scholar 

  16. D. Oestreich, Singular integral equations applied to free surface seepage from nonlinear channels, Math. Meth. Appl. Sci., 12 (1990), 209–219.

    Article  MathSciNet  MATH  Google Scholar 

  17. Y. Saad and M.H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), 856–869.

    Article  MathSciNet  MATH  Google Scholar 

  18. G. Steidl and M. Tasche, A polynomial approach to fast algorithms for discrete Fourier-cosine and Fourier-sine transforms, Math. Comp., 56 (1991), 281–296.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Tasche, Fast algorithms for discrete Chebyshev-Vandermonde transforms and applications, Numerical Algorithms, 5 (1993), 453–464.

    Article  MathSciNet  MATH  Google Scholar 

  20. U. Weber, Weighted Polynomial Approximation Methods for Cauchy Singular Integral Equations in the Non-Periodic Case, PhD-Thesis, Logos Verlag, Berlin, 2000.

    Google Scholar 

  21. E. Zeidler, Nonlinear Functional Analysis and its Applications, I: Fixed-Point Theorems, Springer Verlag, New York, 1986.

    Book  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Chemnitz University of Technology, D-09107, Chemnitz, Germany

    Peter Junghanns, Katja Müller & Karla Rost

Authors
  1. Peter Junghanns
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  2. Katja Müller
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  3. Karla Rost
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Editor information

Editors and Affiliations

  1. Faculty of Mathematics, Technical University Chemnitz, 09107, Chemnitz, Germany

    Albrecht Böttcher  & Peter Junghanns  & 

  2. School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69978, Israel

    Israel Gohberg

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© 2002 Springer Basel AG

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Junghanns, P., Müller, K., Rost, K. (2002). On Collocation Methods for Nonlinear Cauchy Singular Integral Equations. In: Böttcher, A., Gohberg, I., Junghanns, P. (eds) Toeplitz Matrices and Singular Integral Equations. Operator Theory: Advances and Applications, vol 135. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8199-9_13

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  • DOI: https://doi.org/10.1007/978-3-0348-8199-9_13

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9471-5

  • Online ISBN: 978-3-0348-8199-9

  • eBook Packages: Springer Book Archive

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