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Developments in Solving Integral Equations Numerically

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Part of the book series: NATO ASI Series ((ASIC,volume 357))

Abstract

After a short survey over the main methods for treating Fredholm integral equations practically, this article gives some information on the degenerate substitution kernel method and some recent results. In particular, the extension to Hammerstein integral equations, an investigation of Bateman’s method as applied to eigenvalue problems and its convergence behaviour for Green’s kernels are considered.

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References

  1. K.E. Atkinson: A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind. Soc. Ind. Appl. Math. Philadelphia 1976, VII + 230 p.

    Google Scholar 

  2. C.T.H. Baker: The Numerical Treatment of Integral Equations. Monographs on Numerical Analysis, Clarendon Press Oxford 1977, XII + 1034 p.

    Google Scholar 

  3. L. Bamberger and G. Hämmerlin: Spline-blended substitution kernels of optimal convergence. Treatment of Integral Equations by Numerical Methods, ed. by C.T.H. Baker and G.F. Miller, 47–57, Acad. Press London 1982.

    Google Scholar 

  4. H. Bateman: On the Numerical Solution of Linear Integral Equations. Proc. Roy. Soc. London, Ser. A 100, 441–449 (1922).

    Article  Google Scholar 

  5. H. Brakhage: Über die numerische Behandlung von Integralgleichungen nach der Quadraturformelmethode. Numer. Math. 2, 183–196 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  6. H. Brakhage: Zur Fehlerabschätzung für die numerische Eigenwertbestimmung bei Integralgleichungen, Numer. Math. 3, 174–179 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  7. L.M. Delves and J.L. Mohamed: Computational methods for integral equations. Cambridge University Press 1985, XII + 376 p.

    Google Scholar 

  8. S. Fenyö-H.W. Stolle: Theorie und Praxis der linearen Integralgleichungen 4, Math. Reihe Bd. 77, Birkhäuser Verlag Basel 1984, 708 p.

    Google Scholar 

  9. W.J. Gordon: Spline-Blended Surface Interpolation through Curve Networks. J. of Math, and Mech. 18, 931–952 (1969).

    MATH  Google Scholar 

  10. G. Hämmerlin: Ein Ersatzkernverfahren zur numerischen Behandlung von Integralgleichungen 2. Art. Z. Angew. Math. Mech. 42, 439–463 (1962).

    Article  MATH  Google Scholar 

  11. G. Hämmerlin: Zur numerischen Behandlung von homogenen Fredholmschen Integralgleichungen 2. Art mit Splines. Spline Functions Karlsruhe 1975, Lecture Notes in Math. vol. 501, ed. by K. Böhmer, G. Meinardus and W. Schempp, 92-98, Springer-Verlag 1976.

    Google Scholar 

  12. G. Hämmerlin and W. Lückemann: The Numerical Treatment of Integral Equations by a Substitution Kernel Method Using Blending-Splines. Memorie della Accademia Nazionale di Scienze, Lettere e Arti di Modena — Serie VI, Vol. XXI-1979, 1–15 (1981).

    Google Scholar 

  13. G. Hämmerlin and L.L. Schumaker: Error Bounds for the Approximation of Green’s Kernels by Splines. Numer. Math. 33, 17–22 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  14. G. Hämmerlin and L.L. Schumaker: Procedures for Kernel Approximation and Solution of Fredholm Integral Equations of the Second Kind. Handbook Series Approximations, Numer. Math. 34, 125–141 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Hammerstein: Nichtlineare Integralgleichungen nebst Anwendungen. Acta Mathematica 54, 117–176 (1930).

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Joe and I.H. Sloan: On Bateman’s Method for Second Kind Integral Equations. Numer. Math. 49, 499–510 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Kumar and I.H. Sloan: A New Collocation-Type Method for Hammerstein Integral Equations. Math. Comp. 48, 585–593 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  18. H. Kaneko and Y. Xu: Degenerate Kernel Method for Hammerstein Equations. Math. Comp. 56, 141–148 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  19. E.J. Nyström: Über die praktische Auflösung von linearen Integralgleichungen mit Anwendungen auf Randwertaufgaben der Potentialtheorie. Soc. Scient. Fenn. Comm. Phys.-Math. IV. 15, 1–52 (1928).

    Google Scholar 

  20. W. Prock: Zur Theorie des Bateman-Verfahrens für homogene Integralgleichungen zweiter Art. Master’s thesis, Math. Inst. Univ. of Munich, 96 p. (1989).

    Google Scholar 

  21. E. Schäfer: Fehlerabschätzungen für Eigenwertnäherungen nach der Ersatzkernmethode bei Integralgleichungen. Numer. Math. 32, 281–290 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  22. E. Schäfer: Spectral Approximation for Compact Integral Operators by Degenerate Kernel Methods, Numer. Funct. Anal, and Optimiz. 2, 43–63 (1980).

    Article  MATH  Google Scholar 

  23. H. Weyl: Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Hohlraumstrahlung). Math. Annalen 71, 441–479 (1912).

    Article  MathSciNet  MATH  Google Scholar 

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© 1992 Springer Science+Business Media Dordrecht

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Hämmerlin, G. (1992). Developments in Solving Integral Equations Numerically. In: Espelid, T.O., Genz, A. (eds) Numerical Integration. NATO ASI Series, vol 357. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2646-5_15

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  • DOI: https://doi.org/10.1007/978-94-011-2646-5_15

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-5169-9

  • Online ISBN: 978-94-011-2646-5

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