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
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.
C.T.H. Baker: The Numerical Treatment of Integral Equations. Monographs on Numerical Analysis, Clarendon Press Oxford 1977, XII + 1034 p.
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.
H. Bateman: On the Numerical Solution of Linear Integral Equations. Proc. Roy. Soc. London, Ser. A 100, 441–449 (1922).
H. Brakhage: Über die numerische Behandlung von Integralgleichungen nach der Quadraturformelmethode. Numer. Math. 2, 183–196 (1960).
H. Brakhage: Zur Fehlerabschätzung für die numerische Eigenwertbestimmung bei Integralgleichungen, Numer. Math. 3, 174–179 (1961).
L.M. Delves and J.L. Mohamed: Computational methods for integral equations. Cambridge University Press 1985, XII + 376 p.
S. Fenyö-H.W. Stolle: Theorie und Praxis der linearen Integralgleichungen 4, Math. Reihe Bd. 77, Birkhäuser Verlag Basel 1984, 708 p.
W.J. Gordon: Spline-Blended Surface Interpolation through Curve Networks. J. of Math, and Mech. 18, 931–952 (1969).
G. Hämmerlin: Ein Ersatzkernverfahren zur numerischen Behandlung von Integralgleichungen 2. Art. Z. Angew. Math. Mech. 42, 439–463 (1962).
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.
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).
G. Hämmerlin and L.L. Schumaker: Error Bounds for the Approximation of Green’s Kernels by Splines. Numer. Math. 33, 17–22 (1979).
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).
A. Hammerstein: Nichtlineare Integralgleichungen nebst Anwendungen. Acta Mathematica 54, 117–176 (1930).
S. Joe and I.H. Sloan: On Bateman’s Method for Second Kind Integral Equations. Numer. Math. 49, 499–510 (1986).
S. Kumar and I.H. Sloan: A New Collocation-Type Method for Hammerstein Integral Equations. Math. Comp. 48, 585–593 (1987).
H. Kaneko and Y. Xu: Degenerate Kernel Method for Hammerstein Equations. Math. Comp. 56, 141–148 (1991).
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).
W. Prock: Zur Theorie des Bateman-Verfahrens für homogene Integralgleichungen zweiter Art. Master’s thesis, Math. Inst. Univ. of Munich, 96 p. (1989).
E. Schäfer: Fehlerabschätzungen für Eigenwertnäherungen nach der Ersatzkernmethode bei Integralgleichungen. Numer. Math. 32, 281–290 (1979).
E. Schäfer: Spectral Approximation for Compact Integral Operators by Degenerate Kernel Methods, Numer. Funct. Anal, and Optimiz. 2, 43–63 (1980).
H. Weyl: Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Hohlraumstrahlung). Math. Annalen 71, 441–479 (1912).
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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
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