Abstract
A number of methods have been developed for the numerical integration of functions with integrable singularities. Recent references include Osgood and Shisha [7, 8]. For a summary and further references, see Davis and Rabinowitz [4]. Our approach incorporates elements of several of the techniques described there. We approximate a function on neighborhoods of a singularity by bounded functions and then integrate numerically. The given singular function, the neighborhoods of the singularity, the approximations, and the quadrature formula satisfy very limited hypotheses. Convergence results and error bounds are derived. The generality of the results opens up possibilities of constructing particularly efficient schemes for the numerical integration of singular functions.
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References
Anselone, P. M., Collectively Compact Operator Approximation Theory and Applications to Integral Equations, Prentice-Hall, 1971.
Anselone, P. M., Singularity Subtraction in the Numerical Solution of Integral Equations, Journal of Integral Equations, to appear.
Anselone, P. M. and W. Krabs, Approximate Solution of Weakly Singular Integral Equations, Journal of Integral Equations, to appear.
Davis, P. J. and P. Rabinowitz, Methods of Numerical Integrations, Acadamic Press, 1975.
Feldstein, A. and R. K. Miller, Error Bounds for Compound Quadrature of Weakly Singular Integrals, Math. Comp. 25(1971), pp. 505–520.
Miller, R. K., On Ignoring the Singularity in Numerical Quadrature, Math. Comp. 25(1971), pp. 521–532.
Osgood, C. F. and O. Shisha, The Dominated Integral, J. Approximation Theory 17(1976), pp. 150–165.
Osgood, C. F. and O. Shisha, Numerical Quadrature of Improper Integrals and the Dominated Integral, J. Approximation Theory 20(1977), pp. 139–152.
Scheider, C., private communication.
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Dedicated to Professor Dr. Johannes Weissinger, Karlsruhe on the occasion of his 65th birthday
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© 1979 Springer Basel AG
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Anselone, P.M., Opfer, G. (1979). Numerical Integration of Weakly Singular Functions. In: Hämmerlin, G. (eds) Numerische Integration. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’Analyse Numérique, vol 45. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6288-2_1
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DOI: https://doi.org/10.1007/978-3-0348-6288-2_1
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-7643-1014-1
Online ISBN: 978-3-0348-6288-2
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