Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T07:11:04.574Z Has data issue: false hasContentIssue false
  • English
  • Français

Singularity subtraction in the numerical solution of integral equations

Published online by Cambridge University Press:  17 February 2009

P. M. Anselone
P. M. Anselone
Affiliation:
Oregon State University, Department of Mathematics, Corvallis, Oregon 97331, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The singularity subtraction technique described by Kantorovich and Krylov in [11] is designed to reduce or overcome the effect of a weakly singular kernel in the numerical solution of integral equations. First, the equation is rearranged in such a way that the singularity of the kernel is at least partially cancelled by the smoothness of the solution, and then numerical integration is applied. We present convergence results and error bounds under general conditions on the nature of the singularity and the numerical integration procedure. Numerical examples demonstrate the benefit of the singularity subtraction technique.

Type
Research Article
Information
The ANZIAM Journal , Volume 22 , Issue 4 , April 1981 , pp. 408 - 418
Copyright
Copyright © Australian Mathematical Society 1981

References

[1]Anselone, P. M., Collectively compact operator approximation theory and applications to integral equations (Prentice-Hall, Englewood Cliffs, NJ, 1971).Google Scholar
[2]Anselone, P. M. and Krabs, W., “Approximate solution of weakly singular integral equations”, J. of Integral Equations 1 (1979), 61–75.Google Scholar
[3]Anselone, P. M. and Opfer, G., “Numerical integration of weakly singular functions”, in Numerische integration (ed. Hämmerlin, G.) (Birkhäuser Verlag, Basel, 1979).Google Scholar
[4]Atkinson, K. E., “The numerical solution of Fredholm integral equations of the second kind with singular kernels”, Numer. Math. 19 (1972), 248–259.CrossRefGoogle Scholar
[5]Atkinson, K. E., A survey of numerical methods for the solution of Fredholm integral equations of the second kind (SIAM Publications, 1976).Google Scholar
[6]Bechiars, J., “Glattheit und numerische Berechnung der Lösung linearer Integralgleichungen 2. Art mit schwachsingulären Kernen” (Hahn-Meitner-Institut Berlin HMI-B 283, 1978).Google Scholar
[7]Bechiars, J., “Weakly singular integral equations, smoothness properties and numerical solution”, J. of Integral Equations 2 (1980), 259–263.Google Scholar
[8]Borer, D., “Approximate solution of Fredholm integral equations of the second kind with singular kernels”, Thesis, Oregon State University, Corvallis, 1977.Google Scholar
[9]Giraud, G., “Sur certains problèmes non-linéaires de Neumann et sur certain problèmes non-linéaires mixtes”, Ann. Ecole Norm. Suépr. 49 (1932), 3–17.Google Scholar
[10]Graham, I. G., “Singularity expansions for the solutions of second kind integral equations with weakly singular convolution kernels”, J. of Integral Equations, to appear.Google Scholar
[11]Kantorovich, L. V. and Krylov, V. I., Approximate methods of higher analysis (Interscience, New York, 1958).Google Scholar
[12]Kleinman, R. E. and Wendland, W. L., “On Neumann's method for the exterior Neumann problem for the Helmholtz equation”, J. Math. Anal. Appl. 57 (1977), 170–202.CrossRefGoogle Scholar
[13]Krechel, F. W., “Numerische Lösung Fredholmscher Integralgleichungen 2. Art mit schwach singulären Kernen”, Diplomarbeit, Univ. Köln, 1977.Google Scholar
[14]Kussmaul, R. and Werner, P., “Fehlerabschätzungen fur ein numerisches Verfahren zur Auflösung linearer Integralgleichungen mit schwachsingulären Kernen”, Computing 3 (1968), 22–46.CrossRefGoogle Scholar
[15]Miranda, C., Equazioni alle derivate parziali di tipo ellittico (Springer-Verlag, Berlin, 1955).Google Scholar
[16]Schneider, C., “Beiträge zur numerischen Behandlung schwachsingulärer Fredholmscher Integralgleichungen 2. Art”, Dissertation, Univ. Mainz, 1977.Google Scholar
[17]Sloan, I. H., “The numerical solution of Fredholm equations of the second kind by polynomial interpolation”, J. of Integral Equations, to appear.Google Scholar
[18]Sloan, I. H., “On choosing the points in product integration”, J. Math. Phys. 21 (1980), 1032–1039.CrossRefGoogle Scholar
[19]Sloan, I. H., “Analysis of general quadrature methods for integral equations of the second kind”, Univ. of Maryland Tech. Note BN 931, 1979, submitted for publication.Google Scholar
[20]Sloan, I. H. and Burn, B. J., “Collocation and polynomials for integral equations of the second kind: a new approach to the theory”, J. of Integral Equations, 1 (1979), 77–94.Google Scholar
[21]Volk, W., “Die numerische Behandlung Fredholmscher Integralgleichungen zweiter Art mittels Splinefunktionen”, Hahn-Meitner-Institut Berlin, HMI-B 286, 1979.Google Scholar
[22]Wendland, W. L., Elliptic systems in the plane (Pitman Publishing, Ltd, London, 1979).Google Scholar