Keywords
- Integral Equation
- Collocation Method
- Quadrature Formula
- Fredholm Integral Equation
- Volterra Integral Equation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Abdalkhani, J.: Collocation and Runge-Kutta-type methods for Volterra integral equations with weakly singular kernels, Ph. D. Thesis, Dalhousie University, Halifax, N.S. (Canada), 1982.
Atkinson, K., Graham, I., and Sloan, I.: Piecewise continuous collocation for integral equations, SIAM J. Numer. Anal., 20 (1983), 172–186.
Brunner, H.: A survey of recent advances in the numerical treatment of Volterra integral and integro-differential equations, J. Comp. Appl. Math., 8 (1982), 213–229.
Brunner, H.: Nonpolynomial spline collocation for Volterra equations with weakly singular kernels, SIAM J. Numer. Anal., 20 (1983).
Brunner, H.: Iterated collocation methods and their discretizations for Volterra integral equations, SIAM J. Numer. Anal. (to appear).
Brunner, H., and Graham, I.G.: Product integration for weakly singular Volterra integral equation (to appear).
Brunner, H., and Evans, M.D.: Piecewise polynomial collocation for Volterra-type integral equations of the second kind, J. Inst. Math. Appl., 20 (1977), 415–423.
Brunner, H., and Nørsett, S.P.: Superconvergence of collocation methods for Volterra and Abel integral equations of the second kind, Numer. Math., 36 (1981), 347–358.
Chandler, G.A.: Superconvergence of numerical methods to second kind integral equations, Ph. D. Thesis, Australian National University, Canberra, 1979.
Chandler, G.A.: Superconvergence for second kind integral equations, in: The Application and Numerical Solution of Integral Equations (R.S. Anderssen, F.R. de Hoog and M.A. Lukas, eds.), Sijthoff & Noordhoff, Alphen/Rijn (Netherlands), 1980. pp. 103–117.
Chatelin, F., and Lebbar, R.: The iterated projection solution for the Fredholm integral equation of second kind, J. Austral. Math. Soc. Ser. B, 22 (1981), 439–451.
Garey, L.: The numerical solution of Volterra integral equations with singular kernels, BIT, 14 (1974), 33–39.
Graham, I.G.: The numerical solution of Fredholm integral equations of the second kind, Ph. D. Thesis, University of New South Wales, Sydney, 1980.
Graham, I.G.: Singularity expansions for the solutions of second kind Fredholm integral equations with weakly singular convolution kernels, J. Integral Equations, 4 (1982), 1–30.
Graham, I.G.: Galerkin methods for second kind integral equations with singularities, Math. Comp., 39 (1982), 519–533.
Handelsman, R.A., and Olmstead, W.E.: Asymptotic solution to a class of nonlinear Volterra integral equations, SIAM J. Appl. Math., 22 (1972), 373–384.
de Hoog, F., and Weiss, R.: Asymptotic expansions for product integration, Math. Comp., 27 (1973), 295–306.
de Hoog, F.R., and Weiss, R.: High order methods for a class of Volterra integral equations with weakly singular kernels, SIAM J. Numer. Anal., 11 (1974), 1166–1180.
Kershaw, D.: Some results for Abel-Volterra integral equations of the second kind, in: Treatment of Integral Equations by Numerical Methods (C.T.H. Baker and G.F. Miller, eds.), Academic Press, London, 1982, pp. 273–282.
Linz, P.: Numerical methods for Volterra integral equations with singular kernels, SIAM J. Numer. Anal., 6 (1969), 365–374.
Logan, J.E.: The approximate solution of Volterra integral equations of the second kind, Ph. D. Thesis, University of Iowa, Iowa City, 1976.
Lubich, C.: Runge-Kutta theory for Volterra and Abel integral equations of the second kind, Preprint Nr. 154, Sonderforschungsbereich 123, University of Heidelberg, 1982.
Lyness, J.N., and Ninham, B.W.: Numerical quadrature and asymptotic expansions, Math. Comp., 21 (1967), 162–178.
McKee, S.: Generalized discrete Gronwall lemmas, Z. Angew. Math. Mech., 62 (1982), 429–434.
Miller, R.K., and Feldstein, A.: Smoothness of solutions of Volterra integral equations with weakly singular kernels, SIAM J. Math. Anal., 2 (1971), 242–258.
Olmstead, W.E.: A nonlinear integral equation associated with gas absorption in a liquid, Z. Angew. Math. Phys., 28 (1977), 513–523.
Oulès, H.: Résolution numérique d’une équation intégrale singulière, Rev. Française Trait. Inform. (Chiffres), 7 (1964), 117–124.
Pitkäranta, J.: On the differential properties of solutions to Fredholm equations with weakly singular kernels, J. Inst. Math. Appl., 24 (1979), 109–119.
Rice, J.R.: On the degree of convergence of nonlinear spline approximation, in: Approximation with Special Emphasis on Spline Functions (I.J. Schoenberg, ed.), Academic Press, New York, 1969, pp. 349–365.
Richter, G.R.: On weakly singular Fredholm integral equations with displacement kernels, J. Math. Anal. Appl., 55 (1976), 32–42.
te Riele, H.J.J.: Collocation methods for weakly singular second-kind Volterra integral equations with non-smooth solution, IMA J. Numer. Anal., 2 (1982), 437–449.
Schneider, C.: Regularity of the solution to a class of weakly singular Fredholm integral equations of the second kind, Integral Equations Operator Theory, 2 (1979), 62–68.
Schneider, C.: Produktintegration mit nicht-äquidistanten Stützstellen, Numer. Math., 35 (1980), 35–43.
Schneider, C.: Product integration for weakly singular integral equations, Math. Comp., 36 (1981), 207–213.
Sloan, I.H.: Improvement by interation for compact operator equations, Math. Comp., 30 (1976), 758–764.
Sloan, I.H.: A review of numerical methods for Fredholm equations of the second kind, in: The Application and Numerical Solution of Integral Equations (R.S. Anderssen, F.R. de Hoog and M.A. Lukas, eds.), Sijthoff & Noordhoff, Alphen/Rijn (Netherlands), 1980, pp. 51–74.
Vainikko, G., and Pedas, A.: The properties of solutions of weakly singular integral equations, J. Austral. Math. Soc. Ser. B, 22 (1981), 419–430.
Vainikko, G., and Uba, P.: A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel, J. Austral. Math. Soc. Ser. B, 22 (1981), 431–438.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1984 Springer-Verlag
About this paper
Cite this paper
Brunner, H. (1984). The numerical solution of integral equations with weakly singular kernels. In: Griffiths, D.F. (eds) Numerical Analysis. Lecture Notes in Mathematics, vol 1066. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0099518
Download citation
DOI: https://doi.org/10.1007/BFb0099518
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-13344-5
Online ISBN: 978-3-540-38881-4
eBook Packages: Springer Book Archive
