The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes

Brunner, Hermann

doi:10.1090/S0025-5718-1985-0804933-3

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6842(online) ISSN 0025-5718(print)

The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes

Author: Hermann Brunner
Journal: Math. Comp. 45 (1985), 417-437
MSC: Primary 65R20; Secondary 45D05
MathSciNet review: 804933
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Since the solution of a second-kind Volterra integral equation with weakly singular kernel has, in general, unbounded derivatives at the left endpoint of the interval of integration, its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates. In this paper we investigate the convergence rates with respect to graded meshes, and we discuss the problem of how to select the quadrature formulas to obtain the fully discretized collocation equation.

[1] P. R. Beesack, More generalised discrete Gronwall inequalities, Z. Angew. Math. Mech. 65 (1985), no. 12, 583–595 (English, with German and Russian summaries). MR 832170 (87e:26023), http://dx.doi.org/10.1002/zamm.19850651202
[2] Carl de Boor, Good approximation by splines with variable knots, Spline functions and approximation theory (Proc. Sympos., Univ. Alberta, Edmonton, Alta., 1972), Birkhäuser, Basel, 1973, pp. 57–72. Internat. Ser. Numer. Math., Vol. 21. MR 0403169 (53 #6982)
[3] Hermann Brunner, Nonpolynomial spline collocation for Volterra equations with weakly singular kernels, SIAM J. Numer. Anal. 20 (1983), no. 6, 1106–1119. MR 723827 (85d:65069), http://dx.doi.org/10.1137/0720080
[4] Hermann Brunner, The numerical solution of integral equations with weakly singular kernels, Numerical analysis (Dundee, 1983) Lecture Notes in Math., vol. 1066, Springer, Berlin, 1984, pp. 50–71. MR 760457 (85j:65043), http://dx.doi.org/10.1007/BFb0099518
[5] H. Brunner & I. G. Graham, "Product integration for weakly singular Volterra integral equations." (To appear.)
[6] H. G. Burchard, On the degree of convergence of piecewise polynomial approximation on optimal meshes, Trans. Amer. Math. Soc. 234 (1977), no. 2, 531–559. MR 0481758 (58 #1857), http://dx.doi.org/10.1090/S0002-9947-1977-0481758-4
[7] G. A. Chandler, Superconvergence of Numerical Methods to Second Kind Integral Equations, Ph. D. Thesis, Australian National University, Canberra, 1979.
[8] J. Dixon & S. McKee, Singular Gronwall Inequalities, Numerical Analysis Report NA/83/44, Hertford College, University of Oxford, 1983.
[9] I. G. Graham, The Numerical Solution of Fredholm Integral Equations of the Second Kind, Ph. D. Thesis, University of New South Wales, Kensington, 1980.
[10] Ivan G. Graham, Galerkin methods for second kind integral equations with singularities, Math. Comp. 39 (1982), no. 160, 519–533. MR 669644 (84d:65090), http://dx.doi.org/10.1090/S0025-5718-1982-0669644-3
[11] I. G. Graham, Estimates for the Modulus of Smoothness, Research Report No. 22, Dept. of Mathematics, University of Melbourne, 1982.
[12] Frank de Hoog and Richard Weiss, On the solution of a Volterra integral equation with a weakly singular kernel, SIAM J. Math. Anal. 4 (1973), 561–573. MR 0337045 (49 #1818)
[13] Ralph Baker Kearfott, A sinc approximation for the indefinite integral, Math. Comp. 41 (1983), no. 164, 559–572. MR 717703 (85g:65029), http://dx.doi.org/10.1090/S0025-5718-1983-0717703-X
[14] Ch. Lubich, Runge-Kutta theory for Volterra and Abel integral equations of the second kind, Math. Comp. 41 (1983), no. 163, 87–102. MR 701626 (85a:65178), http://dx.doi.org/10.1090/S0025-5718-1983-0701626-6
[15] S. McKee, Generalised discrete Gronwall lemmas, Z. Angew. Math. Mech. 62 (1982), no. 9, 429–434 (English, with German and Russian summaries). MR 682559 (84d:26016), http://dx.doi.org/10.1002/zamm.19820620902
[16] Richard K. Miller and Alan Feldstein, Smoothness of solutions of Volterra integral equations with weakly singular kernels, SIAM J. Math. Anal. 2 (1971), 242–258. MR 0287258 (44 #4465)
[17] John R. Rice, On the degree of convergence of nonlinear spline approximation, Approximations with Special Emphasis on Spline Functions (Proc. Sympos. Univ. of Wisconsin, Madison, Wis., 1969), Academic Press, New York, 1969, pp. 349–365. MR 0267324 (42 #2226)
[18] Herman J. J. te Riele, Collocation methods for weakly singular second-kind Volterra integral equations with nonsmooth solution, IMA J. Numer. Anal. 2 (1982), no. 4, 437–449. MR 692290 (84g:65167), http://dx.doi.org/10.1093/imanum/2.4.437
[19] Claus Schneider, Produktintegration mit nicht-äquidistanten Stützstellen, Numer. Math. 35 (1980), no. 1, 35–43 (German, with English summary). MR 583654 (81j:65049), http://dx.doi.org/10.1007/BF01396368
[20] Claus Schneider, Product integration for weakly singular integral equations, Math. Comp. 36 (1981), no. 153, 207–213. MR 595053 (82c:65090), http://dx.doi.org/10.1090/S0025-5718-1981-0595053-0
[21] Larry L. Schumaker, Spline functions: basic theory, John Wiley & Sons Inc., New York, 1981. Pure and Applied Mathematics; A Wiley-Interscience Publication. MR 606200 (82j:41001)
[22] A. F. Timan, Theory of approximation of functions of a real variable, Translated from the Russian by J. Berry. English translation edited and editorial preface by J. Cossar. International Series of Monographs in Pure and Applied Mathematics, Vol. 34, A Pergamon Press Book. The Macmillan Co., New York, 1963. MR 0192238 (33 #465)
[23] P. R. Uba, The method of piecewise-linear collocation on a nonuniform grid for the solution of integral equations with a singularity, Phys. Lett. A 87 (1981/82), no. 1-2, 52–57 (Russian, with German summary). MR 638855 (83b:45021)
[24] G. Vainikko, A. Pedas, and P. Uba, Metody resheniya slabo-singulyarnykh integralnykh uravnenii, Tartu. Gos. Univ., Tartu, 1984 (Russian). MR 766743 (86f:65221)
[25] G. Vainikko and P. Uba, A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel, J. Austral. Math. Soc. Ser. B 22 (1980/81), no. 4, 431–438. MR 626934 (82h:65100), http://dx.doi.org/10.1017/S0334270000002770
[26] R. DeVore and K. Scherer, Variable knot, variable degree spline approximation to 𝑥^{𝛽}, Quantitative approximation (Proc. Internat. Sympos., Bonn, 1979) Academic Press, New York, 1980, pp. 121–131. MR 588175 (81m:41008)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|