Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

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.

References [Enhancements On Off] (What's this?)

  • [1] P. R. Beesack, "More generalised discrete Gronwall inequalities," Preprint, 1983. MR 832170 (87e:26023)
  • [2] C. de Boor, "Good approximation by splines with variable knots," in Spline Functions and Approximation Theory (A. Meir and A. Sharma, eds.), Birkhäuser-Verlag, Basel, 1973, pp. 57-72. MR 0403169 (53:6982)
  • [3] H. Brunner, "Nonpolynomial spline collocation for Volterra equations with weakly singular kernels," SIAM J. Numer. Anal., v. 20, 1983, pp. 1106-1119. MR 723827 (85d:65069)
  • [4] H. Brunner, "The numerical solution of integral equations with weakly singular kernels," in Numerical Analysis, Dundee 1983 (D. F. Griffiths, ed.), Lecture Notes in Math., Vol. 1066, Springer-Verlag, Berlin, 1984, pp. 50-71. MR 760457 (85j:65043)
  • [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. II," Trans. Amer. Math. Soc., v. 234, 1977, pp. 531-559. MR 0481758 (58:1857)
  • [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] I. G. Graham, "Galerkin methods for second kind integral equations with singularities", Math. Comp., v. 39, 1982, pp. 519-533. MR 669644 (84d:65090)
  • [11] I. G. Graham, Estimates for the Modulus of Smoothness, Research Report No. 22, Dept. of Mathematics, University of Melbourne, 1982.
  • [12] F. de Hoog & R. Weiss, "On the solution of a Volterra integral equation with a weakly singular kernel," SIAM J. Math. Anal., v. 4, 1973, pp. 561-573. MR 0337045 (49:1818)
  • [13] R. B. Kearfott, "A sine approximation for the indefinite integral," Math. Comp., v. 41, 1983, pp. 559-572. MR 717703 (85g:65029)
  • [14] Ch. Lubich, "Runge-Kutta theory for Volterra and Abel integral equations of the second kind," Math. Comp., v. 41, 1983, pp. 87-102. MR 701626 (85a:65178)
  • [15] S. McKee, "Generalised discrete Gronwall lemmas," Z. Angew. Math. Mech., v. 62, 1982, pp. 429-434. MR 682559 (84d:26016)
  • [16] R. K. Miller and A. Feldstein, "Smoothness of solutions of Volterra integral equations with weakly singular kernels," SIAM J. Math. Anal., v. 2, 1971, pp. 242-258. MR 0287258 (44:4465)
  • [17] J. R. Rice, "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. MR 0267324 (42:2226)
  • [18] H. J. J. te Riele, "Collocation methods for weakly singular second-kind Volterra integral equations with non-smooth solution," IMA J. Numer. Anal., v. 2, 1982, pp. 437-449. MR 692290 (84g:65167)
  • [19] C. Schneider, "Produktintegration mit nicht-äquidistanten Stützstellen," Numer. Math., v. 35, 1980, pp. 35-43. MR 583654 (81j:65049)
  • [20] C. Schneider, "Product integration for weakly singular integral equations," Math. Comp., v. 36, 1981, pp. 207-213. MR 595053 (82c:65090)
  • [21] L. L. Schumaker, Spline Functions: Basic Theory, Wiley, New York, 1981. MR 606200 (82j:41001)
  • [22] A. F. Timan, Theory of Approximation of Functions of a Real Variable, Pergamon Press, Oxford, 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," Uchen. Zap. Tartu. Gos. Univ., v. 580, 1981, pp. 52-57. (Russian) MR 638855 (83b:45021)
  • [24] G. Vainikko, A. Pedas & P. Uba, Methods for Solving Weakly Singular Integral Equations, Tartu. Gos. Univ., Tartu, 1984. (Russian) MR 766743 (86f:65221)
  • [25] G. Vainikko & P. Uba, "A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel," J. Austral. Math. Soc. Ser. B., v. 22, 1981, pp. 431-438. MR 626934 (82h:65100)
  • [26] R. DeVore & K. Scherer, "Variable knot, variable degree spline approximation to $ {x^\beta }$," in Quantitative Approximation (R. DeVore and K. Scherer, eds), Academic Press, New York, 1980, pp. 121-131. MR 588175 (81m:41008)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65R20, 45D05

Retrieve articles in all journals with MSC: 65R20, 45D05

Additional Information

Keywords: Volterra integral equations, weakly singular kernels, polynomial spline collocation, graded meshes
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society