Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

On optimal integration methods for Volterra integral equations of the first kind


Author: C. J. Gladwin
Journal: Math. Comp. 39 (1982), 511-518
MSC: Primary 65R20; Secondary 45D05, 45L10
DOI: https://doi.org/10.1090/S0025-5718-1982-0669643-1
MathSciNet review: 669643
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Families of methods depending on free parameters are constructed for the solution of nonsingular Volterra integral equations of the first kind in [5]. These parameters are restricted to certain regions in order that a certain polynomial satisfies both a stability and a consistency condition. In this note an optimal choice of the free parameters is outlined in order that the $ {l_2}$-norm of the roots of the polynomial is minimized.


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

  • [1] C. Andrade & S. Mckee, "On optimal high accuracy linear multistep methods for first kind Volterra integral equations," BIT, v. 19, 1979, pp. 1-11. MR 530109 (83f:65201)
  • [2] H. Brunner, The Approximate Solution of Integral Equations by Projection Methods Based on Collocation, Mathematics and Computation No. 1/78, ISBN 82-7151-022-3, Dept. of Math., The University of Trondheim, Trondheim, Norway, 1978. MR 0411213 (53:14951)
  • [3] R. J. Duffin, "Algorithms for classical stability problems," SIAM Rev., v. 11, 1969, pp. 196-213. MR 0249740 (40:2981)
  • [4] H. Freeman, Finite Differences for Actuarial Students, Cambridge Univ. Press, Cambridge, 1962, p. 113.
  • [5] C. J. Gladwin, "Quadrature rule methods for Volterra integral equations of the first kind," Math. Comp., v. 33, 1979, pp. 705-716. MR 521284 (80f:65144)
  • [6] C. J. Gladwin, Numerical Solution of Volterra Integral Equations of the First Kind, Ph.D. Thesis, Dalhousie Univ., 1975.
  • [7] C. J. Gladwin & R. Jeltsch, "Stability of quadrature rules for first kind Volterra integral equations," BIT, v. 14, 1974, pp. 144-151. MR 0502108 (58:19272)
  • [8] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, 1962. MR 0135729 (24:B1772)
  • [9] P. A. W. Holyhead, S. Mckee & P. J. Taylor, "Multistep methods for solving linear Volterra integral equations of the first kind," SIAM J. Numer. Anal., v. 12, 1975, pp. 698-711. MR 0413564 (54:1678)
  • [10] P. A. W. Holyhead & S. Mckee, "Stability and convergence of multistep methods for linear Volterra integral equations of the first kind," SIAM J. Numer. Anal., v. 13, 1976, pp. 269-292. MR 0471396 (57:11130)
  • [11] M. Kobayasi, "On numerical solutions of Volterra integral equations of the first kind by the trapezoidal rule," Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs., v. 14, 1967, pp. 1-14. MR 0260221 (41:4849)
  • [12] W. Pogorzelski, Integral Equations and their Applications, Vol. 1, Pergamon Press, Oxford, 1966, p. 14. MR 0201934 (34:1811)
  • [13] A. Ralston, A First Course in Numerical Analysis, McGraw-Hill, New York, 1965, p. 152. MR 0191070 (32:8479)
  • [14] P. J. Taylor, "The solution of Volterra integral equations of the first kind using inverted differentiation formulae," BIT, v. 16, 1976, pp. 416-425. MR 0433930 (55:6900)
  • [15] P. H. M. Wolkenfelt, Linear Multistep Methods and the Construction of Quadrature Formulae for Volterra Integral and Integro-Differential Equations, Report No. NW 76/79, Mathematisch Centrum, Amsterdam, 1979. MR 569940 (83e:65216)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65R20, 45D05, 45L10

Retrieve articles in all journals with MSC: 65R20, 45D05, 45L10


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1982-0669643-1
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society