On optimal integration methods for Volterra integral equations of the first kind
Author:
C. J. Gladwin
Journal:
Math. Comp. 39 (1982), 511518
MSC:
Primary 65R20; Secondary 45D05, 45L10
MathSciNet review:
669643
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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 norm of the roots of the polynomial is minimized.
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 [1]
 C. Andrade & S. Mckee, "On optimal high accuracy linear multistep methods for first kind Volterra integral equations," BIT, v. 19, 1979, pp. 111. 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 8271510223, 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. 196213. 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. 705716. 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. 144151. 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. 698711. 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. 269292. 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. 114. 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, McGrawHill, 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. 416425. MR 0433930 (55:6900)
 [15]
 P. H. M. Wolkenfelt, Linear Multistep Methods and the Construction of Quadrature Formulae for Volterra Integral and IntegroDifferential Equations, Report No. NW 76/79, Mathematisch Centrum, Amsterdam, 1979. MR 569940 (83e:65216)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198206696431
PII:
S 00255718(1982)06696431
Article copyright:
© Copyright 1982
American Mathematical Society
