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 -norm of the roots of the polynomial is minimized.

**[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)**

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