Abstract
The numerical solution of Volterra integral equations of the first kind can be achieved via product integration. This paper establishes the asymptotic error expansions of certain product integration rules. The rectangular rules are found to produce expansions containing all powers ofh, and the midpoint product method is found to produce even powers ofh. Extrapolation to the limit is then applied.
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References
A. S. Anderssen and E. T. White,Improved numerical methods for Volterra integral equations of the first kind, Comp. J. 14 (1971), 442–443.
C. Andrade and S. McKee,On optimal high accuracy linear multistep methods for first kind Volterra integral equations, BIT 19 (1979), 1–11.
C. T. H. Baker and M. S. Keech,Stability regions in the numerical treatment of Volterra integral equations, SIAM J. Numer. Anal. 15 (1978), 394–417.
M. Bôcher,An Introduction to the Study of Integral Equations, Hafner Publishing Co., New York, 1908.
H. Brunner,Discretization of Volterra integral equations of the first kind, Math. Comp. 31 (1977), 708–716.
id., Discretization of Volterra integral equations of the first kind, Numer. Math. 30 (1978), 117–136.
id., Superconvergence of collocation methods for Volterra integral equations of the first kind, Computing 21 (1979), 151–157.
F. De Hoog and R. Weiss,On the solution of Volterra integral equations of the first kind, Numer. Math. 21 (1973), 22–32.
id., High order methods for first kind Volterra integral equations, SIAM J. Numer. Anal. 10 (1973), 63–73.
L. Fox,Romberg integration for a class of singular integrands, Comput. J. 10 (1967), 87–93.
C. J. Gladwin,Quadrature rule methods for Volterra integral equations of the first kind, Math. Comp. 33 (1979), 705–716.
id., On optimal integration methods for Volterra integral equations of the first kind, Math. Comp. 39 (1982), 511–518.
C. J. Gladwin and R. Jeltsch,Stability of quadrature rule methods for first kind Volterra integral equations, BIT 14 (1974), 141–151.
P. A. W. Holyhead and S. McKee,Stability and convergence of multistep methods for linear Volterra integral equations of the first kind, SIAM J. Numer. Anal. 13 (1976), 269–292.
P. A. W. Holyhead, S. McKee and P. J. Taylor,Multistep methods for solving linear Volterra integral equations of the first kind, SIAM J. Numer. Anal. 12 (1975), 698–711.
J. G. Jones,On the numerical solution of convolution integral equations and systems of such equations, Math. Comp. 12 (1961), 131–142.
M. Kobayasi,On the numerical solution of the Volterra integral equations of the first kind by the trapezoidal rule, Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs. (1967), 1–14.
P. Linz,The numerical solution of Volterra integral equations by finite difference methods, Tech. Rept. 825, Mathematics Research Center, University of Wisconsin, 1967.
id., Numerical methods for Volterra integral equations of the first kind, Comput. J. 12 (1969), 393–397.
id., Product integration methods for Volterra integral equations of the first kind, BIT 11 (1971), 413–421.
id., Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, 1985.
L. G. McAlevey,On the numerical solution of first kind Volterra integral equations, Ph.D. Thesis, Mathematics Dept., University of Otago, New Zealand, 1982.
G. I. Marchuk and V. V. Shaidurov,Difference Methods and their Extrapolations, Springer-Verlag, New York, 1983.
P. J. Taylor,The solution of Volterra integral equations of the kind using inverted differentiation formulae, BIT 16 (1976), 416–425.
P. H. M. Wolkenfelt,Reducible quadrature methods for Volterra integral equations of the first kind, BIT 21 (1981), 232–241.
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McAlevey, L.G. Product integration rules for volterra integral equations of the first kind. BIT 27, 235–247 (1987). https://doi.org/10.1007/BF01934187
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DOI: https://doi.org/10.1007/BF01934187