Taylor series methods for the solution of Volterra integral and integrodifferential equations
Author:
Alan Goldfine
Journal:
Math. Comp. 31 (1977), 691707
MSC:
Primary 65R05
MathSciNet review:
0440970
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Algorithms based on the use of Taylor series are developed for the numerical solution of Volterra integral and integrodifferential equations of arbitrary order. It is shown that these algorithms are uniformly convergent, bounds are obtained for the truncation error, and an asymptotic error analysis is provided for the integral equation case. The various problems of computer implementation are discussed, and the results of certain experiments suggested by the theory are presented.
 [1]
D. BARTON, I. M. WILLER & R. V. ZAHAR, "Taylor series methods for ordinary differential equations," Mathematical Software, Academic Press, New York, 1971.
 [2]
J. A. BRAUN & R. E. MOORE, A Program for the Solution of Differential Equations Using Interval Arithmetic (DIFEQ) for the CDC 3600 and 1604, MRC Report No. 901, Math. Research Center, Univ. of Wisconsin, Madison, 1968.
 [3]
G.
M. Campbell and J.
T. Day, A block by block method for the numerical solution of
Volterra integral equations, Nordisk Tidskr. Informationsbehandling
(BIT) 11 (1971), 120–124. MR 0281375
(43 #7093)
 [4]
J.
T. Day, A RungeKutta method for the numerical solution of the
Goursat problem in hyperbolic partial differential equations, Comput.
J. 9 (1966), 81–83. MR 0192665
(33 #890)
 [5]
J.
T. Day, On the numerical solution of integrodifferential
equations, Nordisk Tidskr. Informationsbehandling (BIT)
10 (1970), 511–514. MR 0280028
(43 #5749)
 [6]
A. FELDSTEIN & J. SOPKA, "Numerical methods for nonlinear Volterra integrodifferential equations," unpublished (October, 1968).
 [7]
Alan
Goldfine, An algorithm for the numerical solution of
integrodifferential equations, Nordisk Tidskr. Informationsbehandling
(BIT) 12 (1972), 578–580. MR 0324932
(48 #3281)
 [8]
Peter
Henrici, Discrete variable methods in ordinary differential
equations, John Wiley & Sons Inc., New York, 1962. MR 0135729
(24 #B1772)
 [9]
Gershon
Kedem, Automatic differentiation of computer programs, ACM
Trans. Math. Software 6 (1980), no. 2, 150–165.
MR 572087
(81g:68058), http://dx.doi.org/10.1145/355887.355890
 [10]
K. KLINGEN & B. GEISLER, Punching FORMAC Statements, Inst. for Applied Math., 517 Julich, Germany, 1971.
 [11]
P. LINZ, The Numerical Solution of Volterra Integral Equations by Finite Difference Methods, MRC Report No. 825, Math. Res. Center, Univ. of Wisconsin, Madison, 1967.
 [12]
Peter
Linz, Linear multistep methods for Volterra integrodifferential
equations., J. Assoc. Comput. Mach. 16 (1969),
295–301. MR 0239786
(39 #1143)
 [13]
Peter
Linz, A method for solving nonlinear
Volterra integral equations of the second kind, Math. Comp. 23 (1969), 595–599. MR 0247794
(40 #1055), http://dx.doi.org/10.1090/S00255718196902477947
 [14]
R. TOBEY, J. BAKER, R. CREWS, P. MARKS & K. VICTOR, PL/1Interpreter User's Reference Manual, IBM #360D03.33004, 1967.
 [15]
M.
A. Wolfe and G.
M. Phillips, Some methods for the solution of nonsingular Volterra
integrodifferential equations, Comput. J. 11
(1968/1969), 334–336. MR 0235762
(38 #4065)
 [1]
 D. BARTON, I. M. WILLER & R. V. ZAHAR, "Taylor series methods for ordinary differential equations," Mathematical Software, Academic Press, New York, 1971.
 [2]
 J. A. BRAUN & R. E. MOORE, A Program for the Solution of Differential Equations Using Interval Arithmetic (DIFEQ) for the CDC 3600 and 1604, MRC Report No. 901, Math. Research Center, Univ. of Wisconsin, Madison, 1968.
 [3]
 G. M. CAMPBELL & J. T. DAY, "A block by block method for the numerical solution of Volterra integral equations," BIT, v. 11, 1971, pp. 120124. MR 43 #7093. MR 0281375 (43:7093)
 [4]
 J. T. DAY, "A note on the numerical solution of integrodifferential equations," Comput. J., v. 9, 1967, pp. 394395. MR 0192665 (33:890)
 [5]
 J. T. DAY, "On the numerical solution of integrodifferential equations," BIT, v. 10, 1970, pp. 511514. MR 43 #5749. MR 0280028 (43:5749)
 [6]
 A. FELDSTEIN & J. SOPKA, "Numerical methods for nonlinear Volterra integrodifferential equations," unpublished (October, 1968).
 [7]
 A. GOLDFINE, "An algorithm for the numerical solution of integrodifferential equations," BIT, v. 12, 1972, pp. 578580. MR 48 #3281. MR 0324932 (48:3281)
 [8]
 P. HENRICI, Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, 1962. MR 24 #B1772. MR 0135729 (24:B1772)
 [9]
 G. KEDEM, Automatic Differentiation of Computer Programs, MRC Report, Math. Res. Center, Univ. of Wisconsin, Madison. (To appear.) MR 572087 (81g:68058)
 [10]
 K. KLINGEN & B. GEISLER, Punching FORMAC Statements, Inst. for Applied Math., 517 Julich, Germany, 1971.
 [11]
 P. LINZ, The Numerical Solution of Volterra Integral Equations by Finite Difference Methods, MRC Report No. 825, Math. Res. Center, Univ. of Wisconsin, Madison, 1967.
 [12]
 P. LINZ, "Linear multistep methods for Volterra integrodifferential equations," J. Assoc. Comput. Mach., v. 16, 1969, pp. 293301. MR 39 #1143. MR 0239786 (39:1143)
 [13]
 P. LINZ, "A method for solving nonlinear Volterra integral equations of the second kind," Math. Comp., v. 23, 1969, pp. 595599. MR 40 #1055. MR 0247794 (40:1055)
 [14]
 R. TOBEY, J. BAKER, R. CREWS, P. MARKS & K. VICTOR, PL/1Interpreter User's Reference Manual, IBM #360D03.33004, 1967.
 [15]
 M. A. WOLFE & G. M. PHILLIPS, "Some methods for the solution of nonsingular Volterra integrodifferential equations," Comput. J., v. 11, 1968/69, pp. 334336. MR 38 #4065. MR 0235762 (38:4065)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65R05
Retrieve articles in all journals
with MSC:
65R05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197704409704
PII:
S 00255718(1977)04409704
Keywords:
Volterra integral equation,
Volterra integrodifferential equation,
Taylor series method,
initial value problems,
symbolic differentiation
Article copyright:
© Copyright 1977 American Mathematical Society
