Variable step size predictorcorrector schemes for second kind Volterra integral equations
Authors:
H. M. Jones and S. McKee
Journal:
Math. Comp. 44 (1985), 391404
MSC:
Primary 65R20; Secondary 45D05
MathSciNet review:
777271
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Abstract: In this paper a family of implicit multistep methods for the solution of Volterra integral equations is derived. These methods together with an explicit Euler predictor permit the use of a variable step size when solving integral equations. Means of controlling the error and stability by varying the step size and the order of the method are described. Extensive numerical results are presented.
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 [1]
 C. T. H. Baker, The Numerical Treatment of Integral Equations, Oxford Univ. Press, Oxford, 1977. MR 0467215 (57:7079)
 [2]
 C. T. H. Baker & M. S. Keech, "Stability regions in the numerical treatment of Volterra integral equations," SIAM J. Numer. Anal., v. 15, 1978, pp. 394417. MR 0502101 (58:19265)
 [3]
 L. M. Delves & J. Walsh (Eds.), Numerical Solution of Integral Equations, Oxford Univ. Press, Oxford, 1974. MR 0464624 (57:4551)
 [4]
 P. Henrici, Elements of Numerical Analysis, Wiley, New York, 1964. MR 0166900 (29:4173)
 [5]
 H. M. Jones, "A variable step variable order package for solving Volterra integral equations of the second kind," (to be submitted to ACM Trans. Math. Software).
 [6]
 H. M. Jones & S. McKee, "The numerical stability of multistep methods for convolution Volterra integral equations: nonsingular equations," Comput. Math. Appl., v. 8, 1982, pp. 291303. MR 679402 (84a:65104)
 [7]
 Ch. Lubich, "On the stability of linear multistep methods for Volterra integral equations of the second kind," in Treatment of Integral Equations by Numerical Methods (C. T. H. Baker and G. F. Miller, eds.), Academic Press, LondonNew York, 1982. MR 755358
 [8]
 D. F. Mayers, "Equations of Volterra type," in Numerical Solution of Ordinary and Partial Differential Equations (L. Fox, ed.), Pergamon Press, New York, 1962, pp. 165173.
 [9]
 L. F. Shampine & M. K. Gordon, Computer Solution of Ordinary Differential Equations, Freeman. San Francisco, 1975. MR 0478627 (57:18104)
 [10]
 F. Smithies, Integral Equations: Cambridge Tracts in Mathematics and Mathematical Physics, No. 49, Cambridge Univ. Press, New York, 1958. MR 0104991 (21:3738)
 [11]
 F. G. Tricomi, Integral Equations, Interscience, New York, 1957. MR 0094665 (20:1177)
 [12]
 P. J. van der Houwen & P. H. M. Wolkenfelt, "On the stability of multistep formulas for Volterra integral equations of the second kind," Computing, v. 24, 1980, pp. 341347. MR 620232 (82e:65132)
 [13]
 H. J. J. te Riele, Proposal for a Test Set for Comparing Accuracy and Efficiency of Algorithms for the Numerical Solution of Volterra Integral Equations of the Second Kind With Nonsingular Kernels, M/C Rep. Math. Centrum, Amsterdam, 1978.
 [14]
 H. M. Williams, Variable StepSize PredictorCorrector Schemes for Volterra Second Kind Integral Equations, D.Phil. thesis, University of Oxford, Oxford, 1980.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198507772715
PII:
S 00255718(1985)07772715
Article copyright:
© Copyright 1985
American Mathematical Society
