Variable step size predictor-corrector schemes for second kind Volterra integral equations

Authors:
H. M. Jones and S. McKee

Journal:
Math. Comp. **44** (1985), 391-404

MSC:
Primary 65R20; Secondary 45D05

DOI:
https://doi.org/10.1090/S0025-5718-1985-0777271-5

MathSciNet review:
777271

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]**Christopher T. H. Baker,*The numerical treatment of integral equations*, Clarendon Press, Oxford, 1977. Monographs on Numerical Analysis. MR**0467215****[2]**Christopher T. H. Baker and Malcolm S. Keech,*Stability regions in the numerical treatment of Volterra integral equations*, SIAM J. Numer. Anal.**15**(1978), no. 2, 394–417. MR**0502101**, https://doi.org/10.1137/0715025**[3]**L. M. Delves (ed.),*Numerical solution of integral equations*, Clarendon Press, Oxford, 1974. A collection of papers based on the material presented at a joint Summer School in July 1973, organized by the Department of Mathematics, University of Manchester, and the Department of Computational and Statistical Science, University of Liverpool. MR**0464624****[4]**Peter Henrici,*Elements of numerical analysis*, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR**0166900****[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 and S. McKee,*The numerical stability of multistep methods for convolution Volterra integral equations. Nonsingular equations*, Comput. Math. Appl.**8**(1982), no. 4, 291–303. MR**679402**, https://doi.org/10.1016/0898-1221(82)90011-6**[7]**Ch. Lubich,*On the stability of linear multistep methods for Volterra integral equations of the second kind*, Treatment of integral equations by numerical methods (Durham, 1982) Academic Press, London, 1982, pp. 233–238. 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. 165-173.**[9]**L. F. Shampine and M. K. Gordon,*Computer solution of ordinary differential equations*, W. H. Freeman and Co., San Francisco, Calif., 1975. The initial value problem. MR**0478627****[10]**F. Smithies,*Integral equations*, Cambridge Tracts in Mathematics and Mathematical Physics, no. 49, Cambridge University Press, New York, 1958. MR**0104991****[11]**F. G. Tricomi,*Integral equations*, Pure and Applied Mathematics. Vol. V, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957. MR**0094665****[12]**P. J. van der Houwen and P. H. M. Wolkenfelt,*On the stability of multistep formulas for Volterra integral equations of the second kind*, Computing**24**(1980), no. 4, 341–347 (English, with German summary). MR**620232**, https://doi.org/10.1007/BF02237819**[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 Step-Size Predictor-Corrector Schemes for Volterra Second Kind Integral Equations*, D.Phil. thesis, University of Oxford, Oxford, 1980.

Retrieve articles in *Mathematics of Computation*
with MSC:
65R20,
45D05

Retrieve articles in all journals with MSC: 65R20, 45D05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1985-0777271-5

Article copyright:
© Copyright 1985
American Mathematical Society