## Modified multilag methods for Volterra functional equations

HTML articles powered by AMS MathViewer

- by P. H. M. Wolkenfelt PDF
- Math. Comp.
**40**(1983), 301-316 Request permission

## Abstract:

Linear multistep methods for ordinary differential equations in conjunction with a family of computationally efficient quadrature rules are employed to define a class of so-called multilag methods for the solution of Volterra integral and integro-differential equations. In addition, modified multilag methods are proposed which have the property that the stability behavior is independent of the choice of the quadrature rules. High order convergence of the methods is established. In particular, a special class of high order convergent methods is presented for the efficient solution of first-kind Volterra equations. Numerical experiments are reported.## References

- Celia Andrade and S. McKee,
*On optimal high accuracy linear multistep methods for first kind Volterra integral equations*, BIT**19**(1979), no. 1, 1–11. MR**530109**, DOI 10.1007/BF01931215 - Christopher T. H. Baker,
*The numerical treatment of integral equations*, Monographs on Numerical Analysis, Clarendon Press, Oxford, 1977. MR**0467215** - 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**502101**, DOI 10.1137/0715025 - Christopher T. H. Baker, Athena Makroglou, and Edward Short,
*Regions of stability in the numerical treatment of Volterra integro-differential equations*, SIAM J. Numer. Anal.**16**(1979), no. 6, 890–910. MR**551314**, DOI 10.1137/0716066 - H. Brunner and J. D. Lambert,
*Stability of numerical methods for Volterra integro-differential equations*, Computing (Arch. Elektron. Rechnen)**12**(1974), no. 1, 75–89 (English, with German summary). MR**418490**, DOI 10.1007/bf02239501 - Charles J. Gladwin,
*Quadrature rule methods for Volterra integral equations of the first kind*, Math. Comp.**33**(1979), no. 146, 705–716. MR**521284**, DOI 10.1090/S0025-5718-1979-0521284-2 - C. J. Gladwin and R. Jeltsch,
*Stability of quadrature rule methods for first kind Volterra integral equations*, Nordisk Tidskr. Informationsbehandling (BIT)**14**(1974), 144–151. MR**502108**, DOI 10.1007/bf01932943 - Peter Henrici,
*Discrete variable methods in ordinary differential equations*, John Wiley & Sons, Inc., New York-London, 1962. MR**0135729** - Wilhelm Hock,
*Asymptotic expansions for multistep methods applied to nonlinear Volterra integral equations of the second kind*, Numer. Math.**33**(1979), no. 1, 77–100. MR**545744**, DOI 10.1007/BF01396497 - 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), no. 5, 698–711. MR**413564**, DOI 10.1137/0712052 - P. J. van der Houwen,
*On the numerical solution of Volterra integral equations of the second kind. I. Stability*, Mathematisch Centrum, Afdeling Numerieke Wiskunde, No. NW 42/77. [Mathematical Center, Numerical Mathematics Section, No. NW 42/77], Mathematisch Centrum, Amsterdam, 1977. MR**0495061** - P. J. van der Houwen,
*Convergence and stability results in Runge-Kutta type methods for Volterra integral equations of the second kind*, BIT**20**(1980), no. 3, 375–377. MR**595219**, DOI 10.1007/BF01932780 - P. J. van der Houwen, P. H. M. Wolkenfelt, and C. T. H. Baker,
*Convergence and stability analysis for modified Runge-Kutta methods in the numerical treatment of second-kind Volterra integral equations*, IMA J. Numer. Anal.**1**(1981), no. 3, 303–328. MR**641312**, DOI 10.1093/imanum/1.3.303 - J. D. Lambert,
*Computational methods in ordinary differential equations*, John Wiley & Sons, London-New York-Sydney, 1973. Introductory Mathematics for Scientists and Engineers. MR**0423815** - Peter Linz,
*Linear multistep methods for Volterra integro-differential equations*, J. Assoc. Comput. Mach.**16**(1969), 295–301. MR**239786**, DOI 10.1145/321510.321521 - J. Matthys,
*$A$-stable linear multistep methods for Volterra integro-differential equations*, Numer. Math.**27**(1976/77), no. 1, 85–94. MR**436638**, DOI 10.1007/BF01399087 - Malcolm S. Keech,
*A third order, semi-explicit method in the numerical solution of first kind Volterra integral equations*, Nordisk Tidskr. Informationsbehandling (BIT)**17**(1977), no. 3, 312–320. MR**474918**, DOI 10.1007/bf01932151 - P. H. M. Wolkenfelt, P. J. van der Houwen, and Chr. T. H. Baker,
*Analysis of numerical methods for second kind Volterra equations by imbedding techniques*, J. Integral Equations**3**(1981), no. 1, 61–82. MR**604316** - P. H. M. Wolkenfelt,
*Reducible quadrature methods for Volterra integral equations of the first kind*, BIT**21**(1981), no. 2, 232–241. MR**627884**, DOI 10.1007/BF01933168 - P. H. M. Wolkenfelt,
*The construction of reducible quadrature rules for Volterra integral and integro-differential equations*, IMA J. Numer. Anal.**2**(1982), no. 2, 131–152. MR**668589**, DOI 10.1093/imanum/2.2.131 - P. H. M. Wolkenfelt,
*Reducible quadrature methods for Volterra integral equations of the first kind*, BIT**21**(1981), no. 2, 232–241. MR**627884**, DOI 10.1007/BF01933168

## Additional Information

- © Copyright 1983 American Mathematical Society
- Journal: Math. Comp.
**40**(1983), 301-316 - MSC: Primary 65R20
- DOI: https://doi.org/10.1090/S0025-5718-1983-0679447-2
- MathSciNet review: 679447