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Algorithm 689: Discretized collocation and iterated collocation for nonlinear Volterra integral equations of the second kind

Authors:

J. G. Blom,

H. BrunnerAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 17, Issue 2

Pages 167 - 177

https://doi.org/10.1145/108556.108562

Published: 01 June 1991 Publication History

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Abstract

This paper describes a FORTRAN code for calculating approximate solutions to systems of nonlinear Volterra integral equations of the second kind. The algorithm is based on polynomial spline collocation, with the possibility of combination with the corresponding iterated collocation. It exploits certain local superconvergence properties for the error estimation and the stepsize strategy.

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References

[1]

A~DERSON, R. L. Problem-solving software system for mathematical and statistical FOR- TRAN programming. IMSL user's manual, 1985.

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[2]

BLOM, J. G., AND BRUNNER, H. The numerical solution of nonlinear Volterra integral equations of the second kind by collocation and iterated collocation methods. SIAM J. Sc~. Statist. Comput. 8, 5 (Sept. 1987), 806-830.

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[3]

HETHCOTE, H. W., AND TUDOR, D. W. Integral equation models for endemic infectious diseases. J. Math. Biol. 9 (1980), 37-47.

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Hoppensteadt FJackiewicz ZZubik-Kowal B(2022)Numerical solution of Volterra integral and integro-differential equations with rapidly vanishing convolution kernels BIT10.1007/s10543-007-0122-347:2(325-350)Online publication date: 11-Mar-2022
https://dl.acm.org/doi/10.1007/s10543-007-0122-3
Zhao T(2014)Numerical Solution to Volterra-Type Integro-Differential Equations of the Second Kinds by Legendre Collocation MethodApplied Mechanics and Materials10.4028/www.scientific.net/AMM.687-691.1522687-691(1522-1527)Online publication date: Nov-2014
https://doi.org/10.4028/www.scientific.net/AMM.687-691.1522
Mazzia FNagy A(2014)Solving Volterra integro-differential equations by variable stepsize block BS methods: Properties and implementation techniquesApplied Mathematics and Computation10.1016/j.amc.2014.04.030239(198-210)Online publication date: Jul-2014
https://doi.org/10.1016/j.amc.2014.04.030
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Index Terms

Algorithm 689: Discretized collocation and iterated collocation for nonlinear Volterra integral equations of the second kind
1. Mathematics of computing
  1. Mathematical analysis
  2. Mathematical software
2. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 17, Issue 2

June 1991

131 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/108556

Editor:
John Rice
Purdue Univ., West Lafayette, IN

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Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

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Publication History

Published: 01 June 1991

Published in TOMS Volume 17, Issue 2

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Hoppensteadt FJackiewicz ZZubik-Kowal B(2022)Numerical solution of Volterra integral and integro-differential equations with rapidly vanishing convolution kernels BIT10.1007/s10543-007-0122-347:2(325-350)Online publication date: 11-Mar-2022
https://dl.acm.org/doi/10.1007/s10543-007-0122-3
Zhao T(2014)Numerical Solution to Volterra-Type Integro-Differential Equations of the Second Kinds by Legendre Collocation MethodApplied Mechanics and Materials10.4028/www.scientific.net/AMM.687-691.1522687-691(1522-1527)Online publication date: Nov-2014
https://doi.org/10.4028/www.scientific.net/AMM.687-691.1522
Mazzia FNagy A(2014)Solving Volterra integro-differential equations by variable stepsize block BS methods: Properties and implementation techniquesApplied Mathematics and Computation10.1016/j.amc.2014.04.030239(198-210)Online publication date: Jul-2014
https://doi.org/10.1016/j.amc.2014.04.030
Isaacson SKirby R(2011)Numerical solution of linear Volterra integral equations of the second kind with sharp gradientsJournal of Computational and Applied Mathematics10.1016/j.cam.2011.03.029235:14(4283-4301)Online publication date: 1-May-2011
https://dl.acm.org/doi/10.1016/j.cam.2011.03.029
Frishman EShapiro M(2006)Control of spontaneous emission in the presence of collisionsThe Journal of Chemical Physics10.1063/1.2173264124:8Online publication date: 27-Feb-2006
https://doi.org/10.1063/1.2173264
Conte DPrete I(2006)Fast collocation methods for Volterra integral equations of convolution typeJournal of Computational and Applied Mathematics10.1016/j.cam.2005.10.018196:2(652-663)Online publication date: 15-Nov-2006
https://dl.acm.org/doi/10.1016/j.cam.2005.10.018
Finlayson BBiegler LGrossmann I(2006)Mathematics in Chemical EngineeringUllmann's Encyclopedia of Industrial Chemistry10.1002/14356007.b01_01.pub2Online publication date: 15-Dec-2006
https://doi.org/10.1002/14356007.b01_01.pub2
Baker C(2001)A perspective on the numerical treatment of Volterra equationsNumerical Analysis: Historical Developments in the 20th Century10.1016/B978-0-444-50617-7.50017-3(415-447)Online publication date: 2001
https://doi.org/10.1016/B978-0-444-50617-7.50017-3
Baker C(2000)A perspective on the numerical treatment of Volterra equationsJournal of Computational and Applied Mathematics10.1016/S0377-0427(00)00470-2125:1-2(217-249)Online publication date: Dec-2000
https://doi.org/10.1016/S0377-0427(00)00470-2
Diethelm KFreed A(1999)On the Solution of Nonlinear Fractional-Order Differential Equations Used in the Modeling of ViscoplasticityScientific Computing in Chemical Engineering II10.1007/978-3-642-60185-9_24(217-224)Online publication date: 1999
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The Numerical Solution of Nonlinear Volterra Integral Equations of the Second Kind by Collocation and Iterated Collocation Methods

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