Multistep Hermite collocation methods for solving Volterra Integral Equations

Fazeli, Somayyeh; Hojjati, Gholamreza; Shahmorad, S.

doi:10.1007/s11075-011-9510-5

Cited by 11 publications

(3 citation statements)

References 14 publications

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“…We also underline that they have high uniform order, thus they do not suffer from the order reduction phenomenon, well-known in the ODEs context [9]. Other approaches, based on multistep collocation, have been proposed in [25][26][27][28][29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review

Cardone

Conte

D’Ambrosio

et al. 2018

Axioms

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We present a collection of recent results on the numerical approximation of Volterra integral equations and integro-differential equations by means of collocation type methods, which are able to provide better balances between accuracy and stability demanding. We consider both exact and discretized one-step and multistep collocation methods, and illustrate main convergence results, making some comparisons in terms of accuracy and efficiency. Some numerical experiments complete the paper.

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Section: Introductionmentioning

confidence: 99%

Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review

Cardone

Conte

D’Ambrosio

et al. 2018

Axioms

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“…(1) have been explained by Seyed Allaei et al in Allaei et al (2017), and an analysis of the collocation methods for nonlinear Volterra integral equations of the third kind has been presented by Song et al Song et al (2019). The general multistep collocation methods for solving second-kind Volterra integral equations have been established in Conte and Paternoster (2009), and moreover the multistep Hermite collocation methods have been studied in Fazeli et al (2012).…”

Section: Introductionmentioning

confidence: 99%

A numerical method for solving Volterra integral equations of the third kind by multistep collocation method

2019

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In this paper a multistep collocation method for solving Volterra integral equations of the third kind is explained and analyzed. The structure of the method, its solvability and convergence analysis are investigated. Moreover to show the applicability of the presented method and to confirm our theoretical results some numerical examples are given. Keywords Volterra integral equations of the third kind • Collocation method • Multistep collocation method Mathematics Subject Classification 45A05 • 45D05 • 45E99 where 0 ≤ α < 1, 0 < β ≤ 1, f (x) = x β g(x) with g(x) ∈ C(I), k is a real continuous function defined on D = {(x, t) : 0 ≤ t ≤ x ≤ T } and y(x) is an unknown function. Volterra integral equations of the third kind have appeared in modeling numerous problems in various branches of science and engineering, such as heat transfer, population growth models Communicated by Hui Liang.

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“…These integral equations are called stiff [2,13], in analogy with stiff ODEs. To solve stiff VIEs numerically, the applied method must have some reasonably wide region of absolute stability [6,8,9,13]. In this regard, A-and V 0 -stable RungeKutta methods for VIEs have been introduced in [1,5,11].…”

Section: Introductionmentioning

confidence: 99%

Construction of efficient general linear methods for stiff Volterra integral equations

Abdi

Fazeli

Hojjati

2016

Journal of Computational and Applied Mathematics

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General linear methods in the Nordsieck form have been introduced for the numerical solution of Volterra integral equations. In this paper, we introduce general linear methods of order p and stage order q = p for the numerical solution of Volterra integral equations in general form, rather than Nordsieck form. A-and V 0 (α)-stable methods are constructed and applied on stiff problems to show their efficiency.

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Multistep Hermite collocation methods for solving Volterra Integral Equations

Cited by 11 publications

References 14 publications

Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review

Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review

A numerical method for solving Volterra integral equations of the third kind by multistep collocation method

Construction of efficient general linear methods for stiff Volterra integral equations

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