Journal ArticleDOI
Multistep multiderivative methods for volterra integro-differential equations
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In this article, the ideas and theory of multistep multiderivative methods for solving ordinary differential equations are extended to Volterra integro-differential equations and a concept of zero-stability is given which, together with consistency, permits the derivation of an order-of-convergence result.Abstract:
The ideas and theory of multistep multiderivative methods for solving ordinary differential equations are extended to Volterra integro-differential equations. A concept of zero-stability is given which, together with consistency, permits the derivation of an order-of-convergence result. It is noted that lower order quadrature formulae may be used in the evaluation of the integrals involved in the derivatives of the function F without any deterioration in the global order of convergence. Numerical results are presented.read more
Citations
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Journal ArticleDOI
Applications and numerical analysis of partial differential Volterra equations : A brief survey
Simon Shaw,John Robert Whiteman +1 more
TL;DR: A concise survey of the numerical analysis of Volterra equations can be found in this article, which leads up to some recent results on a posteriori error estimation for finite element approximations.
Journal ArticleDOI
Best predictor-corrector methods for first order VIDEs with solutions damped at infinity
L. E. Garey,C. J. Gladwin +1 more
TL;DR: Here (explicit, implicit) pairs of linear multistep methods are considered for first order Volterra integro-differential equations, which attempt to reconcile several irreconcilable factors.
References
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Journal ArticleDOI
Second Derivative Multistep Methods for Stiff Ordinary Differential Equations
TL;DR: In this article, the difficulty associated with the numerical solution of stiff ordinary differential equations is considered and the stability requirements of methods suitable for stiff equations are described, where the authors consider the case of the case where the solution is known to be stable.
Journal ArticleDOI
High order a-stable methods for the numerical solution of systems of D.E.'s
Byron L. Ehle,Byron L. Ehle +1 more
TL;DR: In this article, one can obtain one-step methods of arbitrarily high order which satisfy Dahlquist's requirements of a-stability, and the authors show how to obtain one step methods of arbitrary high order that satisfy these requirements.
Journal ArticleDOI
On the Stability of Linear Multistep Methods for Volterra Convolution Equations
TL;DR: Les methodes multipas lineaires for des equations differentielles ordinaires engendrent des regles de quadrature de convolution as discussed by the authors, and le stabilite de telles methodes appliquees aux equations integrales de Volterra de seconde espece and aux equations of Volterras integrodifferentielle integrodeterminélles
Journal ArticleDOI
Linear Multistep Methods for Volterra Integro-Differential Equations
TL;DR: The Dahlquist stability analysis for ordinary differential equations is extended to the case of Volterra integro-differential equations, and the standard multistep methods can be generalized to furnish algorithms for solving Integro- differential equations.
Journal ArticleDOI
The Construction of Reducible Quadrature Rules for Volterra Integral and Integro-differential Equations
TL;DR: In this paper, a formal relationship between quadrature rules and linear multistep methods for ordinary differential equations is exploited for the generation of quadratures weights, and step-by-step methods for second kind Volterra integral equations and integro-differential equations are defined.
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