Volterra Integro-Differential Equations (VIDEs) are models of evolutionary problems with memory in many applications. Generally, the numerical treatment of such equations leads to an high computational cost, especially due to the computation of the “lag-term", which contains the past history of the phenomenon. As it is well known, a collocation method is based on the idea of approximating the exact solution of a given equation with a suitable function belonging to a chosen finite dimensional space, usually a piecewise algebraic polynomial, which satisfies the equation exactly on a certain subset of the integration interval (called the set of collocation points). Multistep collocation methods and two-step Runge-Kutta methods have been already developed in the context of Ordinary Differential Equations  and Volterra Integral Equations [1,2]. In this talk we describe the derivation of multistep collocation methods for VIDEs, where the numerical solution depends on the approximation of the solution in a fixed number of previous time steps, instead of only one step, as in the standard collocation. This approach allows to increase the order of classical one{step methods, without any increase of the computational cost. We analyze the order of convergence of the proposed methods and provide the stability analysis.  D. Conte, Z. Jackiewicz, B. Paternoster, Two-step almost collocation methods for Volterra integral equations, Appl. Math. Comput., 204 (2008), pp. 839-853.  D. Conte, B. Paternoster, Multistep collocation methods for Volterra Integral Equations,Appl. Num. Math., 59, n. 8, (2009), pp. 1721-1736.  R. D'Ambrosio, M. Ferro, Z. Jackiewicz, B.Paternoster, Two-step almost collocation methods for Ordinary Di_erential Equations, Numerical Algorithms (2009), in press.

### Multistep collocation methods for Volterra Integro-Differential Equations

#### Abstract

Volterra Integro-Differential Equations (VIDEs) are models of evolutionary problems with memory in many applications. Generally, the numerical treatment of such equations leads to an high computational cost, especially due to the computation of the “lag-term", which contains the past history of the phenomenon. As it is well known, a collocation method is based on the idea of approximating the exact solution of a given equation with a suitable function belonging to a chosen finite dimensional space, usually a piecewise algebraic polynomial, which satisfies the equation exactly on a certain subset of the integration interval (called the set of collocation points). Multistep collocation methods and two-step Runge-Kutta methods have been already developed in the context of Ordinary Differential Equations  and Volterra Integral Equations [1,2]. In this talk we describe the derivation of multistep collocation methods for VIDEs, where the numerical solution depends on the approximation of the solution in a fixed number of previous time steps, instead of only one step, as in the standard collocation. This approach allows to increase the order of classical one{step methods, without any increase of the computational cost. We analyze the order of convergence of the proposed methods and provide the stability analysis.  D. Conte, Z. Jackiewicz, B. Paternoster, Two-step almost collocation methods for Volterra integral equations, Appl. Math. Comput., 204 (2008), pp. 839-853.  D. Conte, B. Paternoster, Multistep collocation methods for Volterra Integral Equations,Appl. Num. Math., 59, n. 8, (2009), pp. 1721-1736.  R. D'Ambrosio, M. Ferro, Z. Jackiewicz, B.Paternoster, Two-step almost collocation methods for Ordinary Di_erential Equations, Numerical Algorithms (2009), in press.
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2009
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11386/4505866`
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