General linear methods with large stability regions for Volterra integral equations
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General linear methods with large stability regions for Volterra integral equations Ali Abdi1 Received: 13 October 2018 / Revised: 21 January 2019 / Accepted: 22 January 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract General linear methods are well-known as a large family of methods for the numerical solution of Volterra integral equations of the second kind. This paper is concerned with the construction of such methods with a large region of order p and high stage order q = p with a large region of absolute stability. Some numerical results are presented which indicate the effectiveness of the proposed schemes. Keywords Volterra integral equations · General linear methods · Order conditions · Region of absolute stability Mathematics Subject Classification 65R20 · 45M10
1 Introduction The modeling of certain physical phenomena with history can usually be modeled by a system of Volterra integral equations (VIEs) of the second kind t y(t) = f (t) +
K (t, s, y(s)) ds,
t ∈ I := [t0 , T ],
(1)
t0
is the unknown multivalue function, f : I → Rm and K : S × Rm → in which y : I → m R are assumed to be given functions where m is the dimensionality of the system and S := {(t, s) : t0 ≤ s ≤ t ≤ T }. We assume that the given functions f and K are smooth enough such that system (1) has a unique solution y (Brunner and van der Houwen 1986; Linz 1985). Numerous numerical methods have been introduced to approximate the solution of VIEs (1) (see, for instance, Blom and Brunner 1987; Brunner and Evans 1977; Brunner and Nørsett Rm
Communicated by Hui Liang.
B 1
Ali Abdi [email protected] Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
123
A. Abdi
1981; Conte and Paternoster 2009; Hock 1981; Hoppensteadt et al. 2007; Izzo et al. 2010). Runge–Kutta formulae of Bel’tyukov type (Brunner and van der Houwen 1986) are the most popular and efficient methods for (1). Also, collocation methods, two-step almost collocation methods and multistep collocation methods for the numerical solution of VIEs are investigated in Brunner (2004), Conte et al. (2008) and Conte and Paternoster (2009). Another familiar class of methods are those based on a direct quadrature rule. Two direct quadrature methods (Global and composite methods) based on the linear barycentric rational quadrature rule have been introduced by Berrut et al. (2014) and stability properties for the composite one has been analyzed by Hosseini and the author in Hosseini and Abdi (2016). We refer to Abdi et al. (2018), Abdi and Hosseini (2018), Berrut and Trefethen (2004) and Klein and Berrut (2012) for discussions of barycentric interpolation and its applications. General linear methods (GLMs) as a unifying framework for the traditional numerical methods for solving initial value problems (IVPs)
y (t) = f (y(t)), y(t0 ) = y0 ,
t ∈ [t0 , T ],
(2)
where f : Rm → Rm , were introduced by Butcher (1966, 2016) and extended to second derivative general linear methods (SGLMs) to cover second derivative methods
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