A posteriori error estimation in maximum norm for a system of singularly perturbed Volterra integro-differential equatio
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A posteriori error estimation in maximum norm for a system of singularly perturbed Volterra integro-differential equations Ying Liang1 · Li-Bin Liu1 · Zhongdi Cen2 Received: 12 May 2020 / Revised: 16 July 2020 / Accepted: 14 August 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, a system of singularly perturbed Volterra integro-differential equations is considered. The backward Euler formula is used to discretize the differential part and the right-hand rectangle rule is applied to approximate the integral term. The stabilities of the continuous and discrete solutions are carried out using the Grönwall’s inequality, respectively. The a posterior error bounds are given to design an adaptive grid generation algorithm. Numerical results complement the theoretical results. Keywords Singularly perturbed · Volterra integro-differential equations · A posterior error estimate · Adaptive grid Mathematics Subject Classification 65L05 · 65L20 · 65L50
1 Introduction Volterra integro-differential and systems of Volterra integro-differential equations have been a major source of research work because of their fruitful territory for applications such as physics, biology, ecology, and so on (see, e.g., Rashidinia and Tahmasebi 2012; Markowich and Renardy 1983; Zhao 2003). For this reason, the numerical solution of such problems has attracted much attention over the last decades; see, for example, the monograph (Brunner 2004). Recently, Liang and Brunner (2020) proposed collocation methods for systems of linear Volterra-integro-differential algebraic equations. Berenguer et al. (2013) presented an approximation method to solve a system of Volterra integro-differential equations. These problems with a small parameter ε multiplying the highest order derivative term are said to be singularly perturbed Volterra integro-differential equations. As far as we know,
Communicated by Hui Liang.
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Li-Bin Liu [email protected]
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School of Mathematics and Statistics, Nanning Normal University, Nanning 530029, China
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Institute of Mathematics, Zhejiang Wanli University, Ningbo 315100, China 0123456789().: V,-vol
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many robust numerical methods have been developed for the case of single singularly perturbed Volterra integro-differential equation (see, e.g., Kauthen 1993; Horvat and Rogina 2002; Amiraliyev and Sevgin ¸ 2005; Sevgin ¸ 2014; Huang et al. 2020; Iragi and Munyakazi 2020; Yapman and Amiraliyev 2020; Yapman et al. 2019). It should be pointed out that most of these methods were constructed on the meshes of Bakhvalov and Shishkin types based on some priori information. In Kauthen (1995), Kauthen considered a system of singularly perturbed Volterra integro-differential equations with one perturbation parameter ε and developed a class of implicit Pouzet–Volterra–Runge–Kutta methods on a uniform mesh. However, the adaptive grid methods based on the a posteriori error estimation for a system of singularly perturbed Volterra integro-diffe
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