Abstract
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.
Highlights
It is the purpose of this paper to illustrate recent results on collocation methods for Volterra integral equations (VIEs) and Volterra integro-differential equations (VIDEs), mainly due to the authors
We present some numerical results which confirm that, differently from one step collocation methods, the two step almost collocation (TSAC) methods do not suffer form the order reduction in the integration of stiff systems, as we expect from the uniform order of convergence stated in Theorem 5
Recently two-step collocation methods have been proposed for fractional differential equations [44], and further developments may be achieved for other fractional models, as time fractional differential equations [45]
Summary
It is the purpose of this paper to illustrate recent results on collocation methods for Volterra integral equations (VIEs) and Volterra integro-differential equations (VIDEs), mainly due to the authors. Such equations model evolutionary problems with memory in many applications, such as dynamics of viscoelastic materials with memory, electrodynamics with memory, heat conduction in materials with memory [1,2,3,4,5,6]. A crucial point is finding accurate and efficient numerical methods
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