The aim of this paper is the analysis of an extension of collocation based numerical methods for solving Ordinary Differential Equations (ODEs) and Volterra Integral Equations (VIEs), which turn out to be at the heart of many modern applications of Mathematics to natural phenomena and are used more and more for the description of complex systems, in particular evolutionary problems and problems with memory. We describe the new classes of continuous numerical methods solving systems of ODEs and VIEs. The developed methods have strong stability properties and higher order of convergence than the classical ones, without any increase of the computational cost, which is an important request in order to approach real problems.
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