However, unlike ODE, even for linear FDE there are no general methods of finding solutions in explicit forms. <...> So elaboration of numericalmethods for FDE is the very important problem. <...> Nevertheless, thought there is a great deal of works on theoretical analysis of FDE numerical algorithms and their software realization, however at present there are no books (textbooks) completely devoted to numerical methods of solving FDE. <...> Here we mention some of them: 1) analogs of the most effective numerical methods for ODE, for instance, Runge-Kutta-Fehlbergmethods with automatic step size control which are used in general purpose ODE software packages were modified only for FDE with special types of delays, 2) the problem of handling the discontinuities of FDE solutions requires elaborating laborious and complicated numerical algorithms; 1Under the term a general form FDE we mean an equation in which concrete types of delays are not specified, i.e., these equations can involve discrete and/or distributed delays of various forms. 249 3) creation of methods, taking into account infinite dimensional structure of FDE, for instance continuous methods [199], brings to special, more complicated algorithms. <...> Our aim is to give a reader practical methodology of elaborating numerical algorithms and software programs for solving FDE. <...> The distinguishing features of the described numerical methods are the following: 1) the numerical methods for FDE are direct analogies of the corresponding classical numerical methods ofODE theory, i.e., if delays disappear, then the methods coincide with ODE methods; 2) the proposed numerical algorithms do not depend on specific forms of delays, so the corresponding numerical schemes are the same for different classes of FDE. <...> The approach, presented in the book, is based on: 1) a separation principle that consists of distinguishing finite and infinite dimensional components in the structure of functionals and FDE, 2) an interpolation and an extrapolation of the discrete model prehistory, 3) techniques of i-smooth calculus of functionals. <...> Taking the separation principle as the basis we construct an approximate discrete (numerical) model as a sequence of shifts of the finite dimensional component of FDE state. <...> Interpolation and extrapolation of the prehistory of the discrete model with predefined properties allows <...>
i-гладкий_анализ_и_численные_методы_решения_функционально-дифференциальных_уравнений.pdf
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