517 Modeling of nonlinear dynamic and stationary systems based on Volterra integro–functional series and various classes of quadrature formulas

Висам Махди Абас А. (SRSPU (NPI)), Arutyunyan R. V. (Bauman Moscow State Technical University)


doi: 10.18698/2309-3684-2021-2-6885

The article deals with the analysis of nonlinear dynamic and stationary systems based on Volterra integro–functional series and various classes of quadrature formulas. A mathematical model of the input–output type is used, which does not take into account the specific physical nature of the dynamic process, which is commonly called a black box. The methods of the article are applicable to the main variants of the Volterra integral–functional decomposition, including for the case of stationary dynamical systems, a vector input signal. An example of an optimization problem based on the considered integrative series is given. It is noted that when analyzing and optimizing nonlinear dynamical systems by the method of integro–functional series, the problem of calculating multidimensional integrals may arise. The article considers the application of the combined method based on the Volterra integrative series and grid methods for solving the corresponding one -— and multidimensional integral equations for the analysis of nonlinear dynamic and stationary systems. This article considers the case when a certain set of implementations of input and output signals is known, which can be in principle random processes. According to these data, the kernels are found in the decomposition based on the solution of the corresponding linear multidimensional Fredholm integral equation of the first kind. The corresponding problem belongs to the incorrectly posed ones and the regularization method according to A.N. Tikhonov is used to solve it. The article proposes to apply the quasi Monte–Carlo method, characterized by satisfactory convergence, in this problem in the case of large dimensions. The computational qualities in the considered problem of a semi-statistical method for solving integral equations of large dimension, the quasi Monte–Carlo method, the method of central rectangles (cells) and the quadrature formulas of Gauss–Legendre are studied. The approaches under consideration allow us to expand the range of problems to be solved in the theory of analysis and optimization of systems, since methods are proposed that are practically acceptable for large dimensions of integral equations in conditions of limited information about the system.

Volterra V. Theory of functionals and of integral and integro–differential equations. New York, Dover Publ., 2005, 226 p.
Dimitrienko Yu.I. Mekhanika sploshnoj sredy. T. 1. Tenzornyj analiz [Continuum Mechanics. Vol.1. Tensor analysis]. Moscow, BMSTU Publ., 2011, 367 p.
Dimitrienko Yu.I. Mekhanika sploshnoy sredy. Tom 4. Osnovy mekhaniki tverdogo tela [Continuum Mechanics. Vol.4. Fundamentals of solid mechanics]. Moscow, BMSTU Publ., 2013, 624 p.
Arutyunyan R.V. Modeling of stochastic filtration processes in lattice systems. Маthematical Modeling and Coтputational Methods, 2017, no.4, pp.17–30.
Milov V.R. Recovery of multivariate nonlinear functions on the basis of experimental data. Bulletin of VSAWT, 2003, no.4, pp.163–168.
Apartsyn A.S., Solodusha S.V., Spiryaev V.F. Modeling of nonlinear dynamic systems with Volterra polynomials: elements of theory and applications. International Journal of Energy Optimization and Engineering (IJEOE), 2013, vol.2, iss.4. DOI:10.4018/ijeoe.2013100102
Bobreshov A.M., Mymrikova N.N. The problems of strongly nonlinear analysis for electron circuits based on Volterra series. Proceedings of Voronezh State University. Series: Physics. Mathematics, 2013, no.2, pp.15–25.
Abas V.M.A., Harutyunyan R.V. The calculation of solutions of integral equations on random and pseudorandom grid and its application. University News. North-Caucasian Region. Technical Sciences Series, 2021, no.1 (209), pp.27–37.
Sidorov D.N. Metody analiza integral'nyh dinamicheskih modelej: teoriya i prilozheniya [Methods of analysis of integral dynamic models: theory and applications]. Irkutsk, ISU Publ., 2013, 293 p.
Nekrasov S.A., Abas V.M.A. The calculation of solutions of integral equations on random and pseudorandom grid and its application. Rezul'taty issledovanij — 2021. Materialy VI Nacional'noj konferencii professorsko–prepodavatel'skogo sostava i nauchnyh rabotnikov [Research results — 2021. Materials of the VI National Conference of Faculty and Researchers], Novocherkassk, 2021, pp.44–47.
Abas V.M.A., Harutyunyan R.V. Analysis and optimization of nonlinear systems with memory based on Volterra integro–functional series and Monte–Carlo methods. University News. North-Caucasian Region. Technical Sciences Series, 2021, no.3 (211), pp.30–34.
Ivanov V.M., Kulchitsky O.Yu., Korenevsky M.L. A combined method for solving integral equations. Journal Differential Equations and Control Processes, 1998, no.1, pp.1–40.
Ermakov S.M. Metod Monte–Karlo v vychislitel'noj matematike (vvodnyj kurs) [Monte–Carlo method in computational mathematics (introductory course)]. St. Petersburg, Binom Publ., 2011, 192 p.
Ivanov V.M., Kulchitsky O.Yu. Method of numerical solution of integral equations on a random grid. Differential Equations, 1990, vol.26, iss.2, pp.259–265.
Berkovsky N.A. Modernizaciya polustatisticheskogo metoda chislennogo resheniya integral'nyh uravnenij [Modernization of the semi-statistical method of numerical solution of integral equations]. Abstract of dissertation of the Cand. Sc. (Phys. — Math.). St. Petersburg, 2006, 15 p.

Абас Висам Махди Абас, Арутюнян Р.В. Моделирование нелинейных динамических и стационарных систем на основе интегро–функциональных рядов Вольтерры и различных классов квадратурных формул. Математическое моделирование и численные методы, 2021, № 2, с. 68–85.

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