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PresentationsDevelopment of the method of computer analogy for solving nonlinear systems of differential equationsFederal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 44, b 2, Vavilov st, Moscow, 119333, Russia, E-mail: aristovvl@yandex.ru 1Federal State Budget Educational Institution of Higher Education “MIREA – Russian Technological University", 78, b 2, Vernadsky Avenue, Moscow, 119454, Russia, E-mail: vispoftheblayor@gmail.com New method for constructing solutions to differential equations based on the formalization of number representation in computing devices was proposed in [1]. This method was called the method of computer analogy (MCA). This approach eliminates intermediate steps in the recurrence formulas of difference schemes for solving differential equations. To achieve this, the fundamental properties of a digital computer—a fixed number of digits and the transfer from one digit to the next one—are formalized. The solution is obtained as a segment of a series in powers of the argument step. The formulas for the series coefficients take the form of a linear congruential generator, which allows the coefficients of higher powers to be considered pseudorandom and the application of statistics and probability theory to eliminate intermediate layers. This approach was used to solve systems of nonlinear differential equations [2]. This study develops a technique for obtaining semianalytical approximations for nonlinear systems of differential equations based on the MCA, which is an intermediate goal of the research aimed at constructing explicit solutions for such systems. Using known statistical distributions (Bernoulli distributions) to model the transfers of values in a bit grid containing pseudorandom coefficients allows one to replace calculations at each layer with a known mathematical expectation, eliminating intermediate calculations. Semi-analytical approximations were obtained for the Van der Pol equation, which can be reduced to a system of equations. The approach under consideration allows one to obtain analytical solutions for some systems of kinetic equations.
References 1. Aristov, V.V., Stroganov, A.V.: A method of formalizing computer operations for solving nonlinear differential equations. Applied Mathematics and Computation. 2012, Vol. 218, pp. 8083-8089. 2. 2. Аристов В.В., Музыка А.А., Строганов А.В.: Применение метода компьютерной аналогии для решения сложных нелинейных систем дифференциальных уравнений. Компьютерные исследования и моделирование. 2025 Т. 15 №6. (в печати)
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