Search for a stable state of a continuous medium with a non-local interaction depending on the interaction potential using the method of asymptotic dynamics

Anatoliy Sadkov, Vasiliy Popov


The problems of modeling spatial configurations of a continuous medium in several stable states, the interaction between spatially distant elements of which is characterized by a potential energy proportional to the masses of the elements and the interaction potential, which depends only on the distance between the elements, are considered.

The study of the equation of equilibrium of a continuous medium makes it possible to form an algorithm of a mathematical model for the formation of detailed data on the density of a continuous medium in various phase states. Depending on the value of the Fourier  transform of the interaction potential at the minimum point, and also in accordance with the properties of a one-dimensional continuous medium, the density of the medium assumes the corresponding phase state. The basis for choosing numerical methods for solving the problem according to a given algorithm significantly affects the accuracy of searching for detailed data. The «bottlenecks» in this algorithm are the calculation of integral values, where a particular method of searching for values in the grid nodes is selected to form the final result value or set of values.

The calculation of detailed data according to the formulated algorithm was carried out in the Microsoft Visual Studio 2017 software development environment, and the algorithm itself is described using the C++ programming language. For graphical visualization of the solution, Maple software applications were used. To accurately calculate the final detailed data with an accuracy of up to five decimal places, the Simpson method was used to find the value of the integral.

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