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No 1 (2025)
COMPUTER ALGEBRA
METHOD FOR CHECKING THE REGULARITY OF A SINGULAR POINT OF A SYSTEM OF LINEAR DIFFERENTIAL EQUATIONS WITH MEROMORPHIC COEFFICIENTS
Abstract
This paper proposes a program written in a symbolic computing package that allows one to check whether a singular point of a linear meromorphic system of arbitrary order is regular. The program is based on the known method for reducing this system by linear substitution to a linear differential equation with meromorphic coefficients.
Programmirovanie. 2025;(1):5-9
5-9
9-20
ON CALCULATION OF ABELIAN DIFFERENTIALS
Abstract
This paper considers the construction of the fundamental function and Abelian differentials of the third kind on a plane algebraic curve over the field of complex numbers that has no singular points. The algorithm for constructing differentials of the third kind was described in Weierstrass’s lectures. The paper discusses its implementation in the Sage computer algebra system. The specifics of this algorithm, as well as the very concept of the differential of the third kind, implies the use of both rational numbers and algebraic numbers, even when the equation of a curve has integer coefficients. Sage has a built-in tool for computations in algebraic number fields, which allows the Weierstrass algorithm to be implemented almost literally. The simplest example of an elliptic curve shows that it requires too many resources, far beyond the capabilities of an office computer. A symmetrization of the method is proposed and implemented, which makes it possible to solve the problem while saving a significant amount of computational resources.
Programmirovanie. 2025;(1):21-25
21-25
DEVELOPMENT OF ALGORITHMIC AND SOFTWARE SUPPORT FOR SYMBOLIC COMPUTATIONS IN PROBLEMS OF CONSTRUCTING CONTROLLED COMPARTMENTAL MODELS OF DYNAMIC SYSTEMS
Abstract
The analysis of epidemic spreading processes and the development of the corresponding algorithmic and software support for their mathematical modeling are important areas of research. The purpose of this work is to develop tools for symbolic computations in problems of constructing controlled compartmental models of dynamic systems. As a programming language, Julia is used in combination with scientific computing libraries. A software package for compartmental modeling based on schemes of inter-compartment interactions is developed. Several controlled compartmental models — SIRU, SEIRU, and SIDARTHEU — are constructed and investigated. Control is implemented in the form of additional rules with variable transition rates. The developed software complex implements a domain-specific language for compartmental model construction based on interaction schemes. A simulation algorithm for controlled compartmental models is proposed. Computational experiments on controlled simulation of epidemic spreading are carried out, and the trajectory dynamics of the simulation models and corresponding differential models is analyzed. The results can be useful in modeling epidemiological, ecological, physicochemical, and other processes with one-step interactions.
Programmirovanie. 2025;(1):26-39
26-39
SYMBOLIC CALCULATIONS IN THE STUDY OF SECULAR PERTURBATIONS IN THE MANY-BODY PROBLEM WITH VARIABLE MASSES
Abstract
The problem of deriving differential equations determining secular perturbations of orbital elements in a multiplanetary system is considered in the case when the central star loses its mass isotropically, while the masses of the planets can change anisotropically, which leads to the appearance of reactive forces. As a model of a multiplanetary system, the classical problem of variable-mass bodies is used, when bodies move around the central star along quasi-elliptical nonintersecting orbits and interact with each other in accordance with the law of universal gravitation. It is assumed that the masses of the bodies change at different rates and the laws of mass change are considered to be arbitrary given functions of time. Differential equations of motion of bodies in osculating elements of aperiodic motion along quasi-conical orbits corresponding to the planetary Lagrange equations are derived. An algorithm for calculating the perturbing functions in the form of power series in small parameters and the derivation of differential equations determining secular perturbations of orbital elements are discussed. All necessary symbolic calculations are performed using the Wolfram Mathematica computer algebra system.
Programmirovanie. 2025;(1):40-50
40-50
51-58