Ability to calculate integrals of rapidly oscillating functions is crucial for solving many problems in optics, electrodynamics, quantum mechanics, nuclear physics, and many other areas. The article considers the method of computing oscillatory integrals using the transition to the numerical solution of the system of ordinary differential equations. Using the Levin's collocation method, we reduce the problem to solving a system of linear algebraic equations.In the case where the phase function has stationary points (its derivative vanishes on the interval of integration), the solution of the corresponding system becomes an ill-posed task. The regularized algorithm presented in the article describes the stable method of integration of rapidly oscillating functions at the presence of stationary points. Performance and high accuracy of the algorithms are illustrated by various examples. (C) 2018 Published by Elsevier B.V.

Ability to calculate integrals of rapidly oscillating functions is crucial for solving many problems in optics, electrodynamics, quantum mechanics, nuclear physics, and many other areas. The article considers the method of computing oscillatory integrals using the transition to the numerical solution of the system of ordinary differential equations. Using the Levin's collocation method, we reduce the problem to solving a system of linear algebraic equations.In the case where the phase function has stationary points (its derivative vanishes on the interval of integration), the solution of the corresponding system becomes an ill-posed task. The regularized algorithm presented in the article describes the stable method of integration of rapidly oscillating functions at the presence of stationary points. Performance and high accuracy of the algorithms are illustrated by various examples. (C) 2018 Published by Elsevier B.V.

When modeling such phenomena as population dynamics, controllable flows, etc., a problem arises of adapting the existing models to a phenomenon under study. For this purpose, we propose to derive new models from the first principles by stochastization of one-step processes. Research can be represented as an iterative process that consists in obtaining a model and its further refinement. The number of such iterations can be extremely large. This work is aimed at software implementation (by means of computer algebra) of a method for stochastization of one-step processes. As a basis of the software implementation, we use the SymPy computer algebra system. Based on a developed algorithm, we derive stochastic differential equations and their interaction schemes. The operation of the program is demonstrated on the Verhulst and Lotka-Volterra models.

The investigation of the electromagnetic field in a regular waveguide filled with a homogeneous substance reduces to the study of two independent boundary value problems for the Helmholtz equation. In the case of a waveguide filled with an inhomogeneous substance, a relationship arises between the modes of these two problems, which in numerical experiments can not always be fully taken into account. In this paper, we will show how to rewrite the Helmholtz equations in the Hamiltonian form to express this connection explicitly. In this case, the problem of finding monochromatic waves in a waveguide with arbitrary filling will be reduced to an infinite system of ordinary differential equations in a properly constructed Hilbert space. The results of numerical experiments on finding normal waves, realized in the computer algebra system Sage, are presented.

SymPy computer algebra library is used for automatic generation of ordinary and stochastic systems of differential equations from the schemes of kinetic interaction. Schemes of this type are used not only in chemical kinetics but also in biological, ecological and technical models. This paper describes the automatic generation algorithm with an emphasis on application details.

Nowadays the geometric approach in optics is often used to find out media parameters based on propagation paths of the rays because in this case it is a direct problem. However inverse problem in the framework of geometrized optics is usually not given attention. The aim of this work is to demonstrate the work of the proposed the algorithm in the framework of geometrized approach to optics for solving the problem of finding the propagation path of the electromagnetic radiation depending on environmental parameters. The methods of differential geometry are used for effective metrics construction for isotropic and anisotropic media. For effective metric space ray trajectories are obtained in the form of geodesic curves. The introduced algorithm is applied to well-known objects - Maxwell and Luneburg lenses. The similarity of results obtained by classical and geometric approach is demonstrated.

Self-oscillating modes in computer networks control systems quite negatively affect the characteristics of these networks. The problem of finding the areas of self-oscillations is actual and important as the study of parameters of self-oscillations. Due to the significant nonlinearity of control characteristics, the study of the oscillatory modes presents certain difficulties. This paper describes the technique of research of self-oscillating modes on the basis of the control theory. This material is rather methodical than exploratory one. © Springer International Publishing AG 2018.

SymPy computer algebra library is used for automatic generation of ordinary and stochastic systems of differential equations from the schemes of kinetic interaction. Schemes of this type are used not only in chemical kinetics but also in biological, ecological and technical models. This paper describes the automatic generation algorithm with an emphasis on application details.

Self-oscillating modes in computer networks control systems quite negatively affect the characteristics of these networks. The problem of finding the areas of self-oscillations is actual and important as the study of parameters of self-oscillations. Due to the significant nonlinearity of control characteristics, the study of the oscillatory modes presents certain difficulties. This paper describes the technique of research of self-oscillating modes on the basis of the control theory. This material is rather methodical than exploratory one.

Nowadays the geometric approach in optics is often used to find out media parameters based on propagation paths of the rays because in this case it is a direct problem. However inverse problem in the framework of geometrized optics is usually not given attention. The aim of this work is to demonstrate the work of the proposed the algorithm in the framework of geometrized approach to optics for solving the problem of finding the propagation path of the electromagnetic radiation depending on environmental parameters. The methods of differential geometry are used for effective metrics construction for isotropic and anisotropic media. For effective metric space ray trajectories are obtained in the form of geodesic curves. The introduced algorithm is applied to well-known objects - Maxwell and Luneburg lenses. The similarity of results obtained by classical and geometric approach is demonstrated.