# No 3 (2015)

**Year:**2015**Articles:**11**URL:**https://journals.rudn.ru/miph/issue/view/521

## Articles

### On the Approximate Solving of the Differential Equations which General Solutions Depend on a Constant of Integration Algebraically

#### Abstract

Methods of analytically theory of ordinary diﬀerential equations are based on the analysis of singularities, but the most popular method for the numerical solving, that is the method of ﬁnite diﬀerences does not work in neighborhood of singularities. However Painlevé gave an algebraic method for the solution of the diﬀerential equations in ﬁnite terms and general solutions of this equations depend algebraically on a constant of integration. This approach which was presented as Galois theory on the contrary can be well combined with method of ﬁnite diﬀerences. It is well known, the ordinary diﬀerential equation of form y′ = f(x,y) with this property can be algebraically transformed by substitution to Riccati equation. Euler scheme yn+1 = yn + f(xn,yn)Δx always determines (1,k)-correspondence between neighboring layers. But exact solution of Riccati equation determines (1,1)-correspondence between any layers and thus we can write a scheme which determines (1,1)-correspondence between neighboring layers. In this case anharmonic ratio of 4 points does not change from layer to layer not only for exact solution but also for approximate solution. Thus if an exact solution has a pole then the approximate solution passes through inﬁnity without accumulation of an error. In the presented article this property of (1,1)-scheme will be illustrated by two examples: with and without solution in elementary functions. So the cause of destruction of the approximate solution near a pole is put in Euler scheme itself. In more general case when exact solution of ordinary diﬀerential equation depends algebraically on an integration constant we can write a scheme which determines (l,l)-correspondence between neighboring layers. Approximate solution which is found on this way passes through movable algebraic singularities without accumulation of an error.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(3):5-9

5-9

### Recursive Algorithm for Calculating Stationary Probability Distribution of Model with Interruption of Unicast Traffic by Multicast Traffic

#### Abstract

LTE fourth generation networks become one of the most important trends in the modernization of telecommunications systems. According to international standards, LTE networks specify nine types of services, that diﬀer in terms of the bit-rate (guaranteed or non-guaranteed) and priority level. Depending on the bit rate overall traﬃc generated during the service providing is divided into three types: unicast streaming, streaming multicast, and elastic. The priority level can be realized with use of diﬀerent mechanisms - bit rate degradation, service interruption, reservation, threshold and probabilistic management - that form the foundations of the radio admission control schemes. In the paper, a model of RAC scheme is proposed with service interruption of unicast traﬃc and two service disciplines for multicast traﬃc. The recursive algorithm is proposed for calculating main performance measures of model - blocking and pre-emption probabilities for unicast traﬃc.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(3):10-17

10-17

### Stability Research of Population Dynamics Model on the Basis of Construction of the Stochastic Self-Consistent Models and the Principle of the Reduction

#### Abstract

The three-dimensional model of interaction of populations taking into account the competition and diﬀusion of species is considered. For research of model the combination of known methods of synthesis and the analysis of models, the principle of a reduction and the developed method of construction of the stochastic self-consistent models is used. Existence conditions of equilibrium states are obtained and the analysis of stability is made. Stability conditions on the basis of the principle of a reduction of a problem about stability of solutions of diﬀerential inclusion to a problem on stability of other types of the equations are oﬀered. The speciﬁed principle assumes transition from the vector ordinary diﬀerential equations to vector diﬀerential inclusion and the fuzzy diﬀerential equation, taking into account change of parameters of diﬀerent types in the studied models. For the considered model of population dynamics synthesis of the corresponding stochastic model on the basis of application of a method of construction of the stochastic self-consistent models is carried out. The structure of stochastic model is described, Fokker-Planck equation is written out, and the rule of transition to the stochastic diﬀerential equation in the form of Langevin is formulated. The oﬀered approach allowed to carry out the comparative analysis of qualitative properties of the models considering the competition and diﬀusion of species in deterministic and stochastic cases. Stability conditions can be used for studying of dynamic behavior of models of population dynamics. The received results are aimed at the further development of methods of construction and the analysis of stability of nondeterministic mathematical models of natural sciences.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(3):18-29

18-29

### Quantum Field Theory Approach to the Analysis of One-Step Models

#### Abstract

During development of methods for stochastization of one-step processes the attention was focused on obtaining the stochastic equations in the form of the Langevin, since this form of stochastic equations is most usual in the construction and study of one-step processes models. When applying the method there is the problem of justifying the transition from master equation to the Fokker-Planck equation for the diﬀerent versions of the model. However, the forms of partial diﬀerential equations (master equation and the Fokker-Planck equation) wider description of the model to researchers. It is proposed to treat these equations with the help of perturbation theory in the framework of quantum ﬁeld theory. For this purpose the methodology was described and the analytical software complex was constructed to write down put the main kinetic equation in the operator form in the Fock representation. To solve the resulting equation the software complex generates Feynman diagrams for the corresponding order of perturbation theory. The FORM system was applied as a system of symbolic computation. Selecting FORM as the CAS is reasonable because that the given computer algebra system allows for symbolic computation, using the resources of high-performance computing. In particular, it is possible to use parallel computing technologies such as OpenMP and MPI.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(3):30-40

30-40

### Econophysics Formation

#### Abstract

The article traces the evolution of free roaming hypothesis, which plays a prominent role in many areas of human knowledge: mathematics, molecular physics, hydro and gas dynamics, cosmology, chemistry, and biology. This hypothesis is the foundation for the concept of an eﬃcient market. Also it is the source of many modern theories, as well as methods for the ﬁnancial markets analysis and forecasting. Development of free roaming hypothesis was one of the sources which at the end of the 90s. of the last century led to the emergence of a new ﬁeld of knowledge - econophysics, a scientiﬁc discipline that has emerged at the intersection of economics, physics and mathematics.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(3):41-48

41-48

### A New Approach to the Relatively High Probability of a Crisis In the Financial Markets Explanation

#### Abstract

In this paper we propose a model, which is based on the hypothesis of a quantum nature of the impact of information on the ﬁnancial markets. It is shown that pricing emissions are expected to be real in the information-rich, volatile ﬁnancial markets. Numerous works carried out by ﬁnancial analysts as well as mathematicians were devoted to the problem of research of causes of ﬁnancial markets crashes and the methods of this crashes predictions. The topic itself sometimes provokes spectacular statements and conclusions, often without any convincing reason. However, in recent years there were a number of serious approaches that allowed to obtain encouraging results. Most of them, in one way or the other are connected with econophysics - economics, related to physics. Our research, developing surprising aspect of the problem, belongs to this trend.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(3):49-53

49-53

### Analysis of Nonholonomicity Value of Some Hamiltonian Fields

#### Abstract

In classical mechanics such notion as nonholonomicity is applied only to constraints put on a dynamical system. Besides, Pfaﬃan nonholonomic constraints might be associated with vector ﬁelds. The Nonholonomicity value is one of the principal characteristics of such ﬁelds, which determines properties of geometry of these vector ﬁelds. However, the application of this characteristic in the geometry of vector ﬁelds was restricted only to ﬁelds in Euclidean spaces. Some generalization of nonholonomicity value of vector ﬁelds in non-Euclidean spaces is proposed in this paper. For this purpose the nonholonomicity value is considered as a trilinear form. It is obvious that the coeﬃcients of this form are connected with the components of the metric tensor of the space, where a vector ﬁeld is deﬁned. So generalization of metric tensor on non-Euclidean spaces generates the generalization of the coeﬃcients of trilinear form, which in its turn generates the generalization of nonholonomicity value. As an example, the nonholonomicity values of Hamiltonian vector ﬁelds in sympletic spaces are analyzed in this article. Also it is important to ﬁnd out whether a mechanical interpretation of the received results exists and can we actually apply this method to Hamiltonian ﬁelds.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(3):54-60

54-60

### A Particular Case of a Sequential Growth of an X-Graph

#### Abstract

A particular case of discrete spacetime on a microscopic level is considered. The model is a directed acyclic dyadic graph (an x-graph). The dyadic graph means that each vertex possesses no more than two incident incoming edges and two incident outgoing edges. The sequential growth dynamics of this model is considered. This dynamics is a stochastic sequential addition of new vertices one by one. The probabilities of diﬀerent variants of addition of a new vertex depend on the structure of existed x-graph. It is proved that the algorithm to calculate probabilities of this dynamics is a unique solution that satisﬁes some principles of causality, symmetry and normalization. The algorithm of sequential growth can be represented as following tree steps. The ﬁrst step is the choice of the addition of the new vertex to the future or to the past. By deﬁnition, the probability of this choice is 1∕2 for both outcomes. The second step is the equiprobable choice of one vertex number V . Then the probability is 1∕N, where N is a cardinality of the set of vertices of the x-graph. If we choose the direction to the future, the third step is a random choice of two directed paths from the vertex number V . A new vertex is added to the ends of these paths. If we choose the direction to the past, we must randomly choose the two inversely directed paths from the vertex number V . The iterative procedure to calculate probabilities is considered.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(3):61-73

61-73

### Abdelhaq Safiullovich Galiullin

#### Abstract

In November 2014 the 95... anniversary of Abdelhaq Safiullovich Galiullin, Doctor of Technical Sciences, Honored Worker of Science and Technology of the Russian Federation, Academician of the International Academy of High School, Honorary Academician of the Academy of Sciences of the Republic of Tatarstan was marked. A.S. Galiullin was born on November 26, 1919 in the village of Old Arnash under the old town Arsk. He studied in tatar secondary school No 13. In 1938 he began studies at the Faculty of Mathematics and Physics of Kazan State University. From September 1941 to 1944 he was a student at the Air Force Engineering Academy named after N.E. Zhukovsky. In 1942 he hold a front-line training on the Volkhov front as a mechanic for armaments. From 1946 to 1969 he worked at the Kazan Aviation Institute. On the advice of the scientific supervisor G.V. Kamenkov he started researches in the area of stability and control theory. In 1950 he defended his PhD thesis, and in 1958 - his doctoral dissertation. From 1961 to 1964 A.S. Galiullin worked as Dean of the Faculty of Physics, Mathematics and Natural Sciences at Peoples’ Friendship University named after Patrice Lumumba and, until 1988, as Head of the Department of Theoretical Mechanics. He organized the work of the scientific seminar, and first scientific conferences. He was the Chairman of the Academic Council for master’s and doctoral theses. Abdelhak Safiullovich was a renowned scientist in the field of analytical mechanics, the theory of stability and motion control. He has published more than 90 scientific works, including 8 monographs and textbooks which were widely recognized. The result of his investigations were new, non-traditional methods of analytical dynamics, sections related to the symmetry of the dynamics, with generalizations of Hamiltonian mechanics, systems of Helmholtz, Birkhoff and Nambu. A.S. Galiullin died on April 17, 1999.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(3):74-80

74-80

### Our authors

**Discrete and Continuous Models and Applied Computational Science**. 2015;(3):81

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### Guidelines for Authors

**Discrete and Continuous Models and Applied Computational Science**. 2015;(3):82-83

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