scholarly journals Some Connections between Classical and Nonclassical Symmetries of a Partial Differential Equation and Their Applications

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 524
Author(s):  
Chaolu Temuer ◽  
Laga Tong ◽  
George Bluman
Keyword(s):  
Sufficient Conditions ◽  
Partial Differential ◽  
Burgers Equations ◽  
Classical Symmetry ◽  
Nonclassical Symmetry ◽  
Kdv Type Equations ◽  
Nonclassical Symmetries ◽  
The Given

Essential connections between the classical symmetry and nonclassical symmetry of a partial differential equations (PDEs) are established. Through these connections, the sufficient conditions for the nonclassical symmetry of PDEs can be derived directly from the inconsistent conditions of the system determining equations of the classical symmetry of the PDE. Based on the connections, a new algorithm for determining the nonclassical symmetry of a PDEs is proposed. The algorithm make the determination of the nonclassical symmetry easier by adding compatibility extra equations obtained from system of determining equations of the classical symmetry to the system of determining equations of the nonclassical symmetry of the PDE. The findings of this study not only give an alternative method to determine the nonclassical symmetry of a PDE, but also can help for better understanding of the essential connections between classical and nonclassical symmetries of a PDE. Concurrently, the results obtained here enhance the efficiency of the existing algorithms for determining the nonclassical symmetry of a PDE. As applications of the given algorithm, a nonclassical symmetry classification of a class of generalized Burgers equations and the nonclassical symmetries of a KdV-type equations are given within a relatively easier way and some new nonclassical symmetries have been found for the Burgers equations.

scholarly journals Applications of Differential Form Wu’s Method to Determine Symmetries of (Partial) Differential Equations

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 378 ◽  
Author(s):  
Temuer Chaolu ◽  
Sudao Bilige
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Differential Polynomial ◽  
Decomposition Algorithm ◽  
Partial Differential ◽  
Wu’S Method ◽  
Classical Symmetry ◽  
Symmetries Of Differential Equations ◽  
Wu's Method ◽  
Nonclassical Symmetry

In this paper, we present an application of Wu’s method (differential characteristic set (dchar-set) algorithm) for computing the symmetry of (partial) differential equations (PDEs) that provides a direct and systematic procedure to obtain the classical and nonclassical symmetry of the differential equations. The fundamental theory and subalgorithms used in the proposed algorithm consist of a different version of the Lie criterion for the classical symmetry of PDEs and the zero decomposition algorithm of a differential polynomial (d-pol) system (DPS). The version of the Lie criterion yields determining equations (DTEs) of symmetries of differential equations, even those including a nonsolvable equation. The decomposition algorithm is used to solve the DTEs by decomposing the zero set of the DPS associated with the DTEs into a union of a series of zero sets of dchar-sets of the system, which leads to simplification of the computations.

scholarly journals A Potential Constraints Method of Finding Nonclassical Symmetry of PDEs Based on Wu’s Method

2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
Tonglaga Bai ◽  
Temuer Chaolu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Challenging Problem ◽  
Extended System ◽  
Partial Differential ◽  
Wu’S Method ◽  
Wu's Method ◽  
Nonclassical Symmetry ◽  
Nonclassical Symmetries ◽  
Symmetry Method

Solving nonclassical symmetry of partial differential equations (PDEs) is a challenging problem in applications of symmetry method. In this paper, an alternative method is proposed for computing the nonclassical symmetry of PDEs. The method consists of the following three steps: firstly, a relationship between the classical and nonclassical symmetries of PDEs is established; then based on the link, we give three principles to obtain additional equations (constraints) to extend the system of the determining equations of the nonclassical symmetry. The extended system is more easily solved than the original one; thirdly, we use Wu’s method to solve the extended system. Consequently, the nonclassical symmetries are determined. Due to the fact that some constraints may produce trivial results, we name the candidate constraints as “potential” ones. The method gives a new way to determine a nonclassical symmetry. Several illustrative examples are given to show the efficiency of the presented method.

scholarly journals Group Classification and Conservation Laws of a Class of Hyperbolic Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
J. C. Ndogmo
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Linear Dependence ◽  
Hyperbolic Equations ◽  
Group Classification ◽  
The Family ◽  
The Given ◽  
Low Order

Abstracts. A method for the group classification of differential equations is proposed. It is based on the determination of all possible cases of linear dependence of certain indeterminates appearing in the determining equations of symmetries of the equation. The method is simple and systematic and applied to a family of hyperbolic equations. Moreover, as the given family contains several known equations with important physical applications, low-order conservation laws of some relevant equations from the family are computed, and the results obtained are discussed with regard to the symmetry integrability of a particular class from the underlying family of hyperbolic equations.

scholarly journals PETROPHYSICAL MODELS OF TERRIGENOUS RESERVOIRS OF THE CARBONIFEROUS DEPOSITS OF THE CENTRAL PART OF THE DNIEPER-DONETS DEPRESSION

2019 ◽  
Vol 1 (178) ◽  
pp. 63-73
Author(s):  
Yulia LYSAK ◽  
Yuriy SHPOT ◽  
Andriy SHYRA ◽  
Zoriana KUCHER ◽  
Ihor KUROVETS
Keyword(s):  
Petrophysical Properties ◽  
Reservoir Properties ◽  
Lower Carboniferous ◽  
Reservoir Rocks ◽  
Geological Conditions ◽  
Geological Factors ◽  
Characteristic Features ◽  
The Given

The purpose of the work was to construct petrophysical models of reservoir rocks of different rank: typical and unified. Typical models describe connections between the parameters of individual rocks lithotypes occurring in definite geological conditions and serving as the basis for the development of petrophysical classification of reservoir rocks in the oil geology. The principle of unification provides for creation of the models structure for different reservoir lithotypes both in the geological section and in the area. We have studied petrophysical properties of reservoir rocks of Carboniferous deposits in the central part of the Dnieper-Donets depression. Petrophysical properties of rocks in conditions close to the formational ones and relations between them were studied on a number of samples formed by the core samples of different age. Main geological factors that have an influence on reservoir properties of rocks were taken into consideration. While constructing and analysing of petrophysical models we have used a probable-statistic approach with the use of the correlative-regressive analysis. Result of the work is contained in typical petrophysical models for individual areas and in unified models obtained on consolidated samples for Lower Carboniferous deposits of this region. Characteristic features in variations of petrophysical properties of reservoir rocks of Carboniferous deposits and their models have been ascertained. A conclusion has been made that multidimensional models, in which the depth of occurrence of deposits is one of the parameters that are necessary to consider while constructing petrophysical models, are the most informative for determination of petrophysical properties of the studied deposits, and the models obtained by us are known to be a petrophysical basis for quantitative interpretation of data from geophysical studies in the boreholes of the given region.

scholarly journals Formulation and study of the problem of optimal excitation of plate oscillations

2019 ◽  
pp. 62-65
Author(s):  
G. M. Zrazhevsky ◽  
V. F. Zrazhevska
Keyword(s):  
Objective Function ◽  
Sufficient Conditions ◽  
Analytical Determination ◽  
Main Task ◽  
Hermitian Forms ◽  
Complete Study ◽  
Concentric Circles ◽  
The Green Function ◽  
The Given

A model problem of harmonic oscillations of a hinged plate, that is is under the influence of a certain number of point concentrated forces, is considered. The plate model is considered to satisfy Kirchhoff's conditions. The main task of the consideration is to determine the optimal characteristics of excitation - the number of forces, coordinates of their application, amplitudes and phases. The optimality criterion is constructed as the standard deviation of the complex deflections from a given profile function. With the given excitation characteristics, the problem of determining the vibrations is solved in the form of a superposition of the Green functions with singularities at the points of application of forces. The Green function is constructed as a Fourier series by a circular coordinate. By using Parseval equality in L2, the objective function of the optimization problem is represented as a combination of linear and Hermitian forms with respect to complex amplitudes of forces whose matrices are nonlinear (and not convex) dependent on the coordinates of singular points. A complete study of the objective function is performed. Sufficient conditions are determined for reducing the dimension of the control space by analytical determination of the amplitudes of forces. Expressions were obtained to calculate the gradients of the objective function by angular and radial coordinates. A partial case of grouping of excitation forces on concentric circles is considered, that leads to the degeneration of the problem.

scholarly journals Nonclassical Symmetry of Differential Equations

2020 ◽  
pp. 183-189
Author(s):  
T. Jassim Aldhlki
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Classical Symmetry ◽  
Nonclassical Symmetry ◽  
The Difference ◽  
Nonclassical Symmetries

In this paper, we discuss the difference between classical and nonclassical symmetries. In addition, we found the non-classical symmetry of the Benjamin Bona Mahony Equation (BBM). Finally, we found a new exact solution to a Benjamin Bona Mahony Equation (BBM) using nonclassical symmetry.

The Role and Place of Budget Procedure in the System

2003 ◽  
pp. 42-49 ◽  
Author(s):  
E. Bushmin
Keyword(s):  
Budget Process ◽  
Public Expenditures ◽  
Principal Role ◽  
Process Trends ◽  
The Given ◽  
Rules Of The Game

The article is devoted to the analysis of improving budget process trends. The author offers the concept of "financial technologism". Its usage should promote an essential improvement of the budget process. The given concept is based on the fact that the regulation of budget procedure is the process of determination of "rules of the game", and the order of interaction of different institutions within the framework of the budget process, and the trends and volumes of expenses are the strategy of institutions. The procedure within the budget process plays a principal role as compared with the trends and volumes of public expenditures.

THE ESSENCE, THE NEED FOR EVALUATION AND TYPES OF ECONOMIC POTENTIAL OF ENTERPRISES

2017 ◽  
Author(s):  
Olexandr Yemelyanov ◽  
Keyword(s):  
Economic Activity ◽  
Business Strategies ◽  
Enterprise Management ◽  
Value Assessment ◽  
Economic Potential ◽  
Enterprise Value ◽  
Development Assessment ◽  
State Of The Environment

The formation of business strategies of enterprises should be based on a preliminary assessment of their current and future economic opportunities. Such an assessment is to establish the value of the total economic potential of enterprises and its individual varieties. The purpose of this study was to clarify the essence of the economic potential of enterprises, justify the need for its evaluation and selection of its types. The main approaches to interpreting the terms "potential" and "enterprise potential" are identified. These include resource, result, resource-result, resource-target, and result-target concepts. It is established that the potential of any object, including the enterprise, can be interpreted as a set of its external functional properties, which this object shows or can show in a certain state of the environment in which it is located. Accordingly, the assessment of the potential of an object should be based on the identification and determination of its external properties, taking into account the environment in which the object is located. The main situations in which there is a need for information about certain components of the economic potential of the enterprise are identified, and the types of this potential and the consumers of the information about their level, corresponding to these situations, are determined. In particular, such situations include management of production and sales, management of financial and economic results of the enterprise, management of enterprise development, assessment of the company's need for various types of resources, assessment of enterprise value, assessment of current and future impact of the enterprise on the economy of the country (region), etc. The features of classification of types of enterprise potential existing in the scientific literature are supplemented by the following ones: by the main types of economic activity, by the dynamics of changes in the economic and production system of the enterprise, by consequences for the subject of enterprise potential assessment, by the stages of economic activity, enterprise potential, etc. The obtained results make it possible to improve the understanding of the complex patterns that underlie the formation of the economic potential of enterprises.

State Failure and the Political Violence Phenomenon: A Comparative Analysis of Iraq and Syria Cases

2016 ◽  
Vol 4 (2) ◽  
pp. 170 ◽  
Author(s):  
K. Eylem Özkaya Lassalle
Keyword(s):  
Cold War ◽  
Political Violence ◽  
State Failure ◽  
The Political ◽  
Collective Violence ◽  
Short Run ◽  
Definition Of ◽  
The Cold War

The concept of failed state came to the fore with the end of the Cold War, the collapse of the USSR and the disintegration of Yugoslavia. Political violence is central in these discussions on the definition of the concept or the determination of its dimensions (indicators). Specifically, the level of political violence, the type of political violence and intensity of political violence has been broached in the literature. An effective classification of political violence can lead us to a better understanding of state failure phenomenon. By using Tilly’s classification of collective violence which is based on extent of coordination among violent actors and salience of short-run damage, the role played by political violence in state failure can be understood clearly. In order to do this, two recent cases, Iraq and Syria will be examined.

The Potential of the Russian Language Textbook in the Formation of Primary Schoolchildren’s Communicative Literacy

2018 ◽  
Vol 79 (8) ◽  
pp. 35-40
Author(s):  
M. I. Kuznetsova
Keyword(s):  
Primary School ◽  
Russian Language ◽  
Primary School Children ◽  
Written Text ◽  
Russian Literary ◽  
The Given ◽  
The Russian Language ◽  
Speech Situation ◽  
Linguistic Means

One of the goals of the Russian language course in the primary school is the formation of the communicative literacy. The content of the course should be aimed at understanding the wealth of linguistic means by primary school children; the formation of the ability to detect a violation of linguistic norms and the inadequacy of the linguistic means used in the speech situation; the accumulation of the experience in choosing of linguistic means in accordance with the peculiarities of the speech situation; the creation of oral and written texts that meet the criteria of content, connectivity, compliance with the norms of the Russian literary language. The article considers the classification of exercises that contribute to the formation of communicative literacy. The author gives the examples of exercises where the student acts in different roles: the student is an observer of the speech situation and analyzes the adequacy of the choice of linguistic means; the student is a direct participant in the given speech situation and makes a choice of language facilities; the student is offered to create the speech situation himself, to independently construct an oral and written text.

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