Group Analysis Of Differential Equations Research Articles

Nonlinear Modeling of Heat Distribution during a Concreting Beam, Using a Heating Electric Wire

In this article unlike existing linear models, a nonlinear differential model is used to describe the nonlinear heat distribution in a reinforced concrete beam using a heating wire. The results of experiments carried out under different heating modes are adequately described with good approximation by the analytical results obtained using the methods of group analysis of differential equations.

The main tasks of group analysis of differential equations of mechanics

Group analysis of differential equations uses the Lie theory of correspondence between continuous transformation groups and Lie algebras of first-order differentiation operators. Differential equations of mechanics necessarily admit an extensive group of transformations. Lie theory studies the structure of the algebra of this group. The group analysis of the equations of mechanics uses the structure of the admissible algebra to produce submodels and exact solutions, to study the boundary value problems of submodels and the behavior of the mechanical medium for exact solutions. The main tasks of group analysis are formulated and simple examples of their solutions are given.

THE TECHNOLOGY OF WINTER CONCRETING OF MONOLITHIC FRAME STRUCTURES WITH SUBSTANTIATION OF HEAT TREATMENT MODES BY SOLUTIONS OF THE DIFFERENTIAL EQUATION OF THERMAL CONDUCTIVITY OBTAINED BY THE METHOD OF GROUP ANALYSIS

An innovative method for calculating thermal fields inside monolithic structures has been developed, based on the use and analysis of nonlinear differential equations. The innovativeness of the method lies in the approach to the analysis of nonlinear physical processes using nonlinear differential equations. Thanks to the method of group analysis, 13 expressions are obtained from complex mathematical equations, which are easy to use and depend on several empirical coefficients. It is assumed that this calculation method is a priori more accurate than the existing ones, as well as available to people at a construction site without higher mathematical education, which makes it a priority for research. The applicability of this method must be proven by linking empirical coefficients and variables to the conditions of the experiments, while obtaining reliable data that will turn out to be more accurate than the existing calculation methods. This article demonstrates a systematic approach to establishing the suitability of using the method of group analysis of differential equations for problems of winter concreting on the basis of laboratory experiments under stationary conditions. The equations were subject to verification, which, according to the physical description, correspond to the real conditions of the course of thermal processes inside monolithic structures. Based on the obtained processing results, it was decided that it was necessary to further study the innovative method in the conditions of the construction site, but only for some expressions that showed the best results at the stage of laboratory tests.

ReLie: A Reduce Program for Lie Group Analysis of Differential Equations

Lie symmetry analysis provides a general theoretical framework for investigating ordinary and partial differential equations. The theory is completely algorithmic even if it usually involves lengthy computations. For this reason, along the years many computer algebra packages have been developed to automate the computation. In this paper, we describe the program ReLie, written in the Computer Algebra System Reduce, since 2008 an open source program for all platforms. ReLie is able to perform almost automatically the needed computations for Lie symmetry analysis of differential equations. Its source code is freely available too. The use of the program is illustrated by means of some examples; nevertheless, it is to be underlined that it proves effective also for more complex computations where one has to deal with very large expressions.

All-Russian Conference and School for Young Scientists, devoted to 100th Anniversary of Academician L.V. Ovsiannikov “Mathematical problems of continuum mechanics”

PrefaceThe present volume of the Journal of Physics: Conference Series represents contributions from participants of the All-Russian conference and school for young scientists, devoted to 100th anniversary of academician L.V. Ovsiannikov “Mathematical problems of continuum mechanics” held at the Technopark of Novosibirsk Akademgorodok, Russia, on May 13–17, 2019 organized by Lavrentyev Institute of Hydrodynamics and Novosibirsk State University.The conference is traditional and attracts attention of mathematicians and physicists. This year the Conference was dedicated to the 100th anniversary of Academician of the Russian Academy of Science Lev V. Ovsiannikov (22 April 1919 – 23 May 2014), an outstanding Russian scientist in the field of mechanics and mathematics. Results of L.V. Ovsiannikov in the gas dynamics, the theory of fluid motion with free boundaries, in the field of mathematical justification of the models of continuum mechanics became classical and served as the beginning of new scientific directions. The methods of group analysis of differential equations developed by him are widely used in various fields of mathematics, mechanics, and theoretical physics.Scientific activities of the Conference include discussion of actual problems of applied mathematics, mechanics and methods of mathematical modelling, as well as their applications for describing and forecasting natural and technical processes. The program of conference included plenary talks (30 minutes), oral presentations (20 minutes) and poster presentations given by 190 participants from 18 cities of Russia (Moscow, Novosibirsk, Krasnoyarsk, Ufa, Barnaul, Vladivostok, Irkutsk, Tomsk, Yekaterinburg, Kemerovo, Perm, Tyumen, St Petersburg), Kazakhstan, Belarus, France (Marseille), Germany (Darmstadt) and Canada (St Catharines). A significant part of talks (about 30%) were presented by young researchers and postgraduate students. Within the school of young scientists, lectures on topical problems of continuum mechanics were given.We would like to thank the Speakers for their significant contributions to the conference. We also would like to thank the members of the Organizing Committee. We cannot end without expressing our many thanks to our sponsors (Russian Foundation for Basic Research, Ministry of Science and Education of the Russian Federation, Novosibirsk State University) for their financial support.Novosibirsk, May 2019Alexander ChesnokovAlexander KheViktor ShcherbakovAlexey ShutovThe Conference Photograph is available in this PDF.

Algorithmic framework for group analysis of differential equations and its application to generalized Zakharov–Kuznetsov equations

In this paper, we explain in more details the modern treatment of the problem of group classification of (systems of) partial differential equations (PDEs) from the algorithmic point of view. More precisely, we revise the classical Lie algorithm of construction of symmetries of differential equations, describe the group classification algorithm and discuss the process of reduction of (systems of) PDEs to (systems of) equations with smaller number of independent variables in order to construct invariant solutions. The group classification algorithm and reduction process are illustrated by the example of the generalized Zakharov–Kuznetsov (GZK) equations of form ut+(F(u))xxx+(G(u))xyy+(H(u))x=0. As a result, a complete group classification of the GZK equations is performed and a number of new interesting nonlinear invariant models which have non-trivial invariance algebras are obtained. Lie symmetry reductions and exact solutions for two important invariant models, i.e., the classical and modified Zakharov–Kuznetsov equations, are constructed. The algorithmic framework for group analysis of differential equations presented in this paper can also be applied to other nonlinear PDEs.

An invitation to modern group analysis of differential equations

The purpose of this review paper is two-fold. First, we point out one more time the importance and usefulness of the S. Lie theory of symmetries of differential equations. In particular some illustrative examples are presented. Then, we discuss the Noetherian approach to Rellich–Pohozaev type identities.

Symmetry analysis for a class of nonlinear dispersive equations

A class of dispersive equations is studied within the framework of group analysis of differential equations. The enhanced Lie group classification is achieved. The complete list of equivalence transformations is presented. It is shown that certain equations from the class admit nonclassical reductions. Potential and potential nonclassical symmetries are also considered.

Application of Lie Group Analysis to the Plastic Torsion of Rod with the Saint Venant–Mises Yield Criterion

Based on Lie group and Lie algebra theory, the basic principles of Lie group analysis of differential equations in mechanics are analyzed, and its validity in theory of plasticity is explained by example. For the plastic torsion of rod with variable cross section that consists in non-linear Saint Venant-Mises yield criterion, the 10-dimensional Lie algebra admitted by the equilibrium equation and yield criterion is completely solved, and invariants and group invariant solutions relative to different sub-algebras are given. At last, physical explanations of each group invariant solution are discussed by some types of transformations.

General dynamical systems described by first order quasilinear PDEs reducible to homogeneous and autonomous form

Within the framework of Lie group analysis of differential equations, a theorem is determined stating necessary and sufficient conditions allowing one to recover an invertible point transformation mapping a general dynamical system described by nonhomogeneous and nonautonomous first order quasilinear partial differential equations to homogeneous and autonomous form. The proof of the theorem is constructive and the new independent and dependent variables are obtained by determining the canonical variables associated to a suitable subalgebra of the Lie algebra of point symmetries admitted by the source system. The theorem is applied by considering some examples of physical interest arising from different contexts.

Lie group analysis of two-dimensional variable-coefficient Burgers equation

The modern group analysis of differential equations is used to study a class of two-dimensional variable coefficient Burgers equations. The group classification of this class is performed. Equivalence transformations are also found that allow us to simplify the results of classification and to construct the basis of differential invariants and operators of invariant differentiation. Using equivalence transformations, reductions with respect to Lie symmetry operators and certain non-Lie ansatze, we construct exact analytical solutions for specific forms of the arbitrary elements. Finally, we classify the local conservation laws.

Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics

It is shown that Lie group analysis of differential equations provides the exact solutions of two-dimensional stratified rotating Boussinesq equations which are a basic model in geophysical fluid dynamics. The exact solutions are obtained as group invariant solutions corresponding to the translation and dilation generators of the group of transformations admitted by the equations. The comparison with the previous analytic studies and experimental observations confirms that the anisotropic nature of the wave motion allows to associate these invariant solutions with uni-directional internal wave beams propagating through the medium. It is also shown that the direction of internal wave beam propagation is in the transverse direction to one of the invariants which corresponds to a linear combination of the translation symmetries. Furthermore, the amplitudes of a linear superposition of wave-like invariant solutions forming the internal gravity wave beams are arbitrary functions of that invariant. Analytic examples of the latitude-dependent invariant solutions associated with internal gravity wave beams that have different general profiles along the obtained invariant and propagating in the transverse direction are considered. The behavior of the invariant solutions near the critical latitude is illustrated.

Applications of the group analysis of differential equations to some systems of noncommuting C 1-smooth vector fields

Given a canonical basis of C 1-smooth vector fields $$ \{ \tilde X_i \} $$ satisfying certain restrictions on commutators, we prove an existence theorem for their local nilpotent homogeneous approximation at the origin using the methods of the group analysis of differential equations. We study the properties of the quasimetrics induced by some systems of vector fields related to $$ \{ \tilde X_i \} $$ .

Multi-dimensional quasi-simple waves in weakly dissipative flows

A multi-dimensional simple wave formalism is employed to formulate a multi-dimensional quasi-simple wave for a weakly dissipative fluid. This is a natural but nontrivial generalization of the so-called unidirectional quasi-simple wave. The method is more close to multi-orders analysis which is different from the standard perturbation method. Detailed solving up to the second order is presented for a 2D sound simple wave. A new 2D Burgers equation is derived for the wave phase. It essentially differs from the known 2D generalizations of the Burgers equation, e.g., from the Zabolotskaya–Khokhlov equation. The derived equation is investigated in the framework of group analysis of differential equations. Multi-parameter families of its exact solutions are constructed. Simplest solutions are chosen for analysis of their physical relevance in the initial variables.

Analysis of reaction-diffusion systems by the method of linear determining equations

Exact solutions to two-component systems of reaction-diffusion equations are sought by the method of linear determining equations (LDEs) generalizing the methods of the classical group analysis of differential equations. LDEs are constructed for a system of two second-order evolutionary equations. The results of solving the LDEs are presented for two-component systems of reaction-diffusion equations with polynomial nonlinearities in the diffusion coefficients. Examples of constructing noninvariant solutions are presented for the reaction-diffusion systems that possess invariant manifolds.

Jacobi’s Last Multiplier and the Complete Symmetry Group of the Ermakov-Pinney Equation

The Ermakov-Pinney equation possesses three Lie point symmetries with the algebra sl(2, R). This algebra does not provide a representation of the complete symmetry group of the Ermakov-Pinney equation. We show how the representation of the group can be obtained with the use of the method described in Nucci, J. Nonlin. Math. Phys. 12 (2005) (this issue), which is based on the properties of Jacobi’s last multiplier (Bianchi L, Lezioni sulla teoria dei gruppi continui finiti di trasformazioni, Enrico Spoerri, Pisa, 1918), the method of reduction of order (Nucci,J. Math. Phys 37 (1996), 1772–1775) and an interactive code for calculating symmetries (Nucci, Interactive REDUCE programs for calcuating classical, non-classical and Lie-Bäcklund symmetries for differential equations (preprint: Georgia Institute of Technology, Math 062090-051, 1990, and CRC Handbook of Lie Group Analysis of Differential Equations. Vol. 3: New Trends in Theoretical Developments and Computational Methods, Editor: Ibragimov N H, CRC Press, Boca Raton, 1996, 415–481).

Introduction to Symmetry Analysis

Introduction to Symmetry Analysis

Sobre el problema inverso de difusión

Se describe físicamente la infiltraccion para modelarla como un proceso estocástico de difusión. Se enuncia el teorema M-B 1, cuyo objeto principal es el problema inverso de difusión. Se demuestra dicho teorema, en el contexto particular de la inyectividad de la solución y se aplica para resolver el problema inverso de difusión en presencia del grupo de Boltzmann. Se resuelve el problema inverso del exponente de similaridad por los métodos del análisis de grupo. Se aplica a la dispersión de una gota en un medio poroso tridimensional, resultado a su vez aplicable en el caso del riego por goteo.

SYMMETRY GROUP ANALYSIS OF WEAK SOLUTIONS

Methods of Lie group analysis of differential equations are extended to weak solutions of (linear and non-linear) partial differential equations, where the term `weak solution' comprises the following settings: distributional solutions; solutions in generalized function algebras; solutions in the sense of association (corresponding to a number of weak or integral solution concepts in classical analysis). Factorization properties and infinitesimal criteria that allow the treatment of all three settings simultaneously are developed, thereby unifying and extending previous work in this area.2000 Mathematical Subject Classification: 46F30, 22E70, 35Dxx, 35A30.

Inverse dynamic problems, the Focker— Plank—Kolmogorov equation and the Laplace iterative process

Abstract - In this paper the Focker-Plank-Kolmogorov equation based on the Laplace iteration process in the two-dimensional case is reduced to the system of differential equations for the right-hand sides of the dynamic system and system of equations on probability density. In some cases corollary systems can be integrated that allows to obtain the exact formulas for the right-hand sides of the dynamic system and probability density. To deduce some solutions group analysis of differential equations and invariant solutions are used [9, 10]. The solution obtained are functionally arbitrary. In the paper determination of probability density and right-hand sides of the dynamic system by the most probable motion path is also considered. Supposing probability density known, we obtain a differential equation of the second order in the partial derivatives of the right-hand sides of the dynamic system. At last, in the multidimensional case, we use the Focker-Plank-Kolmogorov equation to obtain an equation on the potential of the dynamic system forces.