Integration of ordinary differential equation with a small parameter via approximate symmetries: Reduction of approximate symmetry algebra to a canonical form

Газизов, Р. К.; Ibragimov, Nail H.; Lukashchuk, V. O.

doi:10.1134/s1995080210020058

Cited by 7 publications

(11 citation statements)

References 4 publications

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“…They developed an elegant theory since all the useful properties of exact Lie symmetries are adapted in the approximate sense. Since its introduction, this approach has been applied to many physical models [14,15,16,17,18,19,20]. Nevertheless, the expanded generator is not consistent with the principles of perturbative analysis [21] because the dependent variables are not expanded.…”

Section: Introductionmentioning

confidence: 99%

Approximately invariant solutions of creeping flow equations

Gorgone

2018

International Journal of Non-Linear Mechanics

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In this paper, the steady creeping flow equations of a second grade fluid in cartesian coordinates are considered; the equations involve a small parameter related to the dimensionless non-Newtonian coefficient. According to a recently introduced approach, the first order approximate Lie symmetries of the equations are computed, some classes of approximately invariant solutions are explicitly determined, and a boundary value problem is analyzed. The main aim of the paper is methodological, and the considered mechanical model is used to test the reliability of the procedure in a physically important application.

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Section: Introductionmentioning

confidence: 99%

Approximately invariant solutions of creeping flow equations

Gorgone

2018

International Journal of Non-Linear Mechanics

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“…The resulting theory is quite elegant since all the useful properties of exact Lie symmetries can be adapted in the approximate sense: reduction of order of ordinary differential equations, approximately invariant solutions, approximate conservation laws, etc. Since its introduction, this approach has been applied to many physical models [29,30,31,32,33,34,35,36,37,38,39]. Nevertheless, the expanded generator is not consistent with the principles of perturbation analysis [40] because the dependent variables are not expanded.…”

Section: Introductionmentioning

confidence: 99%

A consistent approach to approximate Lie symmetries of differential equations

2017

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Lie theory of continuous transformations provides a unified and powerful approach for handling differential equations. Unfortunately, any small perturbation of an equation usually destroys some important symmetries, and this reduces the applicability of Lie group methods to differential equations arising in concrete applications. On the other hand, differential equations containing small terms are commonly and successfully investigated by means of perturbative techniques. Therefore, it is desirable to combine Lie group methods with perturbation analysis, i.e., to establish an approximate symmetry theory. There are two widely used approaches to approximate symmetries: the one proposed in 1988 by Baikov, Gazizov and Ibragimov, and the one introduced in 1989 by Fushchich and Shtelen. Moreover, some variations of the Fushchich-Shtelen method have been proposed with the aim of reducing the length of computations. Here, we propose a new approach that is consistent with perturbation theory and allows to extend all the relevant features of Lie group analysis to an approximate context. Some applications are also presented.

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“…In this expression, n x is an overload, a quantity showing how many times intensity of the active field of mass forces exceeds intensity of the field of gravitational forces (acceleration of gravity). The Sh, Eu and Re criteria remain unchanged and will have the form of (8). The Fr criterion is expressed using the same characteristic scales and quantities:…”

Section: Fo T Fo T Ho a T A T T L Lmentioning

confidence: 99%

“…They have necessary dimensionalities and when substituted in (8), they transform all criteria into quantities identical to one. This substitution demonstrates feasibility of obtaining the number of parameters less than that prescribed by π-theorem when the expressions are dimensionless.…”

Section: Fo T Fo T Ho a T A T T L Lmentioning

confidence: 99%

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Development of a method for approximate solution of nonlinear ordinary differential equations using pendulum motion as an example

Brunetkin

Maksymov

Maksymova

et al. 2017

EEJET

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4thermophysical properties, etc. In turn, this causes variation in a wide range of the process parameters subject to control. In such a situation, application of the results obtained by analysis of linear models is very problematic and even unfeasible in most cases.The search for new approaches to analysis of nonlinear models is an important element in solution of the problem of optimal control of power equipment using non-certified fuel of a variable composition. Literature review and problem statementAn effective way to simplify the study of complex models is to make them dimensionless. Reduction of dimensionality of the modeling space makes it possible both to reduce number of experimental (full-scale and numerical) studies by IntroductionWhen studying behavior of control systems, dynamic conditions require special consideration. Apparatus of ordinary differential equations is most often used as a mathematical tool for studying such systems. General methods of solution have only their linear forms. For this reason, linear (linearized) approaches are used to construct models of object and controller. However more exact models are represented by nonlinear differential equations. This situation leads to the fact that the results obtained can only be used within the range of small variation of parameters of the controlled processes nearby the linearization point.The use of fuel of a variable composition [1,2] in power equipment instead of certified fuel with a constant composition brings about a continuous variation of the fuel calorific value, amount of combustion products, their temperature, MATHEMATICS AND CYBERNETICS -APPLIED ASPECTS DEVELOPMENT OF

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Integration of ordinary differential equation with a small parameter via approximate symmetries: Reduction of approximate symmetry algebra to a canonical form

Cited by 7 publications

References 4 publications

Approximately invariant solutions of creeping flow equations

Approximately invariant solutions of creeping flow equations

A consistent approach to approximate Lie symmetries of differential equations

Development of a method for approximate solution of nonlinear ordinary differential equations using pendulum motion as an example

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