| 1. |
- Bobylev, Alexander, 1947-, et al.
(författare)
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Group analysis of the generalized Burnett equations
- 2020
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Ingår i: Journal of Nonlinear Mathematical Physics. - : Taylor & Francis. - 1402-9251 .- 1776-0852. ; 27:3, s. 494-494
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Tidskriftsartikel (refereegranskat)abstract
- In this paper group properties of the so-called Generalized Burnett equations are studied. In contrast to the clas-sical Burnett equations these equations are well-posed and therefore can be used in applications. We considerthe one-dimensional version of the generalized Burnett equations for Maxwell molecules in both Eulerian andLagrangian coordinates and perform the complete group analysis of these equations. In particular, this includesfinding and analyzing admitted Lie groups. Our classifications of the Lie symmetries of the Navier-Stokes equa-tions of compressible gas and generalized Burnett equations provide a basis for finding invariant solutions ofthese equations. We also consider representations of all invariant solutions. Some particular classes of invariantsolutions are studied in more detail by both analytical and numerical methods
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| 2. |
- Gainetdinova, A., et al.
(författare)
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Group classification of ODE y‴ = F (x, y, y′)
- 2014
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Ingår i: Communications in nonlinear science & numerical simulation. - : Elsevier. - 1007-5704 .- 1878-7274. ; 19:2, s. 345-349
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Tidskriftsartikel (refereegranskat)abstract
- In his extensive work of 1884 on the group classification of ordinary differential equations Lie performed, inter alia, the group classification of the particular type of the second-order equations y″ = F (x, y). In the present paper we extend Lie's classification to the third-order equations y‴ = F (x, y, y′).
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| 3. |
- Grigoriev, Yurii, et al.
(författare)
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Delay differential equations
- 2010
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Ingår i: Lecture Notes in Physics. - Dordrecht : Springer. - 0075-8450. ; 806, s. 251-292
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Tidskriftsartikel (refereegranskat)abstract
- In this chapter, applications of group analysis to delay differential equations are considered. Many mathematical models in biology, physics and engineering, where there is a time lag or aftereffect, are described by delay differential equations. These equations are similar to ordinary differential equations, but their evolution involves past values of the state variable. For the sake of completeness the chapter is started with a short introduction into the theory of delay differential equations. The mathematical background of these equations is followed by the section which deals with the definition of an admitted Lie group for them and some examples. The purpose of the next section is to give a complete group classification with respect to admitted Lie groups of a second-order delay ordinary differential equation. The reasonable generalization of the definition of an equivalence Lie group for delay differential equations is considered in the next section. The last section of the chapter is devoted to application of the developed theory to the reaction-diffusion equation with a delay.
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| 4. |
- Grigoriev, Yurii, et al.
(författare)
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Introduction to group analysis and invariant solutions of integro-differential equations
- 2010
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Ingår i: Lecture Notes in Physics. - Dordrecht : Springer. - 0075-8450. ; 806, s. 57-111
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Tidskriftsartikel (refereegranskat)abstract
- In this chapter an introduction into applications of group analysis to equations with nonlocal operators, in particular, to integro-differential equations is given. The most known integro-differential equations are kinetic equations which form a mathematical basis in the kinetic theories of rarefied gases, plasma, radiation transfer, coagulation. Since these equations are directly associated with fundamental physical laws, there is special interest in studies of their solutions. The first section of this chapter contains a retrospective survey of different methods for constructing symmetries and finding invariant solutions of such equations. The presentation of the methods is carried out using simple model equations of small dimensionality, allowing the reader to follow the calculations in detail. In the next section, the classical scheme of the construction of determining equations of an admitted Lie group is generalized for equations with nonlocal operators. In the concluding sections of this chapter, the developed regular method of obtaining admitted Lie groups is illustrated by applications to some known integro-differential equations.
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| 5. |
- Grigoriev, Yurii, et al.
(författare)
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Introduction to group analysis of differential equations
- 2010
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Ingår i: Lecture Notes in Physics. - Dordrecht : Springer. - 0075-8450. ; 806, s. 1-55
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Tidskriftsartikel (refereegranskat)abstract
- The first chapter is a brief, but a sufficiently comprehensive introduction to the methods of Lie group analysis of ordinary and partial differential equations. The chapter presents basic concepts from the theory: continuous transformation groups, their generators, Lie equations, groups admitted by differential equations, integration of ordinary differential equations using their symmetries, group classification and invariant solutions of partial differential equations. New trends in modern group analysis such as the theory of Lie-Bäcklund transformations groups and approximate groups are also reflected. The intention of the chapter is to give the basic ideas of classical and modern group analysis to beginner readers and provide useful materials for advanced specialists
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| 6. |
- Grigoriev, Yurii, et al.
(författare)
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Plasma kinetic theory : Vlasov-maxwell and related equations
- 2010
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Ingår i: Lecture Notes in Physics. - Dordrecht : Springer. - 0075-8450. ; 806, s. 145-208
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Tidskriftsartikel (refereegranskat)abstract
- This chapter is devoted to a group analysis of the Vlasov-Maxwell and related type equations. The equations form the basis of the collisionless plasma kinetic theory, and are also applied in gravitational astrophysics, in shallow-water theory, etc. Nonlocal operators in these equations appear in the form of the functionals defined by integrals of the distribution functions over momenta of particles. In the beginning sections the plasma kinetic theory equations are introduced and the way of looking at the symmetries of nonlocal equations is described. Much of the importance of the approach used in this chapter for calculating symmetries stems from the procedure of solving determining equations using variational differentiation. The set of symmetries obtained in the sections that follow comprises symmetries for the Vlasov-Maxwell equations of the non-relativistic and relativistic electron and electron-ion plasmas in both one- and three-dimensional cases, and symmetries for Benney equations. In the concluding sections of this chapter the procedure for symmetry calculation and the renormalization group algorithm go hand in hand to present illustrations from plasma kinetic theory, plasma dynamics, and nonlinear optics, which demonstrate the potentialities of the method in construction of analytic solutions to nonlocal problems of nonlinear physics.
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| 7. |
- Grigoriev, Yurii, et al.
(författare)
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Symmetries of stochastic differential equations
- 2010
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Ingår i: Lecture Notes in Physics. - Dordrecht : Springer. - 0075-8450. ; 806, s. 209-250
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Tidskriftsartikel (refereegranskat)abstract
- This chapter deals with applications of the group analysis method to stochastic differential equations. These equations are often obtained by including random fluctuations in differential equations, which have been deduced from phenomenological or physical view. In contrast to deterministic differential equations, only few attempts to apply group analysis to stochastic differential equations can be found in the literature. It is worth to note that this theory is still developing. Before defining an admitted symmetry for stochastic differential equations an introduction into the theory of this type of equations is given. The introduction includes the discussion of a stochastic integration, a stochastic differential and a change of the variables (Itô formula) in stochastic differential equations. Applications of the Itô formula are considered in the next section which deals with the linearization problem. The Itô formula and the change of time in stochastic differential equations are the main tools of defining admitted transformations for them. After introducing an admitted Lie group and supporting material of the introduced definition, some examples of applications of the given definition are studied.
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| 8. |
- Grigoriev, Yurii, et al.
(författare)
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The Boltzmann kinetic equation and various models
- 2010
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Ingår i: Lecture Notes in Physics. - Dordrecht : Springer. - 0075-8450. ; 806, s. 113-144
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Tidskriftsartikel (refereegranskat)abstract
- The chapter deals with applications of the group analysis method to the full Boltzmann kinetic equation and some similar equations. These equations form the foundation of the kinetic theory of rarefied gas and coagulation. They typically include special integral operators with quadratic nonlinearity and multiple kernels which are called collision integrals. Calculations of the 11-parameter Lie group G 11 admitted by the full Boltzmann equation with arbitrary intermolecular potential and its extensions for power potentials are presented. The found isomorphism of these Lie groups with the Lie groups admitted by the ideal gas dynamics equations allowed one to obtain an optimal system of admitted subalgebras and to classify all invariant solutions of the full Boltzmann equation. For equations similar to the full Boltzmann equation complete admitted Lie groups are derived by solving determining equations. The corresponding optimal systems of admitted subalgebras are constructed and representations of all invariant solutions are obtained.
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| 9. |
- Ibragimov, Nail H., et al.
(författare)
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A solution to the problem of invariants for parabolic equations
- 2009
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Ingår i: Communications in nonlinear science & numerical simulation. - AMSTERDAM : ELSEVIER SCIENCE BV. - 1007-5704 .- 1878-7274. ; 14:6, s. 2551-2558
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Tidskriftsartikel (refereegranskat)abstract
- The article is devoted to the Solution Of the invariants problem for the one-dimensional parabolic equations written in the two-coefficient canonical form used recently by N.H. Ibragimov: u(t) - u(xx) + a (t, x)u(x) + c(t, x)u = 0. A simple invariant condition is obtained for determining all equations that are reducible to the heat equation by the general group of equivalence transformations. The solution to the problem of invariants is given also in the one-coefficient canonical u(t) - u(xx) + c(t, x)u = 0. One of the main differences between these two canonical forms is that the equivalence group for the two-coefficient form contains the arbitrary linear transformation of the dependent variable whereas this group for the one-coefficient form contains only a special type of the linear transformations of the dependent variable. (C) 2008 Elsevier B.V. All rights reserved.
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| 10. |
- Ibragimov, Nail H., et al.
(författare)
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Group analysis of evolutionary integro-differential equations describing nonlinear waves : General model
- 2011
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Ingår i: Journal of Physics A. - : IOP publishing. - 1751-8113 .- 1751-8121. ; 44:31
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Tidskriftsartikel (refereegranskat)abstract
- The paper deals with an evolutionary integro-differential equation describing nonlinear waves. Particular choice of the kernel in the integral leads to well-known equations such as the Khokhlov-Zabolotskaya equation, the Kadomtsev-Petviashvili equation and others. Since solutions of these equations describe many physical phenomena, analysis of the general model studied in the paper equation is important. One of the methods for obtaining solutions differential equations is provided by the Lie group analysis. However, this method is not applicable to integro-differential equations. Therefore we discuss new approaches developed in modern group analysis and apply them to the general model considered in the present paper. Reduced equations and exact solutions are also presented.
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| 11. |
- Ibragimov, Nail H., et al.
(författare)
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Invariants and invariant description of second-order ODEs with three infinitesimal symmetries. I
- 2007
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Ingår i: Communications in nonlinear science & numerical simulation. - Amsterdam : Elsevier. - 1007-5704 .- 1878-7274. ; 12:8, s. 1370-1378
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Tidskriftsartikel (refereegranskat)abstract
- Lie's group classification of ODEs shows that the second-order equations can possess one, two, three or eight infinitesimal symmetries. The equations with eight symmetries and only these equations can be linearized by a change of variables. Lie showed that the latter equations are at most cubic in the first derivative and gave a convenient invariant description of all linearizable equations. Our aim is to provide a similar description of the equations with three symmetries. There are four different types of such equations. We present here the candidates for all four types. We give an invariant test for existence of three symmetries for one of these candidates.
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| 12. |
- Ibragimov, Nail H., et al.
(författare)
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Invariants and invariant description of second-order ODEs with three infinitesimal symmetries. II.
- 2008
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Ingår i: Communications in nonlinear science & numerical simulation. - The Netherlands : ELSEVIER SCIENCE BV. - 1007-5704 .- 1878-7274. ; 13:6, s. 1015-1020
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Tidskriftsartikel (refereegranskat)abstract
- The second-order ordinary differential equations can have one, two, three or eight independent symmetries. Sophus Lie showed that the equations with eight symmetries and only these equations can be linearized by a change of variables. Moreover he demonstrated that these equations are at most cubic in the first derivative and gave a convenient invariant description of all linearizable equations. We provide a similar description of the equations with three symmetries. There are four different types of such equations. Classes of equations belonging to one of these types were studied in N.H. Ibragimov and S.V. Meleshko, Invariants and invariant description of second-order ODEs with three infinitesimal symmetries. I, Communications in Nonlinear Science and Numerical Simulation, Vol. 12, No. 8, 2007, pp. 1370--1378. Namely, we presented there the candidates for all four types and studied one of these candidates.The present paper is devoted to other three candidates.
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| 13. |
- Ibragimov, Nail H., et al.
(författare)
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Invariants of linear parabolic differential equations
- 2008
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Ingår i: Communications in nonlinear science & numerical simulation. - AMSTERDAM : ELSEVIER SCIENCE BV. - 1007-5704 .- 1878-7274. ; , s. 277-284
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Tidskriftsartikel (refereegranskat)abstract
- The paper is dedicated to construction of invariants for the parabolic equation u(t) + a(t, x)u(xx) + b(t, x)u(x) + c(t, x)u = 0. We consider the equivalence group given by point transformations and find all invariants up to seventh-order, i.e. the invariants involving the derivatives up to seventh-order of the coefficients a, b and c with respect to the independent variables t, x. (c) 2006 Elsevier B.V. All rights reserved.
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| 14. |
- Ibragimov, Nail H., et al.
(författare)
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Linearization of fourth-order ordinary differential equations by point transformations
- 2007
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Ingår i: Archives of ALGA. - Karlskrona, Sweden : ALGA publications, BTH. - 1652-4934. ; 4, s. 113-134
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Tidskriftsartikel (refereegranskat)abstract
- We present here the solution of the problem on linearization of fourth-order equations by means of point transformations. We show that all fourth-order equations that are linearizable by point transformations are contained in the class of equations which is linear in the third-order derivative. We provide the linearization test and describe the procedure for obtaining the linearizing transformations as well as the linearized equation.
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| 15. |
- Ibragimov, Nail H., et al.
(författare)
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Linearization of fourth order ordinary differential equations by point transformations
- 2008
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Ingår i: Journal of Physics A. - : IOP Publishers. - 1751-8113 .- 1751-8121. ; 23:41, s. 206-235
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Tidskriftsartikel (refereegranskat)abstract
- A general methodology for linearization of fourth order ordinary differential equations is developed using point transformations. The solution of the problem on linearization of fourth-order equations by means of point transformations is presented here. We show that all fourth-order equations that are linearizable by point transformations are contained in the class of equations which is linear in the third-order derivative. We provide the linearization test and describe the procedure for obtaining the linearizing transformations as well as the linearized equation. For ordinary differential equations of order greater than 4 we obtain necessary conditions, which separate all linearizable equations into two classes.
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| 16. |
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| 17. |
- Ibragimov, Nail H., et al.
(författare)
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Linearization of third-order ordinary differential equations
- 2004
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Konferensbidrag (refereegranskat)abstract
- We present here the complete solution to the problem on linearization of third-order equations by means of general point transformations. We also formulate the criteria for reducing third-order equations to the equation y''' = 0 by contact transformations.
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| 18. |
- Ibragimov, Nail H., et al.
(författare)
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Linearization of third-order ordinary differential equations by point and contact transformations.
- 2005
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Ingår i: Journal of Mathematical Analysis and Applications. - The Netherlands : ACADEMIC PRESS INC ELSEVIER SCIENCE. - 0022-247X .- 1096-0813. ; 308:1, s. 266-289
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Tidskriftsartikel (refereegranskat)abstract
- We present here the solution of the problem on linearization of third-order ordinary differential equations by means of point and contact transformations. We provide, in explicit forms, the necessary and sufficient conditions for linearization, the equations for determining the linearizing point and contact transformations as well as the coefficients of the resulting linear equations.
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| 19. |
- Ibragimov, Nail H., et al.
(författare)
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Linearization of third-order ordinary differential equations by point transformations.
- 2004
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Ingår i: Archives of ALGA. - Karlskrona; Sweden : ALGA publications, BTH. - 1652-4934. ; 1, s. 95-126
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Tidskriftsartikel (refereegranskat)abstract
- We present here the necessary and sufficient conditions for linearization of third-order equations by means of point transformations. We show that all third-order equations that are linearizable by point transformations are contained either in the class of equations which are linear in the second-order derivative, or in the class of equations which are quadratic in the second-order derivative. We provide the linearization test for each of these classes and describe the procedure for obtaining the linearizing point transformations as well as the linearized equation.
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| 20. |
- Ibragimov, Nail H., et al.
(författare)
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Second-order ordinary differential equations equivalent to y''=H(y).
- 2007
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Ingår i: Archives of ALGA. - Karlskrona, Sweden : ALGA publications, BTH. - 1652-4934. ; 4, s. 101-112
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Tidskriftsartikel (refereegranskat)abstract
- The main feature of equations of the form y''=H(y) is that their solutions can be represented in quadratures. The paper gives criteria for a second-order ordinary differential equation to be equivalent to an equation of this form.
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| 21. |
- Meleshko, Sergey, et al.
(författare)
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Symmetries of Integro-Differential Equations : with applications in mechanics and plasma physics
- 2010
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Bok (övrigt vetenskapligt/konstnärligt)abstract
- This book aims to coherently present applications of group analysis to integro-differential equations in an accessible way. The book will be useful to both physicists and mathematicians interested in general methods to investigate nonlinear problems using symmetries. Differential and integro-differential equations, especially nonlinear, present the most effective way for describing complex processes. Therefore, methods to obtain exact solutions of differential equations play an important role in physics, applied mathematics and mechanics. This book provides an easy to follow, but comprehensive, description of the application of group analysis to integro-differential equations. The book is primarily designed to present both fundamental theoretical and algorithmic aspects of these methods. It introduces new applications and extensions of the group analysis method. The authors have designed a flexible text for postgraduate courses spanning a variety of topics.
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| 22. |
- Suksern, Supaporn, et al.
(författare)
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Criteria for fourth-order ordinary differential equations to be linearizable by contact transformations
- 2009
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Ingår i: Communications in nonlinear science & numerical simulation. - AMSTERDAM : ELSEVIER SCIENCE BV. - 1007-5704 .- 1878-7274. ; 14:6, s. 2619-2628
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Tidskriftsartikel (refereegranskat)abstract
- Solution of linearization problem of fourth-order ordinary differential equations Via contact transformations is presented in the paper. We show that all fourth-order ordinary differential equations that are linearizable by contact transformations are contained in the class of equations which is at most quadratic in the third-order derivative. We provide the linearization test and describe the procedure for obtaining the linearizing transformations as well as the linearized equation. Moreover, we obtain the general form of ordinary differential equations of order greater than four linearizable via contact transformations. (C) 2008 Elsevier B.V. All rights reserved.
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