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  • 101. Ibragimov, Nail H.
    The answer to the question put to me by L.V. Ovsyannikov 33 years ago2007Conference paper (Refereed)
    Abstract [en]

    In 1973, during a discussion of the pioneering works on soliton theory at ``Theoretical seminar of the Institute of Hydrodynamics" in Novosibirsk, Professor Ovsyannikov asked me if the infinite number of conservation laws for the Korteweg-de Vries equation can be obtained from its symmetries. The answer was by no means evident because the KdV equation did not have the usual Lagrangian, and hence the Noether theorem was not applicable. In my talk I give the affirmative answer to Ovsyannikov's question by proving a general theorem on conservation laws for arbitrary differential equations. The new conservation theorem does not require existence of a Lagrangian and is based on a concept of adjoint equations for non-linear equations. For derivation of the infinites series of conservation laws for the KdV equation, I modify the notion of self-adjoint equations and extend it to non-linear equations.

  • 102. Ibragimov, Nail H.
    Time-dependent exact solutions of the nonlinear Kompaneets equation2010In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 43, no 50Article in journal (Refereed)
    Abstract [en]

    Time-dependent exact solutions of the Kompaneets photon diffusion equation are obtained for several approximations of this equation. One of the approximations describes the case when the induced scattering is dominant. In this case, the Kompaneets equation has an additional symmetry which is used for constructing some exact solutions as group invariant solutions.

  • 103. Ibragimov, Nail H.
    Transformation groups and Lie algebras2009Book (Other academic)
    Abstract [en]

    These notes are designed for the graduate course on Transformation groups and Lie algebras that I have been teaching at Blekinge Institute of Technology since 2002. The course is aimed at augmenting a preliminary knowledge in this field obtained by students during the course on Differential equations based on my book "A practical course in differential equations and mathematical modelling".

  • 104.
    Ibragimov, Nail H
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Natural Sciences.
    Transformation groups and Lie algebras2013Book (Other academic)
    Abstract [en]

    This book is based on the extensive experience of teaching for mathematics, physics and engineering students in Russia, USA, South Africa and Sweden. The author provides students and teachers with an easy to follow textbook spanning a variety of topics. The methods of local Lie groups discussed in the book provide universal and effective method for solving nonlinear differential equations analytically. Introduction to approximate transformation groups also contained in the book helps to develop skills in constructing approximate solutions for differential equations with a small parameter. Readership: Professional mathematics researchers and graduated students.

  • 105. Ibragimov, Nail H.
    et al.
    Aitbayev, Rakhim
    Ibragimov, Ranis
    Three-dimensional non-linear rotating surface waves in channels of variable depth in the presence of formation of a small perturbation of atmospheric pressure across the channel2009In: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 14, no 11, p. 3811-3820Article in journal (Refereed)
    Abstract [en]

    We consider three-dimensional free-boundary problem on the propagation of incompressible, homogeneous and inviscid fluid with zero surface tension confined in a channel of variable depth. Since for large-scale flows the fluid motion is affected by the rotation of the earth, the model is considered in rotating reference frame. Additionally, small atmospheric pressure variations across the channel are taken into account. It is shown that the non-trivial solution to the problem represents three-dimensional solitary wave which is given by the rotation modified Korteweg-de Vries equation (fKdV): b(1)xi(xxx) + b(2)xi xi(x) + b(3)(f)xi(x) = 0, where x is the down-channel coordinate and the coefficients b(i) (i = 1,2,3) of the resulting fKdV equation depend on the transverse topography of the channel and, additionally, b(3) depends on the Coriolis parameter f. It is also shown that if the vertical profile of the channel is symmetric about the vertical axis, the small atmospheric variations will not appear in the resulting fKdV equation. The effects of channel's cross-sectional geometry on the shape of the resulting three-dimensional wave profile in a longitudinal direction are studied numerically. Additionally, to better understand the effects of the Earth rotation, the above analysis is performed at different latitudes. (C) 2008 Elsevier B.V. All rights reserved.

  • 106. Ibragimov, Nail H.
    et al.
    Al-Hammadi, A.S.A.
    Introduction to differential equations2006Book (Other academic)
    Abstract [en]

    This book is based on lectures in differential equations delivered by the authors, namely, by A. Al-Hammadi at the University of Bahrain and by N. Ibragimov at the Blekinge Institute of Technology, Sweden. It is designed for a one semester course in differential equations. We assume that the reader has taken standard courses in mathematical analysis and has encountered basic concepts from linear algebra such as vectors, matrices and algebraic linear systems. Our aim is to provide students with an easy to follow introduction to the subject containing classical devices enriched by fundamentals of Lie group methods given in Chapter 9.

  • 107.
    Ibragimov, Nail H.
    et al.
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Natural Sciences.
    Avdonina, Elena D.
    Conservation laws and exact solutions for nonlinear diffusion in anisotropic media2013In: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 18, no 10Article in journal (Refereed)
    Abstract [en]

    Conservation laws and exact solutions of nonlinear differential equations describing diffusion phenomena in anisotropic media with external sources are constructed. The construction is based on the method of nonlinear self-adjointness. Numerous exact solutions are obtained by using the recent method of conservation laws. These solutions are different from group invariant solutions and can be useful for investigating diffusion phenomena in complex media, e.g. in oil industry.

  • 108. Ibragimov, Nail H.
    et al.
    Ibragimov, Ranis
    Applications of Lie group analysis in Geophysical fluid dynamics2011Book (Other academic)
    Abstract [en]

    This book introduces an effective method for seeking local and nonlocal conservation laws and exact solutions for nonlinear two-dimensional equations which provide a basic model in describing internal waves in the ocean. The model consists of non-hydrostatic equations of motion which uses the Boussinesq approximation and linear stratification. The Lie group analysis is used for constructing non-trivial conservation laws and group invariant solutions. It is shown that nonlinear equations in question have remarkable property to be self-adjoint. This property is crucial for constructing physically relevant conservation laws for nonlinear internal waves in the ocean. The comparison with the previous analytic studies and experimental observations confirrms that the anisotropic nature of the wave motion allows to associate some of the obtained invariant solutions with uni-directional internal wave beams propagating through the medium. Analytic examples of the latitude-dependent invariant solutions associated with internal gravity wave beams are considered. The behavior of the invariant solutions near the critical latitude is investigated.

  • 109. Ibragimov, Nail H.
    et al.
    Ibragimov, Ranis
    Integration by quadratures of the nonlinear Euler equations modeling atmospheric flows in a thin rotating spherical shell2011In: Physics Letters A, ISSN 0375-9601, E-ISSN 1873-2429, Vol. 375, no 44, p. 3858-3865Article in journal (Refereed)
    Abstract [en]

    We study the nonlinear incompressible non-viscous fluid flows within a thin rotating atmospheric shell that serve as a simple mathematical description of an atmospheric circulation caused by the temperature difference between the equator and the poles. The model is also superimposed by a particular stationary flow which, under the assumption of no friction and a distribution of temperature dependent only upon latitude, models the zonal west-to-east flows in the upper atmosphere between the Ferrel and Polar cells. Owing to the Coriolis effects, the resulting achievable meteorological flows correspond to the asymptotical stable flows that are being translated along the equatorial plane. The exact solutions in terms of elementary functions are found by using Lie group methods.

  • 110. Ibragimov, Nail H.
    et al.
    Ibragimov, Ranis
    Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics2010In: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 15, no 8, p. 1989-2002Article in journal (Refereed)
    Abstract [en]

    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. © 2009 Elsevier B.V. All rights reserved.

  • 111. Ibragimov, Nail H.
    et al.
    Ibragimov, Ranis
    Invariant solutions as internal singularities of nonlinear differential equations and their use for qualitative analysis of implicit and numerical solutions2009In: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 14, no 9-10, p. 3537-3547Article in journal (Refereed)
    Abstract [en]

    Lie group analysis of nonlinear differential equations reveals existence of singularities provided by invariant solutions and invisible from the form of the equation in question. We call them internal singularities in contrast with external singularities manifested by the form of the equation. It is illustrated by way of examples that internal singularities are useful for analyzing a behaviour of solutions of nonlinear differential equations near external singularities.

  • 112.
    Ibragimov, Nail H.
    et al.
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Natural Sciences.
    Ibragimov, Ranis N.
    Nonlinear Whirlpools Versus Harmonic Waves in a Rotating Column of Stratified Fluid2013In: Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, Vol. 8, no 1, p. 122-131Article in journal (Refereed)
    Abstract [en]

    Propagation of nonlinear baroclinic Kelvin waves in a rotating column of uniformly stratified fluid under the Boussinesq approximation is investigated. The model is constrained. by the Kelvin's conjecture saying that the velocity component normal to the interface between rotating fluid and surrounding medium (e.g. a seashore) is possibly zero everywhere in the domain of fluid motion, not only at the boundary. Three classes of distinctly different exact solutions for the nonlinear model are obtained. The obtained solutions are associated with symmetries of the Boussinesq model. It is shown that one class of the obtained solutions can be visualized as rotating whirlpools along which the pressure deviation from the mean state is zero, is positive inside and negative outside of the whirlpools. The angular velocity is zero at the center of the whirlpools and it is monotonically increasing function of radius of the whirlpools.

  • 113.
    Ibragimov, Nail H.
    et al.
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Natural Sciences.
    Ibragimov, Ranis N.
    Rotationally symmetric internal gravity waves2012In: International Journal of Non-Linear Mechanics, ISSN 0020-7462, E-ISSN 1878-5638, Vol. 47, no 1, p. 46-52Article in journal (Refereed)
    Abstract [en]

    Many mathematical models formulated in terms of non-linear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating non-linear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is to provide the group theoretical modeling of internal waves in the ocean. The approach is based on a new concept of conservation laws that is utilized to systematically derive the conservation laws of non-linear equations describing propagation of internal waves in the ocean. It was shown in our previous publication that uni-directional internal wave beams can be obtained as invariant solutions of non-linear equations of motion. The main goal of the present publication is to thoroughly analyze another physically significant exact solution, namely the rotationally symmetric solution and the energy carried by this solution. It is shown that the rotationally symmetric solution and its energy are presented by means of a bounded oscillating function.

  • 114. Ibragimov, Nail H.
    et al.
    Khabirov, S.V.
    Existence of integrating factors for higher-order ordinary differential equations2007In: Archives of ALGA, ISSN 1652-4934, Vol. 4, p. 61-70Article in journal (Refereed)
    Abstract [en]

    We investigate the concept of integrating factors for higher-order ordinary differential equations introduced recently by one of the authors (NHI). The integrating factors for equations of order higher than one are determined by overdetermined systems. Therefore one can expect that not all higher-order equations have integrating factors. We prove in this paper that in fact they have. Moreover, we demonstrate that every equation of order n has precisely n functionally independent integrating factors.

  • 115.
    Ibragimov, Nail H.
    et al.
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Science.
    Khamitova, Raisa
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Science.
    Computer Algebra Programs in Group Analysis and Education2006Conference paper (Refereed)
    Abstract [en]

    Lie group analysis provides a universal tool for tackling considerable numbers of differential equations even when other means of integration fail. In fact, group analysis is the only effective method for solving nonlinear differential equations analytically. However the philosophy of Lie groups in the theory of differential equations did not enjoy widespread acceptance in the past and the subject has been neglected in university programs. One of the main aims of ALGA is to improve the situation by developing courses based on group analysis. But the problem is that calculation of symmetries is a lengthy process and this can scare students. The calculation can be simplified by using computer algebra packages. We started to use one of them for teaching engineering students at BTH. This project was supported by STINT, Swedish Foundation for International Cooperation in Research and Higher Education.

  • 116.
    Ibragimov, Nail H.
    et al.
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Natural Sciences.
    Khamitova, Raisa
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Natural Sciences.
    Conservation Laws in Thomas’s Model of Ion Exchange in a Heterogeneous Solution2013In: Discontinuity, Nonlinearity and Complexity, ISSN 2164-6376, Vol. 2, no 2Article in journal (Refereed)
    Abstract [en]

    Physically significant question on calculation of conservation laws of the Thomas equation is investigated. It is demonstrated that the Thomas equation is nonlinearly self-adjoint. Using this property and applying the theorem on nonlocal conservation laws the infinite set of conservation laws corresponding to the symmetries of the Thomas equation is computed. It is shown that the Noether theorem provide only one of these conservation laws.

  • 117.
    Ibragimov, Nail H.
    et al.
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Natural Sciences.
    Khamitova, Raisa
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Natural Sciences.
    Cantoni, Antonio
    Self-adjointness of a generalized Camassa-Holm equation2011In: Applied Mathematics and Computation, ISSN 0096-3003, E-ISSN 1873-5649, Vol. 218, no 6, p. 2579-2583Article in journal (Refereed)
    Abstract [en]

    It is well known that the Camassa-Holm equation possesses numerous remarkable properties characteristic for KdV type equations. In this paper we show that it shares one more property with the KdV equation. Namely, it is shown in [1,2] that the KdV and the modified KdV equations are self-adjoint. Starting from the generalization [3] of the Camassa-Holm equation [4], we prove that the Camassa-Holm equation is self-adjoint. This property is important, e.g. for constructing conservation laws associated with symmetries of the equation in question. Accordingly, we construct conservation laws for the generalized Camassa-Holm equation using its symmetries.

  • 118.
    Ibragimov, Nail H.
    et al.
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Science.
    Khamitova, Raisa
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Science.
    Thidé, Bo
    Adjoint systems and conservation laws for symmetrized electromagnetic equations with a dual Ohm´s law2006In: Archives of ALGA, ISSN 1652-4934, Vol. 3Article in journal (Refereed)
  • 119.
    Ibragimov, Nail H.
    et al.
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Science.
    Khamitova, Raisa
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Science.
    Thidé, Bo
    Conservation laws for symmetrised electromagnetic equations with a dual Ohm's law2007Conference paper (Refereed)
    Abstract [en]

    In all areas of physics, conservation laws are essential since they allow us to draw conclusions of our physical system under study in an indirect but efficient way. Electrodynamics, in terms of the standard Maxwell electromagnetic equations for fields in vacuum, exhibit a rich set of symmetries to which conserved quantities are associated. We have derived conservation laws for Dirac's symmetric version of the Maxwell-Lorentz microscopic equations, allowing magnetic charges and magnetic currents, where the latter, just as electric currents, are assumed to be described by a linear relationship between the field and the current, i.e. an Ohm's law. We find that when we use the method of Ibragimov to construct the conservation laws, they will contain two new adjoint vector fields which fulfil Maxwell-like equations. In particular, we obtain conservation laws for the electromagnetic field which are nonlocal in time.

  • 120.
    Ibragimov, Nail H.
    et al.
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Science.
    Khamitova, Raisa
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Science.
    Thidé, Bo
    Conservation laws for the Maxwell-Dirac equations with dual Ohm's law2007In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 48, no 5Article in journal (Refereed)
    Abstract [en]

    Using a general theorem on conservation laws for arbitrary differential equations proved by Ibragimov [J. Math. Anal. Appl. 333, 311-320 (2007)], we have derived conservation laws for Dirac's symmetrized Maxwell-Lorentz equations under the assumption that both the electric and magnetic charges obey linear conductivity laws (dual Ohm's law). We find that this linear system allows for conservation laws which are nonlocal in time. (c) 2007 American Institute of Physics.

  • 121. Ibragimov, Nail H.
    et al.
    Kolsrud, T.
    Lagrangian approach to evolution equations: symmetries and conservation laws.2004In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 36, no 1, p. 29-40Article in journal (Refereed)
    Abstract [en]

    We show that one can apply a Lagrangian approach to certain evolution equations by considering them together with their associated equations. Consequently, one can employ Noether's theorem and derive conservation laws from symmetries of coupled systems of evolution equations. We discuss in detail the linear and non-linear heat equations as well as the Burgers equation and obtain new non-local conservation laws for the non-linear heat and the Burgers equations by extending their symmetries to the associated equations. We also provide Lagrangians for non-linear Schrödinger and Korteweg-de Vries type systems.

  • 122. Ibragimov, Nail H.
    et al.
    Kovalev, Vladimir
    Approximate and renormgroup symmetries2009Book (Other academic)
    Abstract [en]

    Approximate and Renormgroup Symmetries deals with approximate transformation groups, symmetries of integro-differential equations and renormgroup symmetries. It includes a concise and self contained introduction to basic concepts and methods of Lie group analysis, and provides an easy to follow introduction to the theory of approximate transformation groups and symmetries of integro-differential equations. The book is designed for specialists in nonlinear physics --mathematicians and nonmathematicians--interested in methods of applied group analysis for investigating nonlinear problems in physical, engineering and natural sciences.

  • 123. Ibragimov, Nail H.
    et al.
    Kovalev, Vladimir
    Approximate groups and renormgroup symmetries2008Book (Other academic)
    Abstract [en]

    The book deals with so-called renormalization group symmetries considered in the framework of approximate transformation groups. Renormgroup symmetries provide a basis for the renormgroup algorithm for improving solutions to boundary value problems by converting "less applicable solutions" into "more applicable solutions". the algorithm is particularly useful for improving approximate solutions given by the perturbation theory.

  • 124. Ibragimov, Nail H.
    et al.
    Kovalev, Vladimir
    Pustovalov, V.V.
    Symmetries of integro-differential equations: A survey of methods illustrated by the Benny equations2002In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, p. 135-153Article in journal (Refereed)
    Abstract [en]

    Classical Lie group theory provides a universal tool for calculating symmetry groups for systems of differential equations. However Lie's method is not as much effective in the case of integral or integro-differential equations as well as in the case of infinite systems of differential equations. This paper is aimed to survey the modern approaches to symmetries of integro-differential equations. As an illustration, an infinite symmetry Lie algebra is calculated for a system of integro-differential equations, namely the well-known Benny equations. The crucial idea is to look for symmetry generators in the form of canonical Lie-Backlund operators.

  • 125. Ibragimov, Nail H.
    et al.
    Magri, F.
    Geometric proof of Lie's linearization theorem2004In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 36, no 1, p. 41-46Article in journal (Refereed)
    Abstract [en]

    S. Lie found in 1883 the general form of all second-order ordinary differential equations transformable to the linear equation by a change of variables and proved that their solution reduces to integration of a linear third-order ordinary differential equation. He showed that the linearizable equations are at most cubic in the first-order derivative and described a general procedure for constructing linearizing transformations by using an over-determined system of four equations. We present here a simple geometric proof of the theorem, known as Lie's linearization test, stating that the compatibility of Lie's four auxiliary equations furnishes a necessary and sufficient condition for linearization.

  • 126. Ibragimov, Nail H.
    et al.
    Meleshko, Sergey
    A solution to the problem of invariants for parabolic equations2009In: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 14, no 6, p. 2551-2558Article in journal (Refereed)
    Abstract [en]

    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.

  • 127. Ibragimov, Nail H.
    et al.
    Meleshko, Sergey
    Invariants and invariant description of second-order ODEs with three infinitesimal symmetries. I2007In: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 12, no 8, p. 1370-1378Article in journal (Refereed)
    Abstract [en]

    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.

  • 128. Ibragimov, Nail H.
    et al.
    Meleshko, Sergey
    Invariants and invariant description of second-order ODEs with three infinitesimal symmetries. II.2008In: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 13, no 6, p. 1015-1020Article in journal (Refereed)
    Abstract [en]

    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.

  • 129. Ibragimov, Nail H.
    et al.
    Meleshko, Sergey
    Linearization of second-order ordinary differential equations by changing the order2007In: Archives of ALGA, ISSN 1652-4934, Vol. 4, p. 76-100Article in journal (Refereed)
    Abstract [en]

    Criteria for second-order ordinary differential equations be linearizable after differentiating or after the Ricatti substitution are given.

  • 130. Ibragimov, Nail H.
    et al.
    Meleshko, Sergey
    Linearization of third-order ordinary differential equations2004Conference paper (Refereed)
    Abstract [en]

    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.

  • 131. Ibragimov, Nail H.
    et al.
    Meleshko, Sergey
    Linearization of third-order ordinary differential equations by point and contact transformations.2005In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 308, no 1, p. 266-289Article in journal (Refereed)
    Abstract [en]

    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.

  • 132.
    Ibragimov, Nail H.
    et al.
    Blekinge Institute of Technology, Department of Health, Science and Mathematics.
    Meleshko, Sergey
    Linearization of third-order ordinary differential equations by point transformations.2004In: Archives of ALGA, ISSN 1652-4934, Vol. 1, p. 95-126Article in journal (Refereed)
    Abstract [en]

    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.

  • 133. Ibragimov, Nail H.
    et al.
    Meleshko, Sergey
    Second-order ordinary differential equations equivalent to y''=H(y).2007In: Archives of ALGA, ISSN 1652-4934, Vol. 4, p. 101-112Article in journal (Refereed)
    Abstract [en]

    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.

  • 134. Ibragimov, Nail H.
    et al.
    Meleshko, Sergey
    Rudenko, Oleg
    Group analysis of evolutionary integro-differential equations describing nonlinear waves: General model2011In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 44, no 31Article in journal (Refereed)
    Abstract [en]

    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.

  • 135. Ibragimov, Nail H.
    et al.
    Meleshko, Sergey
    Suksern, Supaporn
    Linearization of fourth order ordinary differential equations by point transformations2008In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 23, no 41, p. 206-235Article in journal (Refereed)
    Abstract [en]

    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.

  • 136. Ibragimov, Nail H.
    et al.
    Meleshko, Sergey
    Suksern, Supaporn
    Linearization of fourth-order ordinary differential equations by point transformations2007In: Archives of ALGA, ISSN 1652-4934, Vol. 4, p. 113-134Article in journal (Refereed)
    Abstract [en]

    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.

  • 137. Ibragimov, Nail H.
    et al.
    Meleshko, Sergey
    Thailert, E
    Invariants of linear parabolic differential equations2008In: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, p. 277-284Article in journal (Refereed)
    Abstract [en]

    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.

  • 138. Ibragimov, Nail H.
    et al.
    Rudenko, Oleg
    Principle of an a priori use of symmetries in the theory of nonlinear waves.2004In: Acoustical Physics, ISSN 1063-7710, E-ISSN 1562-6865, Vol. 50, no 4, p. 406-419Article in journal (Refereed)
    Abstract [en]

    The principle of an a priori use of symmetries is proposed as a new approach to solving nonlinear problems on the basis of a reasonable complication of mathematical models. This approach often provides additional symmetries, and hence opens possibilities to find new analytical solutions. The potentialities of the proposed approach are illustrated by applying to problems of nonlinear acoustics.

  • 139. Ibragimov, Nail H.
    et al.
    Sophocleus, C.
    Differential invariants of the one-dimensional quasi-linear second-order evolution equation2007In: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 12, no 7, p. 1133-1145Article in journal (Refereed)
    Abstract [en]

    We consider evolution equations of the form ut = f(x, u, ux)uxx + g(x, u, ux) and ut = uxx + g(x, u, ux). In the spirit of the recent work of Ibragimov [Ibragimov NH. Laplace type invariants for parabolic equations. Nonlinear Dynam 2002;28:125-33] who adopted the infinitesimal method for calculating invariants of families of differential equations using the equivalence groups, we apply the method to these equations. We show that the first class admits one differential invariant of order two, while the second class admits three functional independent differential invariants of order three. We use these invariants to determine equations that can be transformed into the linear diffusion equation.

  • 140. Ibragimov, Nail H.
    et al.
    Svirshchevskii, SR
    Lie-Backlund symmetries of submaximal order of ordinary differential equations2002In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, p. 155-166Article in journal (Refereed)
    Abstract [en]

    It is well known that the maximal order of Lie-Backlund symmetries for any nth-order ordinary differential equation is equal to n-1, and that the whole set of such symmetries forms an infinite-dimensional Lie algebra. Symmetries of the order pless than or equal ton - 2 span a linear subspace (but not a subalgebra) in this algebra. We call them symmetries of submaximal order. The purpose of the article is to prove that for n less than or equal to 4 this subspace is finite-dimensional and it's dimension cannot be greater than 35 for n=4, 10 for n=3 and 3 for n=2. In the case n=3 this statement follows immediately from Lie's result on contact symmetries of third-order ordinary differential equations. The maximal values of dimensions are reached, e.g., on the simplest equations y((n))=0.

  • 141. Ibragimov, Nail H.
    et al.
    Säfström, Niklas
    The equivalence group and invariant solutions of a tumour growth model2004In: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 9, no 1, p. 61-68Article in journal (Refereed)
    Abstract [en]

    Most of mathematical models describing spread of malignant tumours are formulated as systems of nonlinear partial differential equations containing, in general, several unknown functions of dependent variables. Determination of these unknown functions (called in group analysis arbitrary elements) is a complicated problem that challenges researchers. Our aim is to calculate the generators of the equivalence group for one of the known models and, using the equivalence generators, specify arbitrary elements, find additional symmetries and calculate group invariant solutions.

  • 142.
    Ibragimov, Nail H.
    et al.
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Science.
    Thidé, Bo
    Then, H.
    Sjöholm, J.
    Palmer, K.
    Bergman, J.
    Carozzi, T.D.
    Istomin, YaN
    Khamitova, Raisa
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Science.
    Utilization of photon orbital angular momentum in the low-frequency radio domain2007In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 99, no 8Article in journal (Refereed)
    Abstract [en]

    We show numerically that vector antenna arrays can generate radio beams that exhibit spin and orbital angular momentum characteristics similar to those of helical Laguerre-Gauss laser beams in paraxial optics. For low frequencies (1 GHz), digital techniques can be used to coherently measure the instantaneous, local field vectors and to manipulate them in software. This enables new types of experiments that go beyond what is possible in optics. It allows information-rich radio astronomy and paves the way for novel wireless communication concepts.

  • 143. Ibragimov, Nail H.
    et al.
    Torrisi, M.
    Tracinà, R.
    Quasi self-adjoint nonlinear wave equations2010In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 43, no 44Article in journal (Refereed)
    Abstract [en]

    In this paper we generalize the classification of self-adjoint second-order linear partial differential equation to a family of nonlinear wave equations with two independent variables. We find a class of quasi self-adjoint nonlinear equations which includes the self-adjoint linear equations as a particular case. The property of a differential equation to be quasi self-adjoint is important, e.g. for constructing conservation laws associated with symmetries of the differential equation.

  • 144. Ibragimov, Nail H.
    et al.
    Torrisi, M.
    Tracinà, R.
    Self-adjointness and conservation laws of a generalized Burgers equation2011In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 44, no 14Article in journal (Refereed)
    Abstract [en]

    A (2 + 1)-dimensional generalized Burgers equation is considered. Having written this equation as a system of two dependent variables, we show that it is quasi self-adjoint and find a nontrivial additional conservation law.

  • 145. Ibragimov, Nail H.
    et al.
    Torrisi, M.
    Valenti, A.
    Differential invariants of nonlinear equations v_{tt} =f(x,v_x) v_{xx} + g(x,v_x)2004In: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 9, no 1, p. 69-80Article in journal (Refereed)
    Abstract [en]

    We apply the infinitesimal technique for calculating invariants for the family of nonlinear equations formulated in the title. We show that the infinite-dimensional equivalence Lie algebra has three functionally independent differential invariants of the second order. Knowledge of invariants of families of equations is essential for identifying distinctly different equations and therefore for the problem of group classification.

  • 146. Ibragimov, Nail H.
    et al.
    Wessels, Ewald JH.
    Ellis, GFR
    Group classification of the characteristic initial value equations for a radiating axisymmetric, non-rotating, vacuum spacetime2007In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 24, no 23, p. 6007-6017Article in journal (Refereed)
    Abstract [en]

    We carry out a Lie group analysis of the Sachs equations for a time-dependent axisymmetric non-rotating spacetime in which the Ricci tensor vanishes. These equations, which are the first two members of the set of Newman-Penrose equations, define the characteristic initial-value problem for the spacetime. We find that the equations admit a five-dimensional equivalence Lie algebra. The initial value function that allows the equations to admit a non-trivial Lie symmetry separates into three disjoint equivalence classes.

  • 147. Ibragimov, Nail H.
    et al.
    Yacar, E.
    Non-local conservation laws in fluid dynamics2007In: Archives of ALGA, ISSN 1652-4934, Vol. 4, p. 136-143Article in journal (Refereed)
    Abstract [en]

    A general theorem on symmetries and conservation laws is applied to gasdynamic equations considered together with the adjoint equations. In particular, new non-local conservation laws for the polytropic gas are obtained.

  • 148. Ibragimov, Nail H.
    et al.
    Yasar, Emrullah
    Application of the new conservation theorem to gasdynamic equations2007Conference paper (Refereed)
    Abstract [en]

    We apply the general theorem on symmetries and conservation laws proved recently by N.H. Ibragimov to gasdynamic equations considered together with the adjoint equations. In particular, we derive conservation laws for the polytropic gas.

  • 149. Ibragimov, Nail H.
    et al.
    Ünal, Gazanfer
    Jogréus, Claes
    Approximate symmetries and conservation laws for It(o)over-cap and Stratonovich dynamical systems2004In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 297, no 1, p. 152-168Article in journal (Refereed)
    Abstract [en]

    Determining equations for approximate symmetries of Ito and Stratonovich dynamical systems is obtained. It is shown that approximate conservation laws can be found from the approximate symmetries of stochastic dynamical systems which do not arise from a Hamiltonian.

  • 150. Ibragimov, Nail H.
    et al.
    Ünal, Gazanfer
    Jogréus, Claes
    Group analysis of stochastic differential systems: Approximate symmetries and conservation laws.2004In: Archives of ALGA, ISSN 1652-4934, Vol. 1, p. 95-126Article in journal (Refereed)
    Abstract [en]

    Approximate symmetries and conservation laws for deterministic and stochastic differential equations with a small parameter are discussed in detail. Determining equations for infinitesimal approximate symmetries of Ito and Stratanovich dynamical systems are derived. It is shown how to derive conserved quantities for stochastic dynamical systems using their approximate symmetries without recourse to Noether's theorem

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