SwePub - search: WFRF:(Ibragimov Nail H.)

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1.	Ibragimov, Nail H., et al. (author) Applications of Lie group analysis in Geophysical fluid dynamics 2011 Book (other academic/artistic)abstract 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.
2.	Ibragimov, Nail H., et al. (author) Approximate symmetries and solutions of the Kompaneets equation 2014 In: Communications in nonlinear science & numerical simulation. - : MAIK NAUKA/INTERPERIODICA/SPRINGER. - 1007-5704 .- 1878-7274. ; 55:2, s. 220-224 Journal article (peer-reviewed)abstract Different approximations of the Kompaneets equation are studied using approximate symmetries, which allows consideration of the contributions of all terms of this equation previously neglected in the analysis of the limiting cases.
3.	Ibragimov, Nail H., et al. (author) Integration by quadratures of the nonlinear Euler equations modeling atmospheric flows in a thin rotating spherical shell 2011 In: Physics Letters A. - : Elsevier. - 0375-9601 .- 1873-2429. ; 375:44, s. 3858-3865 Journal article (peer-reviewed)abstract 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.
4.	Ibragimov, Nail H., et al. (author) Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics 2010 In: Communications in nonlinear science & numerical simulation. - : Elsevier. - 1007-5704 .- 1878-7274. ; 15:8, s. 1989-2002 Journal article (peer-reviewed)abstract 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.
5.	Ibragimov, Nail H., et al. (author) Invariant solutions as internal singularities of nonlinear differential equations and their use for qualitative analysis of implicit and numerical solutions 2009 In: Communications in nonlinear science & numerical simulation. - : Elsevier Science. - 1007-5704 .- 1878-7274. ; 14:9-10, s. 3537-3547 Journal article (peer-reviewed)abstract 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.
6.	Ibragimov, Nail H., et al. (author) Nonlinear Whirlpools Versus Harmonic Waves in a Rotating Column of Stratified Fluid 2013 In: Mathematical Modelling of Natural Phenomena. - : EDP SCIENCES. - 0973-5348 .- 1760-6101. ; 8:1, s. 122-131 Journal article (peer-reviewed)abstract 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.
7.	Ibragimov, Nail H., et al. (author) Rotationally symmetric internal gravity waves 2012 In: International Journal of Non-Linear Mechanics. - : Elsevier. - 0020-7462 .- 1878-5638. ; 47:1, s. 46-52 Journal article (peer-reviewed)abstract 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.
8.	Ibragimov, Nail H., et al. (author) Symmetries and Conservation Laws of a Spectral Nonlinear Model for Atmospheric Baroclinic Jets 2014 In: Mathematical Modelling of Natural Phenomena. - : EDP SCIENCES S A.. - 0973-5348 .- 1760-6101. ; 9:5, s. 111-118 Journal article (peer-reviewed)abstract In this paper, we shall obtain the symmetries of the mathematical model describing spontaneous relaxation of eastward jets into a meandering state and use these symmetries for constructing the conservation laws. The basic eastward jet is a spectral parameter of the model, which is in geostrophic equilibrium with the basic density structure and which guarantees the existence of nontrivial conservation laws.
9.	Ibragimov, Nail H., et al. (author) 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 channel 2009 In: Communications in nonlinear science & numerical simulation. - : Elsevier Science. - 1007-5704 .- 1878-7274. ; 14:11, s. 3811-3820 Journal article (peer-reviewed)abstract 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.
10.	Ibragimov, Nail H., et al. (author) Utilization of photon orbital angular momentum in the low-frequency radio domain 2007 In: Physical Review Letters. - : American Physical Society. - 0031-9007 .- 1079-7114. ; 99:8 Journal article (peer-reviewed)abstract 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.
11.	Ibragimov, Ranis, et al. (author) Effects of rotation on self-resonant internal gravity waves in the ocean 2010 In: Ocean Modelling. - : Elsevier. - 1463-5003 .- 1463-5011. ; 31:3-4 Journal article (peer-reviewed)abstract The Resonant Triad Model (RTM) developed in (Ibragimov, 2007), is used to study the Thorpe’s problem (Thorpe, 1997) on the existence of self-resonant internal waves, i.e., the waves for which a resonant interaction occurs at second order between the incident and reflected internal waves off slopes. The RTM represents the extension of the McComas and Bretherton’s three wave hydrostatic model (McComas and Bretherton, 1977) which ignores the effects of the earth’s rotation to the case of the non-hydrostatic analytical model involving arbitrarily large number of rotating internal waves with frequencies spanning the range of possible frequencies, i.e., between the maximum of the buoyancy frequency (vertical motion) and a minimum of the inertial frequency (horizontal motion). The present analysis is based on classification of resonant interactions into the sum, middle and difference interaction classes. It is shown in this paper that there exists a certain value of latitude, which is classified as the singular latitude, at which the coalescence of the middle and difference interaction classes occurs. Such coalescence, which apparently had passed unnoticed before, can be used to study the Thorpe’s problem on the existence of selfresonant waves. In particular, it is shown that the value of the bottom slope at which the second-order frequency and wave number components of the incident and reflected waves satisfy the internal wave dispersion relation can be approximated by two latitude-dependent parameters in the limiting case when latitude approaches its singular value. Since the existence of a such singular latitude is generic for resonant triad interactions, a question on application of the RTM to the modeling of enhanced mixing in the vicinity of ridges in the ocean arises.
12.	Archives of ALGA : volume 9 2012 Editorial collection (other academic/artistic)abstract Volume 9 contains articles of different authors.
13.	Archives of ALGA, volume 1 2004 Editorial collection (other academic/artistic)abstract Volume 1 contains 3 articles: Nail H. Ibragimov, Equivalence groups and invariants of linear and non-linear equations; Nail H. Ibragimov and Sergey V. Meleshko, Linearization of third-order ordinary differential equations; Nail H. Ibragimov, Gazanfer Ünal and Claes Jogreús, Group analysis of stochastic differential systems: Approximate symmetries and conservation laws.
14.	Archives of ALGA. Volume 2 2005 Editorial collection (other academic/artistic)abstract Volume 2 contains 3 articles: Ilir Berisha, Translation of Bäcklunds paper ”Surfaces of constant negative curvature”; Johan Erlandsson, "Survey of mathematical models in biology from point of view of Lie group analysis"; Niklas Säfström, "Group analysis of a tumour growth model"
15.	Archives of ALGA. Volume 4 2007 Editorial collection (other academic/artistic)abstract Volume 4 contains articles: Nail H. Ibragimov, Ewald J.H. Wessels and George F.R. Ellis, Group classication of the Sachs equations for a radiating axisymmetric, non-rotating, vacuum space-time; Nail H. Ibragimov A discussion of conservation laws for over-determined systems with application to the Maxwell equations in vacuum; Nail H. Ibragimov, Quasi-self-adjoint differential equations; Nail H. Ibragimov and Salavat V. Khabirov, Existence of integrating factors for higher-order ordinary differential equations; Nail H. Ibragimov, Integration of second-order linear equations via linearization of Riccati's equations; Nail H. Ibragimov and Sergey V. Meleshko Linearization of second-order ordinary differential equations by changing the order; Nail H. Ibragimov and Sergey V. Meleshko Second-order ordinary differential equations equivalent to y''= H(y); Nail H. Ibragimov, Sergey V. Meleshko and Supaporn Suksern, Linearization of fourth-order ordinary differential equations by point transformations; Nail H. Ibragimov and Emrullah Yasar, Non-local conservation laws in fluid dynamics
16.	Archives of ALGA, volume 7/8 2010 Editorial collection (other academic/artistic)abstract Volume 7/8 contains 3 articles by N.H. Ibragimov, an article by N.H. Ibragimov, E. D. Avdonina and an article by N. H. Ibragimov , R. Khamitova.
17.	Avdonina, Elena D., et al. (author) Exact solutions of gasdynamic equations obtained by the method of conservation laws 2013 In: Communications in nonlinear science & numerical simulation. - : Elsevier B.V.. - 1007-5704 .- 1878-7274. ; 18 Journal article (peer-reviewed)abstract In the present paper, the recent method of conservation laws for constructing exact solutions for systems of nonlinear partial differential equations is applied to the gasdynamic equations describing one-dimensional and three-dimensional polytropic flows. In the one-dimensional case singular solutions are constructed in closed forms. In the threedimensional case several conservation laws are used simultaneously. It is shown that the method of conservation laws leads to particular solutions different from group invariant solutions.
18.	Avdonina, Elena D., et al. (author) Heat conduction in anisotropic media : Nonlinear self-adjointness and conservation laws 2012 In: Discontinuity, Nonlinearity and Complexity. - : L & H Scientific Publishing. - 2164-6376. ; 1:3, s. 237-251 Journal article (peer-reviewed)abstract Nonlinear self-adjointness of the anisotropic nonlinear heat equation is investigated. Mathematical models of heat conduction in anisotropic media with a source are considered and a class of self-adjoint models is identified. Conservation laws corresponding to the symmetries of the equations in question are computed.
19.	Boszhkov, Yuri, et al. (author) Group analysis of the Novikov equation 2014 In: Computational and Applied Mathematics. - : Springer. - 2238-3603 .- 1807-0302 .- 0101-8205. ; 33:1, s. 193-202 Journal article (peer-reviewed)abstract We find the Lie point symmetries of the Novikov equation and demonstrate that it is strictly self-adjoint. Using the self-adjointness and the recent technique for constructing conserved vectors associated with symmetries of differential equations, we find the conservation law corresponding to the dilation symmetry and show that other symmetries do not provide nontrivial conservation laws. Then we investigate the invariant solutions.
20.	Bozhkov, Y., et al. (author) Conservation laws for a coupled variable-coefficient modified Korteweg–de Vries system in a two-layer fluid model 2013 In: Communications in nonlinear science & numerical simulation. - : Elsevier. - 1007-5704 .- 1878-7274. ; 18 Journal article (peer-reviewed)abstract We find the Lie point symmetries of a coupled variable-coefficient modified Korteweg–de Vries system in a two-layer fluid model. Then we establish its quasi self-adjointness and corresponding conservation laws.
21.	Bruzon, MS, et al. (author) Self-adjoint sub-classes of generalized thin film equations 2009 In: Journal of Mathematical Analysis and Applications. - : Academic Press. - 0022-247X .- 1096-0813. ; 357:1, s. 307-313 Journal article (peer-reviewed)abstract In this work we consider a class of fourth-order nonlinear partial differential equation containing several tin-specified coefficient functions of the dependent variable which encapsulates various mathematical models used, e.g. for describing the dynamics of thin liquid films. We determine the subclasses of these equations which are self-adjoint. By using a general theorem on conservation laws proved by one of the authors (NHI) we find conservation laws for some of these partial differential equations without Classical Lagrangians.
22.	Gandarias, ML, et al. (author) Equivalence group of a fourth-order evolution equation unifying various non-linear models 2008 In: Communications in nonlinear science & numerical simulation. - AMSTERDAM : ELSEVIER SCIENCE BV. - 1007-5704 .- 1878-7274. ; 13:2, s. 259-268 Journal article (peer-reviewed)abstract A fourth-order non-linear evolutionary partial differential equation containing several arbitrary functions of the dependent variable is considered. This equation arises as a generalization of various non-linear models describing a non-linear heat diffusion, the dynamics of thin liquid films, etc. Equivalence transformations give more flexibility to the unified model. We determine the generators of the equivalence group and use them for specifying certain types of arbitrary functions when the model equation has additional symmetries, and hence admits non-trivial group invariant solutions. (c) 2006 Elsevier B.V. All rights reserved.
23.	Gazizov, Rafail K., et al. (author) Integration of ODE with a small parameter via approximate symmetries : reduction of approximate symmetry algebra to a canonical form 2007 Conference paper (peer-reviewed)abstract The simplest method of integration of second-order differential equations using the Lie's canonical forms of two-dimensional algebras is well-known. We propose a generalization of this method on a case of integration of second-order differential equation with a small parameter having two approximate symmetries. The solution of such problem is reduced to the followings: 1) to classify approximate Lie algebras with two essential operators. As a result, seven different types of such Lie algebras have been obtained; 2) to construct canonical form of basic operators of non-similar algebras of every types for their realization in R2; 3) to set up general forms of invariant equations and formulas of their approximate solutions. The similar problems are solved for systems of two ordinary differential equations with two approximate symmetries. On this way we have constructed representation of non-similar approximated Lie algebras in R3.
24.	Gazizov, Rafail K., et al. (author) Integration of ordinary differential equation with a small parameter via approximate symmetries : Reduction of approximate symmetry algebra to a canonical form 2010 In: Lobachevskii Journal of Mathematics. - : Pleiades Publishing. - 1995-0802. ; 31:2, s. 141-151 Journal article (peer-reviewed)abstract Method of integration of second-order ordinary differential equations with twodimensional Lie symmetry algebras by reducing basic symmetries to canonical forms is extended to second-order equations with a small parameter for their approximate integration using two essential approximate symmetries. Canonical forms of basic operators of corresponding approximate Lie algebras Lr, r = 2, 3, 4, as well as general forms of invariant differential equations and their solutions are presented. The similar problems are also solved for systems of two first-order ordinary differential equations with two approximate symmetries.
25.	Grigoriev, Yurii, et al. (author) Delay differential equations 2010 In: Lecture Notes in Physics. - Dordrecht : Springer. - 0075-8450. ; 806, s. 251-292 Journal article (peer-reviewed)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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