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  • 51. Ibragimov, Nail
    et al.
    Ibragimov, Ranis
    Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences2012Ingår i: Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, E-ISSN 1760-6101, Vol. 7, nr 2, s. 52-65Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Today engineering and science researchers routinely confront problems in mathematical modeling involving solutions techniques for differential equations. Sometimes these solutions can be obtained analytically by numerous traditional ad hoc methods appropriate for integrating particular types of equations. More often, however, the solutions cannot be obtained by these methods, in spite of the fact that, e.g. over 400 types of integrable second-order ordinary differential equations were summarized in voluminous catalogues. On the other hand, many mathematical models formulated in terms of nonlinear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating nonlinear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is, from the one hand, to provide the wide audience of researchers with the comprehensive introduction to Lie's group analysis and, from the other hand, is to illustrate the advantages of application of Lie group analysis to group theoretical modeling of internal gravity waves in stratified fluids.

  • 52.
    Ibragimov, Nail
    et al.
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Ibragimov, Ranis N.
    Univ Wisconsin Parkside, USA.
    Kovalev, Vladimir F.
    Russian Acad Sci, RUS.
    INVARIANT SOLUTIONS AND SHOCK ATMOSPHERIC WAVES IN A THIN CIRCULAR LAYER2018Ingår i: Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, E-ISSN 1760-6101, Vol. 13, nr 2, artikel-id UNSP 19Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The objective of this paper is to investigate the nonlinear mathematical model describing equatorial waves from Lie group analysis point of view in order to understand the nature of shallow water model theory, which is associated to planetary equatorial waves. Such waves correspond to the Cauchy-Poisson free boundary problem on the nonstationary motion of a perfect incompressible fluid circulating around a solid circle of a large radius.

  • 53.
    Ibragimov, Nail
    et al.
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Ibragimov, R.N.
    Bifurcation of Nonlinear Conservation Laws from the Classical Energy Conservation Law for Internal Gravity Waves in Cylindrical Wave Field2013Ingår i: Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, E-ISSN 1760-6101, Vol. 8, nr 5, s. 119-130Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    New conservation laws bifurcating from the classical form of conservation laws are constructed to the nonlinear Boussinesq model describing internal Kelvin waves propagating in a cylindrical wave field of an uniformly stratified water affected by the earth's rotation. The obtained conservation laws are different from the well known energy conservation law for internal waves and they are associated with symmetries of the Boussinesq model. Particularly, it is shown that application of Lie group analysis provide three infinite sets of nontrivial integral conservation laws depending on two arbitrary functions, namely a(t, theta),b(t, r) and an arbitrary function c(t, theta, r) which is given implicitly as a nontrivial solution of a partial differential equation involving a(t, theta) and b(t,r).

  • 54.
    Ibragimov, Nail
    et al.
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Karimova, E. N.
    Ufa State Aviat Tech Univ, Lab Grp Anal Math Models Nat & Engn Sci, Ufa 450000, Russia..
    Galiakberova, L. R.
    Ufa State Aviat Tech Univ, Lab Grp Anal Math Models Nat & Engn Sci, Ufa 450000, Russia..
    Chaplygin gas motions associated with nonlocal conservation laws2017Ingår i: JOURNAL OF COUPLED SYSTEMS AND MULTISCALE DYNAMICS, ISSN 2330-152X, Vol. 5, nr 2-4, s. 63-68Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The recent method of conservation laws for constructing exact solutions of partial differential equations is applied to the nonlocal conservation laws of the Chaplygin gas. The nonlocal conservation laws provide twenty different types of exact solutions. They are listed in three tables. Seven types of these solutions describe isentropic flows satisfying Chaplygin's relation between the pressure and density. All solutions are written in the explicit form and contain either arbitrary functions or arbitrary constants.

  • 55.
    Ibragimov, Nail
    et al.
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Khamitova, Raisa
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Avdonina, E. D.
    Galiakberova, L. R.
    Conservation laws and solutions of a quantum drift-diffusion model for semiconductors2015Ingår i: International Journal of Non-Linear Mechanics, ISSN 0020-7462, E-ISSN 1878-5638, Vol. 77, s. 69-73Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A non-linear system of partial differential equations describing a quantum drift-diffusion model for semiconductor devices is investigated by methods of group analysis. An infinite number of conservation laws associated with symmetries of the model are found. These conservation laws are used for representing the system of equations under consideration in the conservation form. Exact solutions provided by the method of conservation laws are discussed. These solutions are different from invariant solutions. (C) 2015 Published by Elsevier Ltd.

  • 56.
    Ibragimov, Nail
    et al.
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Khamitova, Raisa
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Avdonina, E. D.
    Galiakberova, L.R.
    Group analysis of the drift–diffusion model for quantum semiconductors2015Ingår i: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 20, nr 1, s. 74-78Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In the present paper a quantum drift–diffusion model describing semi-conductor devices is considered. New conservation laws for the model are computed and used to construct exact solutions.

  • 57. Ibragimov, Nail
    et al.
    Sophocleous, C.Damianou, P.A.
    Proceedings of the 10th International conference on “Modern group Analysis”. Larnaca, Cyprus, 20042004Övrigt (Övrigt vetenskapligt)
    Abstract [en]

    This book contains 33 papers presented at the 10th International conference "Modern Group Analysis" held in Larnaca, Cyprus, during 24-31 October 2004. All papers have been reviewed by two independent referees.

  • 58.
    Ibragimov, R. N.
    et al.
    GE Global Res, Niskayuna, NY 12309 USA..
    Ibragimov, Nail
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Galiakberova, L. R.
    Ufa State Aviat Tech Univ, Lab Grp Anal Math Models Nat & Engn Sci, Ufa 450000, Russia..
    Conservation laws and invariant solutions of the non-linear governing equations associated with a thermodynamic model of a rotating detonation engines with Korobeinikov's chemical source term2016Ingår i: International Journal of Non-Linear Mechanics, ISSN 0020-7462, E-ISSN 1878-5638, Vol. 78, s. 29-34Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The non-linear governing gas dynamics equations that are used as a descriptor of a rotating detonation engine are investigated from the group theoretical standpoint. The equations incorporate approximation of Korobeinikov's chemical reaction model that are used to describe the two-dimensional detonation field on a surface of a two-dimensional cylindrical chamber without thickness. The transformations that leave the equations invariant are found. On the basis of these transformations, the conservation equations were constructed and the invariant solutions were obtained for specific form of the equation of state, for which the equations are non-linearly self-adjoint. The invariant solutions are given in terms of the functions that satisfy non-linear ordinary differential equations. The above reduction simplifies the analysis of the original non-linear system of partial differential equations on a surface of rotating cylinder. (C) 2015 Elsevier Ltd. All rights reserved.

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