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  • 1. Avdonina, Elena D.
    et al.
    Ibragimov, Nail H.
    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.
    Exact solutions of gasdynamic equations obtained by the method of conservation laws2013In: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 18Article in journal (Refereed)
    Abstract [en]

    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.

  • 2.
    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.

  • 3.
    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.

  • 4.
    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.

  • 5.
    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)
  • 6.
    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.

  • 7.
    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.

  • 8.
    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.

  • 9.
    Ibragimov, Nail
    et al.
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Khamitova, Raisa
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Avdonina, E. D.
    Galiakberova, L. R.
    Conservation laws and solutions of a quantum drift-diffusion model for semiconductors2015In: International Journal of Non-Linear Mechanics, ISSN 0020-7462, E-ISSN 1878-5638, Vol. 77, p. 69-73Article in journal (Refereed)
    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.

  • 10.
    Ibragimov, Nail
    et al.
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Khamitova, Raisa
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Avdonina, E. D.
    Galiakberova, L.R.
    Group analysis of the drift–diffusion model for quantum semiconductors2015In: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 20, no 1, p. 74-78Article in journal (Refereed)
    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.

  • 11.
    Khamitova, Raisa
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Science.
    Self-Adjointness and Quasi-Self-Adjointness of the Magma Equation2008Conference paper (Refereed)
    Abstract [en]

    A recent theorem on nonlocal conservation laws is applied to a magma equation modelling a melt migration through the Earth´s mantle. It is shown that the equation in question is quasi-self-adjoint. The self-adjoint equations are singled out. Nonlocal and local conservation densities are obtained using the symmetries of the magma equation.

  • 12.
    Khamitova, Raisa
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Science.
    Symmetries and conservation laws2009Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Conservation laws play an important role in science. The aim of this thesis is to provide an overview and develop new methods for constructing conservation laws using Lie group theory. The derivation of conservation laws for invariant variational problems is based on Noether’s theorem. It is shown that the use of Lie-Bäcklund transformation groups allows one to reduce the number of basic conserved quantities for differential equations obtained by Noether’s theorem and construct a basis of conservation laws. Several examples on constructing a basis for some well-known equations are provided. Moreover, this approach allows one to obtain new conservation laws even for equations without Lagrangians. A formal Lagrangian can be introduced and used for computing nonlocal conservation laws. For self-adjoint or quasi-self-adjoint equations nonlocal conservation laws can be transformed into local conservation laws. One of the fields of applications of this approach is electromagnetic theory, namely, nonlocal conservation laws are obtained for the generalized Maxwell-Dirac equations. The theory is also applied to the nonlinear magma equation and its nonlocal conservation laws are computed.

  • 13. Khamitova, Raisa
    Symmetries and Nonlocal Conservation Laws of the General Magma Equation2009In: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 14, no 11, p. 3754-3769Article in journal (Refereed)
    Abstract [en]

    In this paper the general magma equation modelling a melt flow in the Earth's mantle is discussed. Applying the new theorem on nonlocal conservation laws [Ibragimov NH. A new conservation theorem. J Math Anal Appl 2007;333(1):311-28] and using the symmetries of the model equation nonlocal conservation laws are computed. In accordance with Ibragimov [Ibragimov NH. Quasi-self-adjoint differential equations. Preprint in Archives of ALGA, vol. 4, BTH, Karlskrona, Sweden: Alga Publications; 2007. p. 55-60, ISSN: 1652-4934] it is shown that the general magma equation is quasi-self-adjoint for arbitrary m and n and self-adjoint for n = -m. These important properties are used for deriving local conservation laws. © 2008 Elsevier B.V. All rights reserved.

1 - 13 of 13
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