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  • 201. Ibragimov, Nail H.
    et al.
    Meleshko, Sergey
    Linearization of third-order ordinary differential equations by point and contact transformations.2005Ingår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 308, nr 1, s. 266-289Artikel i tidskrift (Refereegranskat)
    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.

  • 202.
    Ibragimov, Nail H.
    et al.
    Blekinge Tekniska Högskola, Institutionen för hälso- och naturvetenskap.
    Meleshko, Sergey
    Linearization of third-order ordinary differential equations by point transformations.2004Ingår i: Archives of ALGA, ISSN 1652-4934, Vol. 1, s. 95-126Artikel i tidskrift (Refereegranskat)
    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.

  • 203. Ibragimov, Nail H.
    et al.
    Meleshko, Sergey
    Second-order ordinary differential equations equivalent to y''=H(y).2007Ingår i: Archives of ALGA, ISSN 1652-4934, Vol. 4, s. 101-112Artikel i tidskrift (Refereegranskat)
    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.

  • 204. Ibragimov, Nail H.
    et al.
    Meleshko, Sergey
    Rudenko, Oleg
    Group analysis of evolutionary integro-differential equations describing nonlinear waves: General model2011Ingår i: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 44, nr 31Artikel i tidskrift (Refereegranskat)
    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.

  • 205. Ibragimov, Nail H.
    et al.
    Meleshko, Sergey
    Suksern, Supaporn
    Linearization of fourth order ordinary differential equations by point transformations2008Ingår i: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 23, nr 41, s. 206-235Artikel i tidskrift (Refereegranskat)
    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.

  • 206. Ibragimov, Nail H.
    et al.
    Meleshko, Sergey
    Suksern, Supaporn
    Linearization of fourth-order ordinary differential equations by point transformations2007Ingår i: Archives of ALGA, ISSN 1652-4934, Vol. 4, s. 113-134Artikel i tidskrift (Refereegranskat)
    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.

  • 207. Ibragimov, Nail H.
    et al.
    Meleshko, Sergey
    Thailert, E
    Invariants of linear parabolic differential equations2008Ingår i: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, s. 277-284Artikel i tidskrift (Refereegranskat)
    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.

  • 208. Ibragimov, Nail H.
    et al.
    Rudenko, Oleg
    Principle of an a priori use of symmetries in the theory of nonlinear waves.2004Ingår i: Acoustical Physics, ISSN 1063-7710, E-ISSN 1562-6865, Vol. 50, nr 4, s. 406-419Artikel i tidskrift (Refereegranskat)
    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.

  • 209. Ibragimov, Nail H.
    et al.
    Sophocleus, C.
    Differential invariants of the one-dimensional quasi-linear second-order evolution equation2007Ingår i: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 12, nr 7, s. 1133-1145Artikel i tidskrift (Refereegranskat)
    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.

  • 210. Ibragimov, Nail H.
    et al.
    Svirshchevskii, SR
    Lie-Backlund symmetries of submaximal order of ordinary differential equations2002Ingår i: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, s. 155-166Artikel i tidskrift (Refereegranskat)
    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.

  • 211. Ibragimov, Nail H.
    et al.
    Säfström, Niklas
    The equivalence group and invariant solutions of a tumour growth model2004Ingår i: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 9, nr 1, s. 61-68Artikel i tidskrift (Refereegranskat)
    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.

  • 212.
    Ibragimov, Nail H.
    et al.
    Blekinge Tekniska Högskola, Sektionen för teknik, Avdelningen för matematik och naturvetenskap.
    Thidé, Bo
    Then, H.
    Sjöholm, J.
    Palmer, K.
    Bergman, J.
    Carozzi, T.D.
    Istomin, YaN
    Khamitova, Raisa
    Blekinge Tekniska Högskola, Sektionen för teknik, Avdelningen för matematik och naturvetenskap.
    Utilization of photon orbital angular momentum in the low-frequency radio domain2007Ingår i: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 99, nr 8Artikel i tidskrift (Refereegranskat)
    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.

  • 213. Ibragimov, Nail H.
    et al.
    Torrisi, M.
    Tracinà, R.
    Quasi self-adjoint nonlinear wave equations2010Ingår i: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 43, nr 44Artikel i tidskrift (Refereegranskat)
    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.

  • 214. Ibragimov, Nail H.
    et al.
    Torrisi, M.
    Tracinà, R.
    Self-adjointness and conservation laws of a generalized Burgers equation2011Ingår i: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 44, nr 14Artikel i tidskrift (Refereegranskat)
    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.

  • 215. 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)2004Ingår i: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 9, nr 1, s. 69-80Artikel i tidskrift (Refereegranskat)
    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.

  • 216. 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 spacetime2007Ingår i: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 24, nr 23, s. 6007-6017Artikel i tidskrift (Refereegranskat)
    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.

  • 217. Ibragimov, Nail H.
    et al.
    Yacar, E.
    Non-local conservation laws in fluid dynamics2007Ingår i: Archives of ALGA, ISSN 1652-4934, Vol. 4, s. 136-143Artikel i tidskrift (Refereegranskat)
    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.

  • 218. Ibragimov, Nail H.
    et al.
    Yasar, Emrullah
    Application of the new conservation theorem to gasdynamic equations2007Konferensbidrag (Refereegranskat)
    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.

  • 219. Ibragimov, Nail H.
    et al.
    Ünal, Gazanfer
    Jogréus, Claes
    Approximate symmetries and conservation laws for It(o)over-cap and Stratonovich dynamical systems2004Ingår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 297, nr 1, s. 152-168Artikel i tidskrift (Refereegranskat)
    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.

  • 220. Ibragimov, Nail H.
    et al.
    Ünal, Gazanfer
    Jogréus, Claes
    Group analysis of stochastic differential systems: Approximate symmetries and conservation laws.2004Ingår i: Archives of ALGA, ISSN 1652-4934, Vol. 1, s. 95-126Artikel i tidskrift (Refereegranskat)
    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

  • 221. 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.

  • 222.
    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.

  • 223.
    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).

  • 224.
    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.

  • 225.
    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.

  • 226.
    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.

  • 227. 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.

  • 228.
    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.

  • 229. Ibragimov, Ranis
    et al.
    Ibragimov, Nail H.
    Effects of rotation on self-resonant internal gravity waves in the ocean2010Ingår i: Ocean Modelling, ISSN 1463-5003, E-ISSN 1463-5011, Vol. 31, nr 3-4Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.

  • 230.
    Ijaz, Muhammad
    Blekinge Tekniska Högskola, Sektionen för teknik, Avdelningen för telekommunikationssystem.
    TRASMISSION CONTROL PROTOCOL (TCP) PERFORMANCE EVALUATION IN MANET2009Självständigt arbete på avancerad nivå (masterexamen)Studentuppsats (Examensarbete)
    Abstract [en]

    Mobile Ad hoc network routing protocols have been divided in several different categories such as Reactive and Proactive Routing Protocol. The performances of these categories are evaluated in different scenario with TCP variants. We present a comprehensive TCP performance evaluation study to understand the nature of the TCP performance in different scenarios with variable amount of payload and number of nodes. The traffic consists of three different packet sizes i.e. 512, 1000, 1500 bytes each. Three different routing protocols (AODV, DSR and TORA) are to be evaluated with three different TCP variants (Tahoe, Reno and New Reno) in three different scenarios having 3, 5 and 8 nodes. The performances parameters on the basis of which routing protocols are to be graded are mainly throughput, congestion window and delay. Conclusions are drawn based on the simulation results and the comparisons between them have been elaborated.

  • 231. Ilie, Dragos
    Gnutella Network Traffic: Measurements and Characteristics2006Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    Wide availability of computing resources at the edge of the network has lead to the appearance of new services based on peer-to-peer architectures. In a peer-to-peer network nodes have the capability to act both as client and server. They self-organize and cooperate with each other to perform more efficiently operations related to peer discovery, content search and content distribution. The main goal of this thesis is to obtain a better understanding of the network traffic generated by Gnutella peers. Gnutella is a well-known, heavily decentralized file-sharing peer-to-peer network. It is based on open protocol specifications for peer signaling, which enable detailed measurements and analysis down to individual messages. File transfers are performed using HTTP. An 11-days long Gnutella link-layer packet trace collected at BTH is systematically decoded and analyzed. Analysis results include various traffic characteristics and statistical models. The emphasis for the characteristics has been on accuracy and detail, while for the traffic models the emphasis has been on analytical tractability and ease of simulation. To the author's best knowledge this is the first work on Gnutella that presents statistics down to message level. The results show that incoming requests to open a session follow a Poisson distribution. Incoming messages of mixed types can be described by a compound Poisson distribution. Mixture distribution models for message transfer rates include a heavy-tailed component.

  • 232. Ilie, Dragos
    Optimization Algorithms with Applications to Unicast QoS Routing in Overlay Networks2007Rapport (Övrigt vetenskapligt)
    Abstract [en]

    The research report is focused on optimization algorithms with application to quality of service (QoS) routing. A brief theoretical background is provided for mathematical tools in relation to optimization theory. The rest of the report provides a survey of different types of optimization algorithms: several numerical methods, a heuristics and a metaheuristic. In particular, we discuss basic descent methods, gradient-based methods, particle swarm optimization (PSO) and a constrained-path selection algorithm called Self-Adaptive Multiple Constraints Routing Algorithm (SAMCRA).

  • 233.
    Irshad, Saba
    et al.
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap.
    Nepal, Purna Chandra
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap.
    Rise Over Thermal Estimation Algorithm Optimization and Implementation2013Självständigt arbete på avancerad nivå (masterexamen)Studentuppsats (Examensarbete)
    Abstract [en]

    The uplink load for the scheduling of Enhanced-Uplink (E-UL) channels determine the achievable data rate for Wideband Code Division Multiple Access (WCDMA) systems, therefore its accurate measurement carries a prime significance. The uplink load also known as Rise-over-Thermal (RoT), which is the quotient of the Received Total Wideband Power (RTWP) and the Thermal Noise Power floor. It is a major parameter which is calculated at each Transmission Time Interval (TTI) for maintaining cell coverage and stability. The RoT algorithm for evaluation of uplink load is considered as a complex and resource demanding among several Radio Resource Management (RRM) algorithms running in a radio system. The main focus of this thesis is to study RoT algorithm presently deployed in radio units and its possible optimization by reducing complexity of the algorithm in terms of memory usage and processing power. The calculation of RoT comprises three main blocks a Kalman filter, a noise floor estimator and the RoT computation. After analyzing the complexity of each block it has been established that the noise floor estimator block is consuming most of the processing power producing peak processor load since it involves many complex floating point calculations. However, the other blocks do not affect the processing load significantly. It was also observed that some block updates can be reduced in order to decrease the average load on the processor. Three techniques are proposed for reducing the complexity of the RoT algorithm, two for the reduction of peak load and one for the reduction of average load. For reducing the peak load, an interpolation approach is used instead of performing transcendental mathematical calculations. Also, the calculations involving noise floor estimation are extended over several TTIs by keeping in view that the estimation is not time critical. For the reduction of average load, the update rate for the Kalman Filter block is reduced. Based on these optimization steps, a modified algorithm for RoT computation with reduced complexity is proposed. The proposed changes are tested by means of MATLAB simulations demonstrating the improved performance with consistency in the output results. Finally, an arithmetic operation count is done using the hardware manual of Power PC (PPC405) used in Platform 4, which gives a rough estimate of decrease in the percentage of calculations after optimization.

  • 234.
    Ishaq, Rizwan
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap.
    Adaptive Gain Equalizer and Modulation Frequency Domain for Noise Reduction2010Självständigt arbete på avancerad nivå (masterexamen)Studentuppsats (Examensarbete)
    Abstract [en]

    Speech quality during the communication is generally e ected by the surrounding noise and interference. To improve the quality of speech signals and to reduce the amount of disturbing noise, speech enhancement is one of the emerging and most used branches of signal processing. For the reduction of noise from speech signals, methods are continuously developed, one method is the Adaptive Gain Equalizer (AGE) which is a single-channel speech enhancement, method that has the particular focus on enhancement of speech instead suppression of noise. Modulation decomposition of the speech signals brought the idea of a modulation system which is useful for modeling of speech and other signals. The purpose of this thesis is to implement the AGE within modulation system, for the purpose of enhancing speech signal, by reducing noise. The successful implementation of the system has been validated with di erent performance measurements, i.e., Signal to Noise Ratio Improvement(SNRI), Mean Opinion Score(MOS), Spectral Distortion(SD). The system has been checked with male and female speaker and with the noise signals Engine Noise(EN), Factory Noise(FN), Gaussian Noise(GN), Tonal Noise(TN) and Impulse Noise(IN) at 0dB, 5dB, 10dB and -5dB Signal to Noise Ratio(SNR). The system has provided the 10dB SNRI for the TN, and around 6dB SNRI for EN and FN. The system has some compromises on the GN and IN in a sense it gives good sound but low SNRI. MOS has been shown between 4 and 3 for all the test cases.

  • 235.
    ISLAM, MD. JAHERUL
    Blekinge Tekniska Högskola, Sektionen för datavetenskap och kommunikation.
    Performance Analysis of Diversity Techniques for Wireless Communication System2012Självständigt arbete på avancerad nivå (masterexamen)Studentuppsats (Examensarbete)
    Abstract [en]

    Different diversity techniques such as Maximal-Ratio Combining (MRC), Equal-Gain Combining (EGC) and Selection Combining (SC) are described and analyzed. Two branches (N=2) diversity systems that are used for pre-detection combining have been investigated and computed. The statistics of carrier to noise ratio (CNR) and carrier to interference ratio (CIR) without diversity assuming Rayleigh fading model have been examined and then measured for diversity systems. The probability of error (p_e) vs CNR and (p_e) versus CIR have also been obtained. The fading dynamic range of the instantaneous CNR and CIR is reduced remarkably when diversity systems are used [1]. For a certain average probability of error, a higher valued average CNR and CIR is in need for non-diversity systems [1]. But a smaller valued of CNR and CIR are compared to diversity systems. The overall conclusion is that maximal-ratio combining (MRC) achieves the best performance improvement compared to other combining methods. Diversity techniques are very useful to improve the performance of high speed wireless channel to transmit data and information. The problems which considered in this thesis are not new but I have tried to organize, prove and analyze in new ways.

  • 236.
    Islam, Md. Samsul
    et al.
    Blekinge Tekniska Högskola, Sektionen för management.
    Zhou, Lin
    Blekinge Tekniska Högskola, Sektionen för management.
    Li, Fei
    Blekinge Tekniska Högskola, Sektionen för management.
    Application of Artificial Intelligence (Artificial Neural Network) to Assess Credit Risk: A Predictive Model For Credit Card Scoring2009Självständigt arbete på avancerad nivå (magisterexamen)Studentuppsats (Examensarbete)
    Abstract [en]

    Credit Decisions are extremely vital for any type of financial institution because it can stimulate huge financial losses generated from defaulters. A number of banks use judgmental decisions, means credit analysts go through every application separately and other banks use credit scoring system or combination of both. Credit scoring system uses many types of statistical models. But recently, professionals started looking for alternative algorithms that can provide better accuracy regarding classification. Neural network can be a suitable alternative. It is apparent from the classification outcomes of this study that neural network gives slightly better results than discriminant analysis and logistic regression. It should be noted that it is not possible to draw a general conclusion that neural network holds better predictive ability than logistic regression and discriminant analysis, because this study covers only one dataset. Moreover, it is comprehensible that a “Bad Accepted” generates much higher costs than a “Good Rejected” and neural network acquires less amount of “Bad Accepted” than discriminant analysis and logistic regression. So, neural network achieves less cost of misclassification for the dataset used in this study. Furthermore, in the final section of this study, an optimization algorithm (Genetic Algorithm) is proposed in order to obtain better classification accuracy through the configurations of the neural network architecture. On the contrary, it is vital to note that the success of any predictive model largely depends on the predictor variables that are selected to use as the model inputs. But it is important to consider some points regarding predictor variables selection, for example, some specific variables are prohibited in some countries, variables all together should provide the highest predictive strength and variables may be judged through statistical analysis etc. This study also covers those concepts about input variables selection standards.

  • 237.
    Islam, Md. Zameari
    et al.
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap.
    Sajjad, G.M. Sabil
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap.
    Single Channel Speech Enhancement Using Spectral Subtraction Based on Minimum Statistics2011Självständigt arbete på avancerad nivå (masterexamen)Studentuppsats (Examensarbete)
    Abstract [en]

    Speech is an elementary source of human interaction. The quality and intelligibility of speech signals during communication are generally degraded by the surrounding noise. Corrupted speech signals need therefore to be enhanced to improve quality and intelligibility. In the field of speech processing, much effort has been devoted to develop speech enhancement techniques in order to restore the speech signal by reducing the amount of disturbing noise. This thesis focuses on a single channel speech enhancement technique that performs noise reduction by spectral subtraction based on minimum statistics. Minimum statistics means that the power spectrum of the non-stationary noise signal is estimated by finding the minimum values of a smoothed power spectrum of the noisy speech signal and, thus, circumvents the speech activity detection problem. The performance of the spectral subtraction method is evaluated using single channel speech data and for a wide range of noise types with various noise levels. This evaluation is used in order to find optimum method parameter values, thereby improving this algorithm to make it more appropriate for speech communication purposes. The system is implemented in MATLAB and validated by considering different performance measure and for different Signal to Noise Ratio Improvement (SNRI) and Spectral Distortion (SD). The SNRI and SD were calculated for different filter bank settings such as different number of subbands and for different decimation and interpolation ratios. The method provides efficient speech enhancement in terms of SNRI and SD performance measures.

  • 238.
    Islam, Muhammad Adeel
    et al.
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap.
    Shah, Khan Ahmad
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap.
    Performance analysis of dual hop cellular networks2011Självständigt arbete på avancerad nivå (masterexamen)Studentuppsats (Examensarbete)
    Abstract [en]

    Explosive growth in wireless technology caused by development in digital and RF circuit fabrications put some serious challenges on wireless system designers and link budget planning. Low transmit power, system coverage and capacity, high data rates, spatial diversity and quality of services (QOS) are the key factors in future wireless communication system that made it attractive. Dual-hop relaying is the promising underlying technique for future wireless communication to address such dilemmas. Based on dual-hop relaying this thesis addresses two scenarios. In the first case the system model employs dual-hop amplify and forward (AF) multiple input multiple output (MIMO) relay channels with transmit and receive antenna selection over independent Rayleigh fading channels where source and destination contain multiple antennas and communicate with each other with help of single antenna relay. It is assumed that the source and destination has perfect knowledge of channel state information (CSI). Our analysis shows that full spatial diversity order can be achieved with minimum number of antennas at source and destination i.e. min{N_s N_d }. In the second case the performance analysis of dual-hop amplify and forward (AF) multiple relay cooperative diversity network with best relay selection schemes over Rayleigh fading channels is investigated where the source and destination communicate with each other through direct and indirect links. Only the performance of best relay is investigated which participates in the transmission alone. The relay node that achieves highest SNR at the destination is selected as a best relay. Once again our analysis shows that full diversity order can be achieved with single relay with fewer resources compare to the regular cooperative diversity system.

  • 239.
    Jacobsson, Mattias
    Blekinge Tekniska Högskola, Fakulteten för datavetenskaper, Institutionen för datalogi och datorsystemteknik.
    Bitwise relations between n and φ(n): A novel approach at prime number factorization2018Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
    Abstract [en]

    Cryptography plays a crucial role in today’s society. Given the influence, cryptographic algorithms need to be trustworthy. Cryptographic algorithms such as RSA relies on the problem of prime number factorization to provide its confidentiality. Hence finding a way to make it computationally feasible to find the prime factors of any integer would break RSA’s confidentiality.

    The approach presented in this thesis explores the possibility of trying to construct φ(n) from n. This enables factorization of n into its two prime numbers p and q through the method presented in the original RSA paper. The construction of φ(n) from n is achieved by analyzing bitwise relations between the two.

    While there are some limitations on p and q this thesis can in favorable circumstances construct about half of the bits in φ(n) from n. Moreover, based on the research a conjecture has been proposed which outlines further characteristics between n and φ(n).

  • 240.
    Jaffal, Hussein
    et al.
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap.
    Tao, Cheng
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap.
    Multiple Attributes Group Decision Making by Type-2 Fuzzy Sets and Systems2011Självständigt arbete på avancerad nivå (masterexamen)Studentuppsats (Examensarbete)
    Abstract [en]

    We are living in a world full of uncertainty and ambiguity. We usually ask ourselves questions that we are uncertain about their answers. Is it going to rain tomorrow? What will be the exchange rate of euro next month? Why, where and how should I invest? Type-1 Fuzzy sets are characterized by the membership function whose value for a given element x is said to be the grade of membership having a value in the interval [0, 1]. In addition, type-1 fuzzy sets have limited capabilities to deal with uncertainty. In our thesis, we study another concept of a fuzzy description of uncertainty which is called Type-2 fuzzy sets. According to this concept, for any given element x, we can’t speak of an unambiguously specified value of the membership function. Moreover, Type-2 fuzzy sets constitute a powerful tool for handling uncertainty. The aim of our thesis is to examine the potential of the Type-2 fuzzy sets especially in decision making. So, we present basic definitions concerning Type-2 fuzzy sets, and operations on these sets are to be discussed too. Then, Type-2 fuzzy relations and methods of transformation of Type-2 fuzzy sets will be introduced. Also, the theory of Type-2 Fuzzy sets will serve for the construction of the fuzzy inference system. Finally, we utilize interval type-2 fuzzy sets in the application of Multiple Attributes Group Decision Making which is called TOPSIS.

  • 241.
    Jamil, Momin
    et al.
    Blekinge Tekniska Högskola, Sektionen för datavetenskap och kommunikation.
    Yang, Xin-She
    Zepernick, Hans-Jürgen
    Blekinge Tekniska Högskola, Sektionen för datavetenskap och kommunikation.
    Test Functions for Global Optimization: A Comprehensive Survey2013Ingår i: Swarm Intelligence and Bio-Inspired Computation: Theory and Applications / [ed] Yang, Xin-She; Cui, Zhihua; Xiao, Renbin; Gandomi, Amir Hossein; Karamanoglu, Mehmet, London: Elsevier , 2013, s. 193-222Kapitel i bok, del av antologi (Refereegranskat)
    Abstract [en]

    In this book chapter, we present a comprehensive collection of unconstrained optimization test problems that can be used as a benchmark function to validate and compare different optimization algorithms.

  • 242.
    Jamil, Momin
    et al.
    Blekinge Tekniska Högskola, Sektionen för datavetenskap och kommunikation.
    Zepernick, Hans-Jürgen
    Blekinge Tekniska Högskola, Sektionen för datavetenskap och kommunikation.
    Levy Flights and Global Optimization2013Ingår i: Swarm Intelligence and Bio-Inspired Computation: Theory and Applications / [ed] Yang, Xin-She; Cui, Zhihua; Xiao, Renbin; Gandomi, Amir Hossein; Karamanoglu, Mehmet, London: Elsevier , 2013, s. 49-72Kapitel i bok, del av antologi (Refereegranskat)
    Abstract [en]

    A review of metaheuristic algorithms (MAs) is presented. Levy flights (LFs) in the context of global optimization, the Levy distribution, a method to generate random numbers within this distribution are discussed. Subsequently, diversification and intensification in global optimization based on LFs are discussed. Then, MAs using LFs as a search mechanism to solve global optimization problems are presented along with highlighting the differences and similarities of different MAs.

  • 243.
    Jamil, Momin
    et al.
    Blekinge Tekniska Högskola, Sektionen för datavetenskap och kommunikation.
    Zepernick, Hans-Jürgen
    Blekinge Tekniska Högskola, Sektionen för datavetenskap och kommunikation.
    Multimodal function optimisation with cuckoo search algorithm2013Ingår i: International Journal of Bio-Inspired Computation (IJBIC), ISSN 1758-0366, E-ISSN 1758-0374, Vol. 5, nr 2, s. 73-83Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Modern engineering and scientific optimisation problems are becoming complicated. In order to cope with the increasing level of difficulty of these problems, optimisation methods are required to find more than one solution to these problems. The aim of this paper is to gain an insight into the ability of cuckoo search to locate more than one solution for multimodal problems. We also study the performance of this algorithm in the additive white Gaussian noise. Numerical results are presented to show that the cuckoo search algorithm can successfully locate multiple solutions in both non-noise and additive white Gaussian noise with relatively high degree of accuracy.

  • 244.
    Jamil, Momin
    et al.
    Harman/Becker Automotive Systems GmbH.
    Zepernick, Hans-Jürgen
    Blekinge Tekniska Högskola, Fakulteten för datavetenskaper, Institutionen för kreativa teknologier.
    Syntheszizing Cross-Ambiguity Function Using Improved BAt Algorithm2017Ingår i: Recent Advances in Swarm Intelligence and Evolutionary Computation / [ed] Xin-She Yang, Springer Publishing Company, 2017, s. 179-202Kapitel i bok, del av antologi (Refereegranskat)
  • 245. Jena, Ajit K.
    et al.
    Popescu, Adrian
    Nilsson, Arne A.
    Modeling and Evaluation of Internet Applications2002Rapport (Övrigt vetenskapligt)
    Abstract [sv]

    Det handlar om trafik mättningar, analys samt modellering av några av Internets applikationer (SMTP, HTTP, FTP).

  • 246. Jena, Ajit K.
    et al.
    Popescu, Adrian
    Pruthi, Parag
    Modeling and Analysis of HTTP Traffic2000Konferensbidrag (Refereegranskat)
    Abstract [en]

    The global Internet has seen tremendous growth in terms of nodes and user base as well as of types of applications. One of the most important consequences of this growth is related to an increased complexity of the traffic experienced in these networks. Each application has a set of unique characteristics in terms of performance characteristics, transactions as well as the way the transaction processing profile maps onto unique network resource requirements. In order to support Internet applications effectively,it is therefore important to understand and to characterize the application level transactions as well as the effect of different TCP/IP control mechanisms on application-level parameters. It is the goal of this paper to model and to evaluate the characteristics of World Wide Web traffic. Results are reported on measuring, modeling and analysis of specific Hyper Text Transfer Protocol traffic collected from different (classes of) sites together with methodologies used for capturing HTTP flows as well as for modeling. The paper concludes with a discussion on the structure of Web pages and a model for the generation of the number of embedded pages in a Web page is suggested.

  • 247.
    Jiayuan, Dai
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap.
    Equivalence Group of Lie's Second-Order Equation of Type III2012Självständigt arbete på avancerad nivå (masterexamen)Studentuppsats (Examensarbete)
    Abstract [en]

    Based on symmetry and invariance principles, Lie group analysis is the only systematic method for solving nonlinear differential equations analytically. Nonlinear second-order ordinary differential equations admitting two-dimensional Lie algebras can be transformed into one of the four types of canonical forms via Lie's integration method. In this thesis, Lie's second-order equation of type III is considered from the point of view of equivalence transformations. The generators of the equivalence group and the principal Lie algebra are calculated.

  • 248. Kao-Walter, Sharon
    et al.
    Mfoumou, Etienne
    Laksman, Efraim
    Stress Relaxation of a Paper Sheet under Cyclic Load: An Experimental and Theoretical Model2010Ingår i: Materials Sciences and Applications, ISSN 2153-117X, E-ISSN 2153-1188, Vol. 1, nr 6, s. 317-322Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Mechanical experiments have been performed to study the dynamic stress relaxation of a paper sheet material mainly used in food packaging industry. The material was cyclically tensile-loaded with a strain range between 2.4% and 4%. The time period for each cycle was 400 seconds. It was found that stress will decrease when the number of cycles increases in the case of upper load and vice versa in the case of lower load. At the same time, the stress to strain curves followed the same pattern as the one from the previous cycle. The stress relaxation behavior of each cycle has been analyzed and the dynamic relaxation modulus was derived. An improved model is proposed to describe the dynamic relaxation behavior of the paper sheet. This model shows a very good fit to the experimental results and trends of prediction are observed. Furthermore, the physical description of this model and the variation by the cycles is discussed.

  • 249.
    Kaowichakorn, Peerachai
    Blekinge Tekniska Högskola, Sektionen för datavetenskap och kommunikation.
    Probabilistic Analysis of Quality of Service2013Självständigt arbete på avancerad nivå (masterexamen)Studentuppsats (Examensarbete)
    Abstract [en]

    Current complex service systems are usually comprised of many other components which are often external services performing particular tasks. The quality of service (QoS) attributes such as availability, cost, response time are essential to determine usability and eciency of such system. Obviously, the QoS of such compound system is dependent on the QoS of its components. However, the QoS of each component is naturally unstable and di erent each time it is called due to many factors like network bandwidth, workload, hardware resource, etc. This will consequently make the QoS of the whole system be unstable. This uncertainty can be described and represented with probability distributions. This thesis presents an approach to calculate the QoS of the system when the probability distributions of QoS of each component are provided by service provider or derived from historical data, along with the structure of their compositions. In addition, an analyzer tool is implemented in order to predict the QoS of the given compositions and probability distributions following the proposed approach. The output of the analyzer can be used to predict the behavior of the system to be implemented and to make decisions based on the expected performance. The experimental evaluation shows that the estimation is reliable with a minimal and acceptable error measurement.

  • 250.
    Karagözlü, Erkan
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap.
    Investigation of the Mathematical Model of Acoustic Waves in a Fluid with Bubbles Using Lie Group Analysis Method2012Självständigt arbete på avancerad nivå (masterexamen)Studentuppsats (Examensarbete)
    Abstract [en]

    This thesis comprises investigation of the mathematical model of acoustic waves in a fluid with bubbles for nonlinear self-adjointness, Lie point symmetries, conservation laws and invariant solutions. It is based on the theory developed recently by Prof. Nail Ibragimov. It is shown that the systems of differential equations describing the model are nonlinearly self-adjoint. The symmetries are calculated and the conservation laws are constructed using the formal Lagrangian. In addition, invariant solutions are derived.

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