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  • 1. Alexandrova, A. A.
    et al.
    Ibragimov, Nail
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Lukashchuk, V. O.
    Group classification and conservation laws of nonlinear filtration equation with a small parameter2014Inngår i: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 19, nr 2, s. 364-370Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Group classification of the perturbed nonlinear filtration equation is performed assuming that the perturbation is an arbitrary function of the dependent variable. The nonlinear self-adjointness of the equation under consideration is investigated. Using these results, the approximate conservation laws are constructed.

  • 2.
    Anco, S.
    et al.
    Brock Univ, Dept Math & Stat, St Catharines, ON L2S 3A1, Canada..
    Avdonina, E. D.
    Ufa State Aviat Tech Univ, Lab Grp Anal Math Models Nat & Engn Sci, Ufa 450000, Russia..
    Gainetdinova, A.
    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..
    Ibragimov, Nail
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Wolf, T.
    Brock Univ, Dept Math & Stat, St Catharines, ON L2S 3A1, Canada..
    Symmetries and conservation laws of the generalized Krichever-Novikov equation2016Inngår i: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 49, nr 10, artikkel-id 105201Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A computational classification of contact symmetries and higher-order local symmetries that do not commute with t, x, as well as local conserved densities that are not invariant under t, x is carried out for a generalized version of the Krichever-Novikov (KN) equation. Several new results are obtained. First, the KN equation is explicitly shown to have a local conserved density that contains t, x. Second, apart from the dilational point symmetries known for special cases of the KN equation and its generalized version, no other local symmetries with low differential order are found to contain t, x. Third, the basic Hamiltonian structure of the KN equation is used to map the local conserved density containing t, x into a nonlocal symmetry that contains t, x. Fourth, a recursion operator is applied to this nonlocal symmetry to produce a hierarchy of nonlocal symmetries that have explicit dependence on t, x. When the inverse of the Hamiltonian map is applied to this hierarchy, only trivial conserved densities are obtained.

  • 3. Anco, S.
    et al.
    Ibragimov, Nail
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Imamutdinova, K.V.
    Karimova, E.N.
    Solutions of gasdynamic equations associated with classical and new conservation laws2015Inngår i: Applied Mathematics and Computation, ISSN 0096-3003, E-ISSN 1873-5649, Vol. 268, s. 52-58Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Exact solutions of the one-dimensional gasdynamic equations are constructed by applying the method of conservation laws to all point-wise conserved vectors of the equations under consideration. © 2015 Elsevier Inc.

  • 4. Avdonina, Elena D.
    et al.
    Ibragimov, Nail H.
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Heat conduction in anisotropic media: Nonlinear self-adjointness and conservation laws2012Inngår i: Discontinuity, Nonlinearity and Complexity, ISSN 2164-6376, Vol. 1, nr 3, s. 237-251Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    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.

  • 5. Avdonina, Elena D.
    et al.
    Ibragimov, Nail H.
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Khamitova, Raisa
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Exact solutions of gasdynamic equations obtained by the method of conservation laws2013Inngår i: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 18Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 6. Baikov, V. A.
    et al.
    Ibragimov, Nail
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Zheltova, I. S.
    Yakovlev, A. A.
    Conservation laws for two-phase filtration models2014Inngår i: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 19, nr 2, s. 383-389Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The paper is devoted to investigation of group properties of a one-dimensional model of two-phase filtration in porous medium. Along with the general model, some of its particular cases widely used in oil-field development are discussed. The Buckley-Leverett model is considered in detail as a particular case of the one-dimensional filtration model. This model is constructed under the assumption that filtration is one-dimensional and horizontally directed, the porous medium is homogeneous and incompressible, the filtering fluids are also incompressible. The model of "chromatic fluid" filtration is also investigated. New conservation laws and particular solutions are constructed using symmetries and nonlinear self-adjointness of the system of equations.

  • 7. Boszhkov, Yuri
    et al.
    Freire, Igor
    Ibragimov, Nail H.
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Group analysis of the Novikov equation2014Inngår i: Computational and Applied Mathematics, ISSN 2238-3603, E-ISSN 1807-0302, Vol. 33, nr 1, s. 193-202Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    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.

  • 8. Bozhkov, Y.
    et al.
    Dimas, S.
    Ibragimov, Nail H.
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Conservation laws for a coupled variable-coefficient modified Korteweg–de Vries system in a two-layer fluid model2013Inngår i: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 18Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    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.

  • 9. Dorodnitsyn, Vladimir
    et al.
    Ibragimov, Nail
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    An extension of the Noether theorem: Accompanying equations possessing conservation laws2014Inngår i: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 19, nr 2, s. 328-336Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    It is shown that the Noether theorem can be extended for some equations associated (accompanying) with Euler-Lagrange equation. Each symmetry of Lagrangian yields a class of accompanying equations possessing conservation law (first integral). The generalization is done for canonical Hamiltonian equations as well.

  • 10. Gainetdinova, A.
    et al.
    Ibragimov, Nail
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Meleshko, Sergey
    Group classification of ODE y‴ = F (x, y, y′)2014Inngår i: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 19, nr 2, s. 345-349Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In his extensive work of 1884 on the group classification of ordinary differential equations Lie performed, inter alia, the group classification of the particular type of the second-order equations y″ = F (x, y). In the present paper we extend Lie's classification to the third-order equations y‴ = F (x, y, y′).

  • 11. Galiakberova, L. R.
    et al.
    Ibragimov, Nail
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Nonlinear self-adjointness of the Krichever-Novikov equation2014Inngår i: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 19, nr 2, s. 361-363Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    It is known that the classification of third-order evolutionary equations with the constant separant possessing a nontrivial Lie-Bäcklund algebra (in other words, integrable equations) results in the linear equation, the KdV equation and the Krichever-Novikov equation. The first two of these equations are nonlinearly self-adjoint. This property allows to associate conservation laws of the equations in question with their symmetries. The problem on nonlinear self-adjointness of the Krichever-Novikov equation has not been solved yet. In the present paper we solve this problem and find the explicit form of the differential substitution providing the nonlinear self-adjointness.

  • 12. Gazizov, R. K.
    et al.
    Ibragimov, N. H.
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Lukashchuk, S. Yu.
    Nonlinear self-adjointness, conservation laws and exact solutions of time-fractional Kompaneets equations2015Inngår i: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 23, nr 1-3, s. 153-163Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Four time-fractional generalizations of the Kompaneets equation are considered. Group analysis is performed for physically relevant approximations. It is shown that all approximations have nontrivial symmetries and conservation laws. The symmetries are used for constructing group invariant solutions, whereas the conservation laws allow to find non-invariant exact solutions. (C) 2014 Elsevier B.V. All rights reserved.

  • 13. Gazizov, Rafail
    et al.
    Ibragimov, Nail
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Rudenko, Oleg
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för maskinteknik.
    Effect of resonant absorption in viscous and dry vibrating contact: Mathematical models and theory connected with slow dynamics and friction welding2014Inngår i: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 19, nr 2, s. 337-344Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Process of heating of thin layer located between two vibrating surfaces is studied. Energy loss goes on due to viscous or dry friction. Optimal quantities of shear viscosity and friction corresponding to maximum energy loss are determined. Resonant behavior of loss must be taken into account in the description of "slow dynamics" of rocks and materials exposed to high-intensity seismic or acoustic irradiation as well as in various technologies. Bonding of materials by linear friction welding, widely used in propulsion engineering, can exemplify such a technology.

  • 14. Ibragimov, Nail
    A survey on integration of parabolic equations by reducing them to the heat equation2011Inngår i: Contemporary mathematics / [ed] Blázquez-Sanz, David; Morales-Ruiz, Juan J.; Lombardero, Jesús Rodríguez, AMS (American Mathematical society) , 2011Kapittel i bok, del av antologi (Fagfellevurdert)
    Abstract [en]

    The present paper is a survey of results [1], [2] on extension of Euler’s method for solving hyperbolic equations with one spatial variable to parabolic equations. The new method, based on the invariants of parabolic equations, allows one to identify all linear parabolic equations reducible to the heat equation and find their general solution. The method is illustrated by several examples.

  • 15. Ibragimov, Nail
    Archives of ALGA. Volume 32006Collection/Antologi (Annet vitenskapelig)
    Abstract [en]

    Volume 3 contains 6 articles: Lars Haikola, Louise Petrén-Overton, min mormor; Lars Haikola, Louise Petr´en-Overton, my grandmother. Traslations: Louise Petrén, Extension of Laplace's method to the equations...; E. Bessel-Hagen, On conservation laws of electrodynamics. Nail H. Ibragimov, The answer to the question put to me by L.V. Ovsyannikov 33 years ago; Nail H. Ibragimov, Raisa Khamitova, Bo Thidé, Adjoint system and conservation laws for symmetrized electromagnetic equations with a dual Ohm's law.

  • 16. Ibragimov, Nail
    Archives of ALGA, volume 52008Collection/Antologi (Annet vitenskapelig)
    Abstract [en]

    Volume 5 contains 3 articles by N.Ibragimov, an article by R. Khamitova and the English translation of V.P.Ermakov's article "Second order differential equations:conditions of complete integrability".

  • 17. Ibragimov, Nail
    Archives of ALGA. Volume 62009Collection/Antologi (Annet vitenskapelig)
    Abstract [en]

    Volume 6 contains 6 articles: Nail H. Ibragimov, Utilization of canonical variables for integration of systems of first-order differential equations; Nail H. Ibragimov and Ranis N. Ibragimov, Group analysis of nonlinear internal waves in oceans. I: Self-adjointness, conservation laws, invariant solutions; Nail H. Ibragimov, Ranis N. Ibragimov and Vladimir F. Kovalev, Group analysis of nonlinear internal waves in oceans. II: The symmetries and rotationally invariant solution; Nail H. Ibragimov and Ranis N. Ibragimov, Group analysis of nonlinear internal waves in oceans. III: Additional conservation laws; Nail H. Ibragimov Alternative presentation of Lagrange's method of variation of parameters; Nail H. Ibragimov Application of group analysis to liquid metal systems.

  • 18.
    Ibragimov, Nail
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Conservation laws and non-invariant solutions of anisotropic wave equations with a source2018Inngår i: Nonlinear Analysis, ISSN 1468-1218, Vol. 40, s. 82-94Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Linear and nonlinear waves in anisotropic media are used in various fields, e.g. in biomechanics, biomedical acoustics, etc. The present paper is devoted to discussion of nonlinear anisotropic wave equations with a source from point of view of their conservation laws and exact solutions associated with conservation laws. Nonlinearly self-adjoint wave equations with special source terms are singled out. The conservation laws associated with symmetries of the nonlinearly self-adjoint wave equations are computed and used for constructing exact solutions. The obtained solutions are different from group invariants solutions, in particular, from steady state and traveling wave solutions. The paper is designed for the application oriented readers. Its main goal is to introduce readers, interested in solutions of mathematical models having real world applications, to the recent method of conservation laws for constructing exact solutions of partial differential equations using conservation laws. © 2017 Elsevier Ltd

  • 19.
    Ibragimov, Nail
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Construction of Conservation Laws Using Symmetries2014Konferansepaper (Fagfellevurdert)
    Abstract [en]

    The concept of nonlinear self-adjointness of differential equations, introduced by the author in 2010, is discussed in detail. All linear equations and systems are nonlinearly self-adjoint. Moreover, the class of nonlinearly self-adjoint equations includes all nonlinear equations and systems having at least one local conservation law. It follows, in particular, that the integrable systems possessing infinite set of Lie-Backlund symmetries (higher-order tangent transformations) are nonlinearly self-adjoint. An explicit formula for conserved vectors associated with symmetries is provided for all nonlinearly self-adjoint differential equations and systems. The number of equations contained in the systems under consideration can be different from the number of dependent variables. A utilization of conservation laws for constructing exact solutions is discussed and illustrated by computing non-invariant solutions of the Chaplygin equations in gas dynamics.

  • 20.
    Ibragimov, Nail
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Integration of dynamical systems admitting nonlinear superposition2016Inngår i: JOURNAL OF COUPLED SYSTEMS AND MULTISCALE DYNAMICS, ISSN 2330-152X, Vol. 4, nr 2, s. 91-106Artikkel, forskningsoversikt (Fagfellevurdert)
    Abstract [en]

    A method of integration of non-stationary dynamical systems admitting nonlinear superpositions is presented. The method does not require knowledge of symmetries of the differential equations under consideration. The integration procedure is based on classification of Vessiot-Guldberg-Lie algebras associated with nonlinear superpositions. It is shown that the systems associated with one-and two-dimensional Lie algebras can be integrated by quadrature upon introducing Lie's canonical variables. It is not necessary to know symmetries of a system in question in this approach. Two-dimensional non-stationary dynamical systems with three-dimensional Vessiot-Guldberg-Lie algebras are classified into thirteen standard forms. Ten of them are integrable by quadrature. The remaining three standard forms lead to the Riccati equations. Integration of perturbed dynamical systems possessing approximate nonlinear superposition is discussed.

  • 21.
    Ibragimov, Nail
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Selected Works: Volume 52014Bok (Fagfellevurdert)
    Abstract [en]

    Advanced topics to be added to my textbook "A Practical Course in Differential Equations and Mathematical Modelling".

    This volume is dedicated to the memory of my teacher and friend Lev Vasilyevich Ovsyannikov (22.04.1919{23.05.2014).

    Volume V contains preprints written during 2008-2014 as advanced topics to be added to the textbook.  They include, e.g. a discussion of a wide class of linear ordinary differential equations whose integration is reducible to solution of algebraic equations. This class contains the constant coefficient equations and Euler’s equations as particular cases. The recent method of nonlinear self-adjointness for constructing conservations laws associated with symmetries of partial differential equations is also presented in this volume.

  • 22.
    Ibragimov, Nail
    et al.
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Avdonina, E.D.
    Nonlinear self-adjointness, conservation laws, and the construction of solutions of partial differential equations using conservation laws2013Inngår i: Russian Mathematical Surveys, ISSN 0036-0279, E-ISSN 1468-4829, Vol. 68, nr 5, s. 889-921Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The method of nonlinear self-adjointness, which was recently developed by the first author, gives a generalization of Noether's theorem. This new method significantly extends approaches to constructing conservation laws associated with symmetries, since it does not require the existence of a Lagrangian. In particular, it can be applied to any linear equations and any nonlinear equations that possess at least one local conservation law. The present paper provides a brief survey of results on conservation laws which have been obtained by this method and published mostly in recent preprints of the authors, along with a method for constructing exact solutions of systems of partial differential equations with the use of conservation laws. In most cases the solutions obtained by the method of conservation laws cannot be found as invariant or partially invariant solutions.

  • 23.
    Ibragimov, Nail
    et al.
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Gainetdinova, A. A.
    Ufa State Aviat Tech Univ, Rus.
    Three-dimensional dynamical systems admitting nonlinear superposition with three-dimensional Vessiot-Guldberg-Lie algebras2016Inngår i: Applied Mathematics Letters, ISSN 0893-9659, E-ISSN 1873-5452, Vol. 52, s. 126-131Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The recent method of integration of non-stationary dynamical systems admitting nonlinear superpositions is applied to the three-dimensional dynamical systems associated with three-dimensional Vessiot-Guldberg-Lie algebras L-3. The investigation is based on Bianchi's classification of real three-dimensional Lie algebras and realizations of these algebras in the three-dimensional space. Enumeration of the Vessiot-Guldberg-Lie algebras L-3 allows to classify three-dimensional dynamical systems admitting nonlinear superpositions into thirty one standard types by introducing canonical variables. Twenty four of them are associated with solvable Vessiot-Guldberg-Lie algebras and can be reduced to systems of first-order linear equations. The remaining seven standard types are nonlinear. Integration of the latter types is an open problem. (C) 2015 Elsevier Ltd. All rights reserved.

  • 24.
    Ibragimov, Nail
    et al.
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Gainetdinova, Aliya
    Ufimskij Gosudarstvennyj Aviacionnyj Tehniceskij Universitet, RUS.
    Three-dimensional dynamical systems with four-dimensional vessiot-guldberg-lie algebras2017Inngår i: The Journal of Applied Analysis and Computation, ISSN 2156-907X, E-ISSN 2158-5644, Vol. 7, nr 3, s. 872-883Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    - Dynamical systems attract much attention due to their wide applications. Many significant results have been obtained in this field from various points of view. The present paper is devoted to an algebraic method of integration of three-dimensional nonlinear time dependent dynamical systems admitting nonlinear superposition with four-dimensional Vessiot-Guldberg-Lie algebras L4. The invariance of the relation between a dynamical system admitting nonlinear superposition and its Vessiot-Guldberg-Lie algebra is the core of the integration method. It allows to simplify the dynamical systems in question by reducing them to standard forms. We reduce the three-dimensional dynamical systems with four-dimensional Vessiot-Guldberg-Lie algebras to 98 standard types and show that 86 of them are integrable by quadratures.

  • 25.
    Ibragimov, Nail
    et al.
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Gainetdinova, Aliya. A.
    Ufa State Aviation Technical University, RUS.
    Classification and integration of four-dimensional dynamical systems admitting non-linear superposition2017Inngår i: International Journal of Non-Linear Mechanics, ISSN 0020-7462, Vol. 90, s. 50-71Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The method of integration of dynamical systems admitting non-linear superpositions is applied to four-dimensional non-linear dynamical systems. All four-dimensional dynamical systems admitting non-linear superpositions with four-dimensional Vessiot-Guldberg-Lie algebras are classified into 160 standard forms. The integration method is described and illustrated.

  • 26.
    Ibragimov, Nail
    et al.
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Gandarias, M.L.
    Galiakberova, L.R.
    Bruzon, M.S.
    Avdonina, E.D.
    Group classification and conservation laws of anisotropic wave equations with a source2016Inngår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 57, nr 8, artikkel-id 083504Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Linear and nonlinear waves in anisotropic media are useful in investigating complex materials in physics, biomechanics, biomedical acoustics, etc. The present paper is devoted to investigation of symmetries and conservation laws for nonlinear anisotropic wave equations with specific external sources when the equations in question are nonlinearly self-adjoint. These equations involve two arbitrary functions. Construction of conservation laws associated with symmetries is based on the generalized conservation theorem for nonlinearly self-adjoint partial differential equations. First we calculate the conservation laws for the basic equation without any restrictions on the arbitrary functions. Then we make the group classification of the basic equation in order to specify all possible values of the arbitrary functions when the equation has additional symmetries and construct the additional conservation laws.

  • 27.
    Ibragimov, Nail H.
    Blekinge Tekniska Högskola, Sektionen för teknik, Avdelningen för matematik och naturvetenskap.
    Approximate transformation groups2008Bok (Annet vitenskapelig)
    Abstract [en]

    These notes provide an easy to follow introduction to the topic and are based on my talks at various conferences, in particular on the plenary lecture at the International Workshop on ``Differential equations and chaos" (University of Witwatersrand, Johannesburg, South Africa, January 1996). The book is prepared for the new graduate course ``Approximate transformation groups" that will be given at Blekinge Institute of Technology during January-March, 2009.

  • 28.
    Ibragimov, Nail H.
    Blekinge Tekniska Högskola, Sektionen för teknik, Avdelningen för matematik och naturvetenskap.
    Archives of ALGA: volume 92012Collection/Antologi (Annet vitenskapelig)
    Abstract [en]

    Volume 9 contains articles of different authors.

  • 29.
    Ibragimov, Nail H.
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Comments on the paper "Conservation laws of the (2+1)-dimensional KP equation and Burgers equation with variable coefficients and cross terms" by Li-Hua Zhang2014Inngår i: Applied Mathematics and Computation, ISSN 0096-3003, E-ISSN 1873-5649, Vol. 226, s. 1-2Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The recent paper mentioned in the title contains a confusing statement on computing conservation laws corresponding to symmetries of nonlinearly self-adjoint differential equations. The present brief article contains clarifying comments.

  • 30.
    Ibragimov, Nail H.
    Blekinge Tekniska Högskola, Sektionen för teknik, Avdelningen för matematik och naturvetenskap.
    Integration of second-order linear equations via linearization of Riccati's equations2007Inngår i: Archives of ALGA, ISSN 1652-4934, Vol. 4, s. 71-75Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The test for linearization of the Riccati equations by a change of the dependent variable, proved by the author in 1989, is utilized for integration of second-order linear equations by quadratures.

  • 31.
    Ibragimov, Nail H.
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Lie Group Analysis of Moffatt´s model in Metallurgical Industry2011Inngår i: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 18, nr Suppl. 1Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The paper is devoted to the Lie group analysis of a nonlinear equation arising in metallurgical applications of Magnetohydrodynamics. Self-adjointness of the basic equations is investigated. The analysis reveals two exceptional values of the exponent playing a significant role in the model.

  • 32.
    Ibragimov, Nail H.
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Method of conservation laws for constructing solutions to systems of PDEs2012Inngår i: Discontinuity, Nonlinearity, and Complexity, ISSN 2164−6376, Vol. 1, nr 4Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In the present paper, a new method is proposed for constructing exact solutions for systems of nonlinear partial differential equations. It is called the method of conservation laws. Application of the method to the Chaplygin gas allowed to construct new solutions containing several arbitrary parameters. It is shown that these solutions cannot be obtained, in general, as group invariant solutions.

  • 33.
    Ibragimov, Nail H.
    Blekinge Tekniska Högskola, Sektionen för teknik, Avdelningen för matematik och naturvetenskap.
    Selected works. Volume 42009Bok (Annet vitenskapelig)
    Abstract [en]

    Volume IV contains papers written during 1996-2007. The main topics in this volume include Equivalence groups and invariants of differential equations, Extension of Eulers’ method of integration of hyperbolic equations to parabolic equations, Invariant and formal Lagrangians, Conservation laws.

  • 34.
    Ibragimov, Nail H
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Transformation groups and Lie algebras2013Bok (Annet vitenskapelig)
    Abstract [en]

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

  • 35.
    Ibragimov, Nail H.
    et al.
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Avdonina, Elena D.
    Conservation laws and exact solutions for nonlinear diffusion in anisotropic media2013Inngår i: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 18, nr 10Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

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

  • 36.
    Ibragimov, Nail H.
    et al.
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Gazizov, Rafail H.
    Approximate symmetries and solutions of the Kompaneets equation2014Inngår i: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 55, nr 2, s. 220-224Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    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.

  • 37.
    Ibragimov, Nail H.
    et al.
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Ibragimov, Ranis N.
    Nonlinear Whirlpools Versus Harmonic Waves in a Rotating Column of Stratified Fluid2013Inngår i: Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, Vol. 8, nr 1, s. 122-131Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

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

  • 38.
    Ibragimov, Nail H.
    et al.
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Ibragimov, Ranis N.
    Rotationally symmetric internal gravity waves2012Inngår i: International Journal of Non-Linear Mechanics, ISSN 0020-7462, E-ISSN 1878-5638, Vol. 47, nr 1, s. 46-52Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

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

  • 39.
    Ibragimov, Nail H.
    et al.
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Ibragimov, Ranis N.
    Galiakberova, L. R.
    Symmetries and Conservation Laws of a Spectral Nonlinear Model for Atmospheric Baroclinic Jets2014Inngår i: Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, E-ISSN 1760-6101, Vol. 9, nr 5, s. 111-118Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    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.

  • 40.
    Ibragimov, Nail H.
    et al.
    Blekinge Tekniska Högskola, Sektionen för teknik, Avdelningen för matematik och naturvetenskap.
    Khamitova, Raisa
    Blekinge Tekniska Högskola, Sektionen för teknik, Avdelningen för matematik och naturvetenskap.
    Computer Algebra Programs in Group Analysis and Education2006Konferansepaper (Fagfellevurdert)
    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.

  • 41.
    Ibragimov, Nail H.
    et al.
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Khamitova, Raisa
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Conservation Laws in Thomas’s Model of Ion Exchange in a Heterogeneous Solution2013Inngår i: Discontinuity, Nonlinearity and Complexity, ISSN 2164-6376, Vol. 2, nr 2Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 42.
    Ibragimov, Nail H.
    et al.
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Khamitova, Raisa
    Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, Avdelningen för matematik och naturvetenskap.
    Cantoni, Antonio
    Self-adjointness of a generalized Camassa-Holm equation2011Inngår i: Applied Mathematics and Computation, ISSN 0096-3003, E-ISSN 1873-5649, Vol. 218, nr 6, s. 2579-2583Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 43.
    Ibragimov, Nail H.
    et al.
    Blekinge Tekniska Högskola, Sektionen för teknik, Avdelningen för matematik och naturvetenskap.
    Khamitova, Raisa
    Blekinge Tekniska Högskola, Sektionen för teknik, Avdelningen för matematik och naturvetenskap.
    Thidé, Bo
    Adjoint systems and conservation laws for symmetrized electromagnetic equations with a dual Ohm´s law2006Inngår i: Archives of ALGA, ISSN 1652-4934, Vol. 3Artikkel i tidsskrift (Fagfellevurdert)
  • 44.
    Ibragimov, Nail H.
    et al.
    Blekinge Tekniska Högskola, Sektionen för teknik, Avdelningen för matematik och naturvetenskap.
    Khamitova, Raisa
    Blekinge Tekniska Högskola, Sektionen för teknik, Avdelningen för matematik och naturvetenskap.
    Thidé, Bo
    Conservation laws for symmetrised electromagnetic equations with a dual Ohm's law2007Konferansepaper (Fagfellevurdert)
    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.

  • 45.
    Ibragimov, Nail H.
    et al.
    Blekinge Tekniska Högskola, Sektionen för teknik, Avdelningen för matematik och naturvetenskap.
    Khamitova, Raisa
    Blekinge Tekniska Högskola, Sektionen för teknik, Avdelningen för matematik och naturvetenskap.
    Thidé, Bo
    Conservation laws for the Maxwell-Dirac equations with dual Ohm's law2007Inngår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 48, nr 5Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 46.
    Ibragimov, Nail H.
    et al.
    Blekinge Tekniska Högskola, Fakulteten för teknikvetenskaper, Institutionen för matematik och naturvetenskap.
    Kovalev, V. F.
    Meleshko, S. V.
    Bychenkov, V. Yu.
    Group analysis of kinetic equations in a non-linear thermal transport problem2015Inngår i: International Journal of Non-Linear Mechanics, ISSN 0020-7462, E-ISSN 1878-5638, Vol. 71, s. 1-7Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    An application of modern group analysis to electron kinetic equations in non-linear thermal transport problem is discussed. The admitted symmetry group is calculated, and the optimal system of one and two-dimensional subalgebras is constructed. Representations of invariant solutions are presented.

  • 47.
    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.2004Inngår i: Archives of ALGA, ISSN 1652-4934, Vol. 1, s. 95-126Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 48.
    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 domain2007Inngår i: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 99, nr 8Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 49. 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)2004Inngår i: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 9, nr 1, s. 69-80Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 50. Ibragimov, Nail
    et al.
    Ibragimov, Ranis
    Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences2012Inngår i: Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, E-ISSN 1760-6101, Vol. 7, nr 2, s. 52-65Artikkel i tidsskrift (Fagfellevurdert)
    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.

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