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  • 51. Ibragimov, Nail
    A survey on integration of parabolic equations by reducing them to the heat equation2011In: Contemporary mathematics / [ed] Blázquez-Sanz, David; Morales-Ruiz, Juan J.; Lombardero, Jesús Rodríguez, AMS (American Mathematical society) , 2011Chapter in book (Refereed)
    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.

  • 52. Ibragimov, Nail
    Archives of ALGA. Volume 32006Collection (editor) (Other academic)
    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.

  • 53. Ibragimov, Nail
    Archives of ALGA, volume 52008Collection (editor) (Other academic)
    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".

  • 54. Ibragimov, Nail
    Archives of ALGA. Volume 62009Collection (editor) (Other academic)
    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.

  • 55.
    Ibragimov, Nail
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Natural Sciences.
    Construction of Conservation Laws Using Symmetries2014Conference paper (Refereed)
    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.

  • 56.
    Ibragimov, Nail
    et al.
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Natural Sciences.
    Avdonina, E.D.
    Nonlinear self-adjointness, conservation laws, and the construction of solutions of partial differential equations using conservation laws2013In: Russian Mathematical Surveys, ISSN 0036-0279, E-ISSN 1468-4829, Vol. 68, no 5, p. 889-921Article in journal (Refereed)
    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.

  • 57. Ibragimov, Nail H.
    A Bridge Between Lie Symmetries and Galois Groups2009Conference paper (Refereed)
    Abstract [en]

    A bridge between Lie symmetry groups for differential equations and Galois groups for algebraic equations is suggested. It is based on calculation of Lie symmetries for algebraic equations and their restriction of the roots of the equations under consideration. The approach is illustrated by several examples. An alternative representation of Lie symmetries, called the Galois representation, is provided for differential equations.

  • 58. Ibragimov, Nail H.
    A discussion of conservation laws for over-determined systems with application to the Maxwell equations in vacuum.2007In: Archives of ALGA, ISSN 1652-4934, Vol. 4, p. 19-54Article in journal (Refereed)
    Abstract [en]

    The evolutionary part of Maxwell's equations has a Lagrangian written in terms of the electric and magnetic fields, but admit neither Lorentz nor conformal transformations. The additional equations on vanishing the divergences of the electric and magnetic fields guarantee the Lorentz and conformal invariance, but the resulting overdetermined system does not have a Lagrangian. The aim of the present paper is to attain a harmony between these two contradictory properties.

  • 59. Ibragimov, Nail H.
    A new conservation theorem2007In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 333, no 1, p. 311-328Article in journal (Refereed)
    Abstract [en]

    A general theorem on conservation laws for arbitrary differential equations is proved. The theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables. The new theorem does not require existence of a Lagrangian and is based on a concept of an adjoint equation for non-linear equations suggested recently by the author. It is proved that the adjoint equation inherits all symmetries of the original equation. Accordingly, one can associate a conservation law with any group of Lie, Lie-Backlund or non-local symmetries and find conservation laws for differential equations without classical Lagrangians.

  • 60. Ibragimov, Nail H.
    A practical course in differential equations and mathematical modeling2009Book (Other academic)
    Abstract [en]

    A practical course in differential equations and mathematical modelling is a unique blend of the traditional methods with Lie group analysis enriched by author’s own theoretical developments. The main objective of the book is to develop new mathematical curricula based on symmetry and invariance principles. This approach helps to make courses in differential equations, mathematical modelling, distributions and fundamental solution, etc. easy to follow and interesting for students. The book is based on author’s long-term experience of teaching at Novosibirsk and Moscow Universities in Russia, Collège de France, Georgia Tech and Stanford University in USA, Universities in South Africa, Cyprus, Turkey, and Blekinge Institute of Technology (BTH) in Sweden. The new curriculum prepares the students for solving modern nonlinear problems and attracts essentially more students than the traditional way of teaching mathematics. The book can be used as a main textbook by undergraduate and graduate students and their teachers in applied mathematics, physics and engineering sciences.

  • 61. Ibragimov, Nail H.
    A practical course in differential equations and mathematical modelling: classical and new methods, nonlinear mathematical models, symmetry and invariance principles2006Book (Other academic)
  • 62. Ibragimov, Nail H.
    A practical course in differential equations and mathematical modelling: Classical and new methods, nonlinear mathematical models, symmetry and invariance principles. 1st Ed.2004Book (Other academic)
    Abstract [en]

    The present textbook on ordinary and partial differential equations is tailored to develop analytic skills and "working knowledge'' in both classical and Lie's methods for solving linear and nonlinear equations. It is based on lectures delivered by the author at Moscow Institute of Physics and Technology, and Blekinge Institute of Technology. The prerequisites are linear algebra and mathematical analysis with one and several variables including elements of ordinary differential equations.

  • 63. Ibragimov, Nail H.
    A Practical Course in Differential Equations and Mathematical Modelling: Classical and New Methods, Nonlinear Mathematical Models, Symmetry and Invariance Principles. 2nd ED2005Book (Other academic)
    Abstract [en]

    The present textbook on ordinary and partial differential equations is tailored to develop analytic skills and "working knowledge'' in both classical and Lie's methods for solving linear and nonlinear equations. It is based on lectures delivered by the author at Moscow Institute of Physics and Technology, and Blekinge Institute of Technology. The prerequisites are linear algebra and mathematical analysis with one and several variables including elements of ordinary differential equations.

  • 64. Ibragimov, Nail H.
    An alternative presentation of the method of variation of parameters for higher-order equations2010In: Lobachevskii Journal of Mathematics, ISSN 1995-0802, Vol. 31, no 2, p. 174-191Article in journal (Refereed)
    Abstract [en]

    An alternative approach to Lagrange's method of variation of parameters is presented. Explicit formulas for solutions of arbitrary initial value problems for linear equations of the first, second and third order are provided. These formulas are as simple for practical using as the formula for roots of quadratic equations.

  • 65. Ibragimov, Nail H.
    Applications of group invariance methods to problems of transport phenomena2008Conference paper (Refereed)
    Abstract [en]

    An overview is given of lie group analysis and symmetry methods applied to transport phenomena

  • 66. Ibragimov, Nail H.
    Approximate representation of the de Sitter group2007In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 333, no 1, p. 329-346Article in journal (Refereed)
    Abstract [en]

    The purpose of this paper is to develop a theory of approximate representations of the de Sitter group considered as a perturbation of the Poincare group. This approach simplifies investigation of relativistic effects pertaining to the mechanics in the de Sitter universe. Utility of the approximate approach is manifest if one compares the transformations of the de Sitter group with their approximate representations.

  • 67.
    Ibragimov, Nail H.
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Science.
    Approximate transformation groups2008Book (Other academic)
    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.

  • 68. Ibragimov, Nail H.
    Archives of ALGA, volume 12004Collection (editor) (Other academic)
    Abstract [en]

    Volume 1 contains 3 articles: Nail H. Ibragimov, Equivalence groups and invariants of linear and non-linear equations; Nail H. Ibragimov and Sergey V. Meleshko, Linearization of third-order ordinary differential equations; Nail H. Ibragimov, Gazanfer Ünal and Claes Jogreús, Group analysis of stochastic differential systems: Approximate symmetries and conservation laws.

  • 69. Ibragimov, Nail H.
    Archives of ALGA. Volume 22005Collection (editor) (Other academic)
    Abstract [en]

    Volume 2 contains 3 articles: Ilir Berisha, Translation of Bäcklunds paper ”Surfaces of constant negative curvature”; Johan Erlandsson, "Survey of mathematical models in biology from point of view of Lie group analysis"; Niklas Säfström, "Group analysis of a tumour growth model"

  • 70. Ibragimov, Nail H.
    Archives of ALGA. Volume 42007Collection (editor) (Other academic)
    Abstract [en]

    Volume 4 contains articles: Nail H. Ibragimov, Ewald J.H. Wessels and George F.R. Ellis, Group classication of the Sachs equations for a radiating axisymmetric, non-rotating, vacuum space-time; Nail H. Ibragimov A discussion of conservation laws for over-determined systems with application to the Maxwell equations in vacuum; Nail H. Ibragimov, Quasi-self-adjoint differential equations; Nail H. Ibragimov and Salavat V. Khabirov, Existence of integrating factors for higher-order ordinary differential equations; Nail H. Ibragimov, Integration of second-order linear equations via linearization of Riccati's equations; Nail H. Ibragimov and Sergey V. Meleshko Linearization of second-order ordinary differential equations by changing the order; Nail H. Ibragimov and Sergey V. Meleshko Second-order ordinary differential equations equivalent to y''= H(y); Nail H. Ibragimov, Sergey V. Meleshko and Supaporn Suksern, Linearization of fourth-order ordinary differential equations by point transformations; Nail H. Ibragimov and Emrullah Yasar, Non-local conservation laws in fluid dynamics

  • 71. Ibragimov, Nail H.
    Archives of ALGA, volume 7/82010Collection (editor) (Other academic)
    Abstract [en]

    Volume 7/8 contains 3 articles by N.H. Ibragimov, an article by N.H. Ibragimov, E. D. Avdonina and an article by N. H. Ibragimov , R. Khamitova.

  • 72. Ibragimov, Nail H.
    Covariant method for solution of Cauchy's problem based on Lie group analysis and Leray's form2003Conference paper (Refereed)
    Abstract [en]

    The Lie group theory and Leray's form provide a covariant (i.e., independent of a coordinate system) method for the calculation of fundamental solutions for linear hyperbolic equations, and hence for the solution of Cauchy's problem. Here, the method is illustrated by the classical wave equation.

  • 73. Ibragimov, Nail H.
    Differentialekvationer och matematisk modellering2009Book (Other academic)
    Abstract [sv]

    Differentialekvationer och matematisk modellering" är avsedd för högskolestuderande på nybörjarnivå, exempelvis inom ingenjörs- och naturvetarprogram. Den kombinerar klassiska lösningsmetoder med moderna metoder från gruppanalys. Texten är generös med exempel och övningsuppgifter, vissa med fullständiga lösningar. Många vanliga läroböcker är begränsade till ett antal speciella differentialekvationer som man råkar kunna lösa med enkla recept. De flesta matematiska modeller inom naturvetenskap och teknik leder dock till ickelinjära differentialekvationer som man inte kan lösa med traditionella metoder. Med de redskap som ges i denna bok ligger många intressanta tillämpningar inom räckhåll.

  • 74. Ibragimov, Nail H.
    Equivalence groups and invariants of linear and non-linear equations2004In: Archives of ALGA, ISSN 1652-4934, Vol. 1, p. 9-69Article in journal (Refereed)
    Abstract [en]

    Recently I developed a systematic method for determining invariants of families of equations. The method is based on the infinitesimal approach and is applicable to algebraic and differential equations possessing finite or infinite equivalence groups. Moreover, it does not depend on the assumption of linearity of equations. The method was applied to variety of ordinary and partial differential equations. The present paper is aimed at discussing the main principles of the method and its applications with emphasis on the use of infinite Lie groups.

  • 75. Ibragimov, Nail H.
    Extension of Euler's method to parabolic equations2009In: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 14, no 4, p. 1157-1168Article in journal (Refereed)
    Abstract [en]

    Euler generalized d'Alembert's solution to a wide class of linear hyperbolic equations with two independent variables. He introduced in 1769 the quantities that were rediscovered by Laplace in 1773 and became known as the Laplace invariants. The present paper is devoted to an extension of Euler's method to linear parabolic equations with two independent variables. The new method allows one to derive an explicit formula for the general solution of a wide class of parabolic equations. In particular, the general solution of the Black-Scholes equation is obtained. (c) 2008 Elsevier B.V. All rights reserved.

  • 76. Ibragimov, Nail H.
    Extension of Laplace´s method2006Other (Other academic)
    Abstract [en]

    Översättning av Louise Petréns doktorsavhandling vid Lunds universitet

  • 77. Ibragimov, Nail H.
    Generalization of Euler´s Equations2009In: Quarterly of Applied Mathematics, ISSN 0033-569X , Vol. 67, no 2, p. 327-341Article in journal (Refereed)
    Abstract [en]

    A wide class of linear ordinary differential equations reducible to algebraic equations is studied. The method for solving all these equations is given. The new class depends essentially on two arbitrary functions and contains the constant coefficient equations and Euler's equations as particular cases.

  • 78. Ibragimov, Nail H.
    Integrating factors, adjoint equations and Lagrangians2006In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 318, no 2, p. 742-757Article in journal (Refereed)
    Abstract [en]

    Integrating factors and adjoint equations are determined for linear and non-linear differential equations of an arbitrary order. The new concept of an adjoint equation is used for construction of a Lagrangian for an arbitrary differential equation and for any system of differential equations where the number of equations is equal to the number of dependent variables. The method is illustrated by considering several equations traditionally regarded as equations without Lagrangians. Noether's theorem is applied to the Maxwell equations.

  • 79. Ibragimov, Nail H.
    Integrating factors for higher-order equations2007Conference paper (Refereed)
    Abstract [en]

    Integrating factors are determined for ordinary differential equations of an arbitrary order. In the case of first-order equations the new definition coincides with the usual definition of integrating factor introduced by A. Clairaut in 1739. It is shown that higher-order equations, unlike the first-order equations, can have an integrating factor even when they do not admit Lie point symmetries. The method is illustrated by several examples.

  • 80.
    Ibragimov, Nail H.
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Science.
    Integration of second-order linear equations via linearization of Riccati's equations2007In: Archives of ALGA, ISSN 1652-4934, Vol. 4, p. 71-75Article in journal (Refereed)
    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.

  • 81. Ibragimov, Nail H.
    INTEGRATION OF SYSTEMS OF FIRST-ORDER EQUATIONS ADMITTING NONLINEAR SUPERPOSITION2009Conference paper (Refereed)
    Abstract [en]

    Systems of two nonlinear ordinary differential equations of the first order admitting nonlinear superpositions are investigated using Lie's enumeration of groups on the plane. It is shown that the systems associated with two-dimensional Vessiot-Guldberg-Lie algebras can be integrated by quadrature upon introducing Lie's canonical variables. The knowledge of a symmetry group of a system in question is not needed in this approach. The systems associated with three-dimensional Vessiot-Guldberg-Lie algebras are classified into 13 standard forms 10 of which are integrable by quadratures and three are reduced to Riccati equations.

  • 82. Ibragimov, Nail H.
    Invariant Lagrangians and a new method of integration of nonlinear equations.2005In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 304, no 1, p. 212-235Article in journal (Refereed)
    Abstract [en]

    A method for solving the inverse variational problem for differential equations admitting a Lie group is presented. The method is used for determining invariant Lagrangians and integration of second-order nonlinear differential equations admitting two-dimensional non-commutative Lie algebras. The method of integration suggested here is different from Lie's classical method of integration of second-order ordinary differential equations based on canonical forms of two-dimensional Lie algebras. The new method reveals existence and significance of one-parameter families of singular solutions to nonlinear equations of second order.

  • 83. Ibragimov, Nail H.
    et al.
    ,
    Invariants of evolution equations2004Other (Other academic)
    Abstract [en]

    We investigate invariants of equations u_t=f(x,u)u_{xx}+g(x,u,u_x). We show that these equations have two functionally independent invariants of the second order.

  • 84. Ibragimov, Nail H.
    Invariants of hyperbolic equations: Solution of the Laplace problem.2004In: Journal of Applied Mechanics and Technical Physics, ISSN 0021-8944 (Print) 1573-8620 (Online), Vol. 45, no 2, p. 158-166Article in journal (Refereed)
    Abstract [en]

    The renown Laplace invariants were used by P.S. Laplace in 1773 in his integration theory of linear hyperbolic differential equations in two variables. In 1960, L.V.Ovsyannikov, tackling the problem of group classification of hyperbolic equations, came across two proper invariants that do not change under the most general equivalence transformation of hyperbolic equations. The question on existence of other invariants remained open. The present paper is dedicated to solution of Laplace's problem which consists in finding all invariants for hyperbolic equations and constructing a basis of invariants. Three new invariants of first and second order as well as operators of invariant differentiations are constructed. It is shown that the new invariants, together with Ovsyannikov's invariants, provide a basis of invariants, so that any invariant of higher order is a function of the basic invariants and their invariant derivatives.

  • 85. Ibragimov, Nail H.
    Laplace type invariants for parabolic equations2002In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, p. 125-133Article in journal (Refereed)
    Abstract [en]

    The Laplace invariants pertain to linear hyperbolic differential equations with two independent variables. They were discovered by Laplace in 1773 and used in his integration theory of hyperbolic equations. Cotton extended the Laplace invariants to elliptic equations in 1900. Cotton's invariants can be obtained from the Laplace invariants merely by the complex change of variables relating the elliptic and hyperbolic equations. To the best of my knowledge, the invariants for parabolic equations were not found thus far. The purpose of this paper is to fill this gap by considering what will be called Laplace type invariants (or seminvariants), i.e. the quantities that remain unaltered under the linear transformation of the dependent variable. Laplace type invariants are calculated here for all hyperbolic, elliptic and parabolic equations using the unified infinitesimal method. A new invariant is found for parabolic equations.

  • 86.
    Ibragimov, Nail H
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Natural Sciences.
    Lectures on the Theory of Group Properties of Differential Equations by L. V. Ovsyannikov2013Collection (editor) (Other academic)
    Abstract [en]

    These lecturers provide a clear introduction to Lie group methods for determining and using symmetries of differential equations, a variety of their applications in gas dynamics and other nonlinear models as well as the author's remarkable contribution to this classical subject. It contains material that is useful for students and teachers but cannot be found in modern texts. For example, the theory of partially invariant solutions developed by Ovsyannikov provides a powerful tool for solving systems of nonlinear differential equations and investigating complicated mathematical models.

  • 87. Ibragimov, Nail H.
    Lie group analysis: Classical Heritage2004Book (Other academic)
    Abstract [en]

    Classical works in Lie group analysis, e.g. important papers of S.Lie and A.V.Bäcklund are written in old German and somewhat old fashioned mathematical language. The present volume comprises translation into English of fundamental papers of S. Lie, A.V.Bäcklund and L.V. Ovsyannikov. I have selected here some of my favorite papers containing profound results significant for modern group analysis. The first paper imparts not only Lie's interesting view on the development of the general theory of differential equations but also contains Lie's theory of group invariant solutions. His second paper is dedicated to group classification of second-order linear partial differential equations in two variables and can serve as a concise practical guide to the group analysis of partial differential equations even today. The translation of Bäcklund's fundamental paper on non-existence of finite-order tangent transformations higher than first-order contains roots of the modern theory of Lie-Bäcklund transformation groups. Finally, Ovsyannikov's paper contains an essential development of the group classification of hyperbolic equations given in Lie's second paper. Moreover, it contains two proper invariants for hyperbolic equations discovered by Ovsyannikov.

  • 88.
    Ibragimov, Nail H.
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Natural Sciences.
    Lie Group Analysis of Moffatt´s model in Metallurgical Industry2011In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 18, no Suppl. 1Article in journal (Refereed)
    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.

  • 89.
    Ibragimov, Nail H.
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Natural Sciences.
    Method of conservation laws for constructing solutions to systems of PDEs2012In: Discontinuity, Nonlinearity, and Complexity, ISSN 2164−6376, Vol. 1, no 4Article in journal (Refereed)
    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.

  • 90. Ibragimov, Nail H.
    Nonlinear self-adjointness and conservation laws2011In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 44, no 43Article in journal (Refereed)
    Abstract [en]

    The general concept of nonlinear self-adjointness of differential equations is introduced. It includes the linear self-adjointness as a particular case. Moreover, it embraces the strict self-adjointness (definition 1) and quasi-self-adjointness introduced earlier by the author. It is shown that the equations possessing nonlinear self-adjointness can be written equivalently in a strictly self-adjoint form by using appropriate multipliers. All linear equations possess the property of nonlinear self-adjointness, and hence can be rewritten in a nonlinear strictly self-adjoint form. For example, the heat equation ut u = 0 becomes strictly self-adjoint after multiplying by u1. Conservation laws associated with symmetries are given in an explicit form for all nonlinearly self-adjoint partial differential equations and systems.

  • 91. Ibragimov, Nail H.
    On conservation laws of electrodynamics2007Other (Other academic)
    Abstract [en]

    Översättning av E. Bessel Hagens artikel om elektrodynamik i Mathematische Annalen 1921

  • 92. Ibragimov, Nail H.
    Optimal system of invariant solutions for the Burgers equation2008Conference paper (Refereed)
    Abstract [en]

    An optimal system of invariant solution is derived for the Burger´s equation.

  • 93. Ibragimov, Nail H.
    Quasi-self-adjoint differential equations2007In: Archives of ALGA, ISSN 1652-4934, Vol. 4, p. 55-60Article in journal (Refereed)
    Abstract [en]

    The concept of self-adjoint equations is useful for calculating conservation laws. The aim of the present paper is extend the property of self-adjoint equations to a wider class of equations called quasi-self-adjoint.

  • 94. Ibragimov, Nail H.
    Selected works. Volume 12006Book (Other academic)
    Abstract [en]

    Volume 1 contains papers written during 1966-1992. The main topics in this volume include Lie group analysis of differential equations, generalized motions in Riemannian spaces, Huygens’ principle, theory of Lie-Bäcklund transformation groups, symmetries in fluid dynamics and mathematical physics.

  • 95. Ibragimov, Nail H.
    Selected works. Volume 22006Book (Other academic)
    Abstract [en]

    Volume II contains the MSc and PhD theses and papers written during 1986-2000. The main topics in this volume include approximate and nonlocal symmetries, method of preliminary group classification, Galilean principle in diffusion problems, dynamics in the de Sitter space, nonlinear superposition.

  • 96. Ibragimov, Nail H.
    Selected works. Volume 32008Book (Other academic)
    Abstract [en]

    Volume III contains Dr.Sci thesis (1972) and papers written during 1987-2004. The main topics in this volume include Lie groups in mathematical physics and mathematical modelling, fluid dynamics, Huygens’ principle and approximate symmetries of equations with small parameter.

  • 97.
    Ibragimov, Nail H.
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Science.
    Selected works. Volume 42009Book (Other academic)
    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.

  • 98. Ibragimov, Nail H.
    Symmetries, Lagrangian and Conservation Laws for the Maxwell Equations2009In: Acta Applicandae Mathematicae, ISSN 0167-8019 , Vol. 105, no 2, p. 157-187Article in journal (Refereed)
    Abstract [en]

    The evolution equations of Maxwell's equations has a Lagrangian written in terms of the electric E and magnetic H fields, but admit neither Lorentz nor conformal transformations. The additional equations del center dot E=0, del center dot H=0 guarantee the Lorentz and conformal invariance, but the resulting system is overdetermined, and hence does not have a Lagrangian. The aim of the present paper is to attain a harmony between these two contradictory properties and provide a correspondence between symmetries and conservation laws using the Lagrangian for the evolutionary part of Maxwell's equations.

  • 99. Ibragimov, Nail H.
    Tensors and Riemannian geometry with applications to differential equations and relativity2008Book (Other academic)
    Abstract [en]

    These notes are based on my lectures delivered in Russia (Novosibirsk and Moscow State Universities, 1972-1973 and 1988-1990), France (Collège de France, 1980), South Africa (University of the Witwatersrand, 1995-1997) and Sweden (Blekinge Institute of Technology, during 2004-2008).

  • 100. Ibragimov, Nail H.
    The answer to the question put to me by L.V. Ovsyannikov 33 years ago2006In: Archives of ALGA, ISSN 1652-4934, Vol. 3Article in journal (Refereed)
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