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  • 1. Bruzon, MS
    et al.
    Gandarias, ML
    Ibragimov, Nail H.
    Self-adjoint sub-classes of generalized thin film equations2009Ingår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 357, nr 1, s. 307-313Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this work we consider a class of fourth-order nonlinear partial differential equation containing several tin-specified coefficient functions of the dependent variable which encapsulates various mathematical models used, e.g. for describing the dynamics of thin liquid films. We determine the subclasses of these equations which are self-adjoint. By using a general theorem on conservation laws proved by one of the authors (NHI) we find conservation laws for some of these partial differential equations without Classical Lagrangians.

  • 2. Ibragimov, Nail H.
    A new conservation theorem2007Ingår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 333, nr 1, s. 311-328Artikel i tidskrift (Refereegranskat)
    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.

  • 3. Ibragimov, Nail H.
    Approximate representation of the de Sitter group2007Ingår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 333, nr 1, s. 329-346Artikel i tidskrift (Refereegranskat)
    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.

  • 4. Ibragimov, Nail H.
    Integrating factors, adjoint equations and Lagrangians2006Ingår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 318, nr 2, s. 742-757Artikel i tidskrift (Refereegranskat)
    Abstract [sv]

    Online: http://www.bth.se/ihn/alga.nsf/sidor/cc32aec8706331d0c125716b0041d97c!OpenDocume nt

  • 5. Ibragimov, Nail H.
    Invariant Lagrangians and a new method of integration of nonlinear equations.2005Ingår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 304, nr 1, s. 212-235Artikel i tidskrift (Refereegranskat)
    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.

  • 6. Ibragimov, Nail H.
    et al.
    Kolsrud, T.
    Lagrangian approach to evolution equations: symmetries and conservation laws.2004Ingår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 36, nr 1, s. 29-40Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We show that one can apply a Lagrangian approach to certain evolution equations by considering them together with their associated equations. Consequently, one can employ Noether's theorem and derive conservation laws from symmetries of coupled systems of evolution equations. We discuss in detail the linear and non-linear heat equations as well as the Burgers equation and obtain new non-local conservation laws for the non-linear heat and the Burgers equations by extending their symmetries to the associated equations. We also provide Lagrangians for non-linear Schrödinger and Korteweg-de Vries type systems.

  • 7. 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.

  • 8. 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.

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