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| 2. |
- Anco, S., et al.
(author)
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Solutions of gasdynamic equations associated with classical and new conservation laws
- 2015
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In: Applied Mathematics and Computation. - : Elsevier. - 0096-3003 .- 1873-5649. ; 268, s. 52-58
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Journal article (peer-reviewed)abstract
- 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.
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| 3. |
- Anco, S., et al.
(author)
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Symmetries and conservation laws of the generalized Krichever-Novikov equation
- 2016
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In: Journal of Physics A. - : Institute of Physics Publishing (IOPP). - 1751-8113 .- 1751-8121. ; 49:10
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Journal article (peer-reviewed)abstract
- 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.
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| 4. |
- Baikov, V. A., et al.
(author)
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Conservation laws for two-phase filtration models
- 2014
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In: Communications in nonlinear science & numerical simulation. - : Elsevier. - 1007-5704 .- 1878-7274. ; 19:2, s. 383-389
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Journal article (peer-reviewed)abstract
- 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.
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| 6. |
- Ibragimov, Nail, et al.
(author)
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Group classification and conservation laws of anisotropic wave equations with a source
- 2016
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In: Journal of Mathematical Physics. - : American Institute of Physics (AIP). - 0022-2488 .- 1089-7658. ; 57:8
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Journal article (peer-reviewed)abstract
- 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.
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| 7. |
- Ibragimov, Nail H., et al.
(author)
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Existence of integrating factors for higher-order ordinary differential equations
- 2007
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In: Archives of ALGA. - Karlskrona, Sweden : ALGA publications, BTH. - 1652-4934. ; 4, s. 61-70
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Journal article (peer-reviewed)abstract
- We investigate the concept of integrating factors for higher-order ordinary differential equations introduced recently by one of the authors (NHI). The integrating factors for equations of order higher than one are determined by overdetermined systems. Therefore one can expect that not all higher-order equations have integrating factors. We prove in this paper that in fact they have. Moreover, we demonstrate that every equation of order n has precisely n functionally independent integrating factors.
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| 8. |
- Ibragimov, Nail H., 1939-, et al.
(author)
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Group analysis of kinetic equations in a non-linear thermal transport problem
- 2015
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In: International Journal of Non-Linear Mechanics. - : Pergamon Press. - 0020-7462 .- 1878-5638. ; 71, s. 1-7
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Journal article (peer-reviewed)abstract
- 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.
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| 9. |
- Ibragimov, Nail H., et al.
(author)
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Introduction to differential equations
- 2006
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Book (other academic/artistic)abstract
- This book is based on lectures in differential equations delivered by the authors, namely, by A. Al-Hammadi at the University of Bahrain and by N. Ibragimov at the Blekinge Institute of Technology, Sweden. It is designed for a one semester course in differential equations. We assume that the reader has taken standard courses in mathematical analysis and has encountered basic concepts from linear algebra such as vectors, matrices and algebraic linear systems. Our aim is to provide students with an easy to follow introduction to the subject containing classical devices enriched by fundamentals of Lie group methods given in Chapter 9.
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