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Ibragimov, Nail
Alternative names
Publications (10 of 57) Show all publications
Ibragimov, N. (2018). Conservation laws and non-invariant solutions of anisotropic wave equations with a source. Nonlinear Analysis, 40, 82-94
Open this publication in new window or tab >>Conservation laws and non-invariant solutions of anisotropic wave equations with a source
2018 (English)In: Nonlinear Analysis, ISSN 1468-1218, Vol. 40, p. 82-94Article in journal (Refereed) Published
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

Place, publisher, year, edition, pages
Elsevier, 2018
Keywords
Anisotropic wave equation, Non-invariant solutions, Acoustics, Anisotropic media, Anisotropy, Nonlinear equations, Physical properties, Application-oriented, Biomedical acoustics, Conservation law, Exact solution, Invariant solutions, Nonlinear waves, Steady state, Traveling wave solution, Wave equations
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:bth-15213 (URN)10.1016/j.nonrwa.2017.08.005 (DOI)2-s2.0-85029605943 (Scopus ID)
Available from: 2017-09-29 Created: 2017-09-29 Last updated: 2017-09-29Bibliographically approved
Ibragimov, N., Ibragimov, R. N. & Kovalev, V. F. (2018). INVARIANT SOLUTIONS AND SHOCK ATMOSPHERIC WAVES IN A THIN CIRCULAR LAYER. Mathematical Modelling of Natural Phenomena, 13(2), Article ID UNSP 19.
Open this publication in new window or tab >>INVARIANT SOLUTIONS AND SHOCK ATMOSPHERIC WAVES IN A THIN CIRCULAR LAYER
2018 (English)In: Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, E-ISSN 1760-6101, Vol. 13, no 2, article id UNSP 19Article in journal (Refereed) Published
Abstract [en]

The objective of this paper is to investigate the nonlinear mathematical model describing equatorial waves from Lie group analysis point of view in order to understand the nature of shallow water model theory, which is associated to planetary equatorial waves. Such waves correspond to the Cauchy-Poisson free boundary problem on the nonstationary motion of a perfect incompressible fluid circulating around a solid circle of a large radius.

Place, publisher, year, edition, pages
EDP SCIENCES S A, 2018
Keywords
Method of conservation laws, nonlinear PDEs, symmetries, exact solutions, chaplygin gas
National Category
Other Mathematics
Identifiers
urn:nbn:se:bth-17016 (URN)10.1051/mmnp/2018021 (DOI)000443894600004 ()
Available from: 2018-09-20 Created: 2018-09-20 Last updated: 2018-09-20Bibliographically approved
Ibragimov, N., Karimova, E. N. & Galiakberova, L. R. (2017). Chaplygin gas motions associated with nonlocal conservation laws. JOURNAL OF COUPLED SYSTEMS AND MULTISCALE DYNAMICS, 5(2-4), 63-68
Open this publication in new window or tab >>Chaplygin gas motions associated with nonlocal conservation laws
2017 (English)In: JOURNAL OF COUPLED SYSTEMS AND MULTISCALE DYNAMICS, ISSN 2330-152X, Vol. 5, no 2-4, p. 63-68Article in journal (Refereed) Published
Abstract [en]

The recent method of conservation laws for constructing exact solutions of partial differential equations is applied to the nonlocal conservation laws of the Chaplygin gas. The nonlocal conservation laws provide twenty different types of exact solutions. They are listed in three tables. Seven types of these solutions describe isentropic flows satisfying Chaplygin's relation between the pressure and density. All solutions are written in the explicit form and contain either arbitrary functions or arbitrary constants.

Place, publisher, year, edition, pages
AMER SCIENTIFIC PUBLISHERS, 2017
Keywords
Chaplygin Gas, Nonlocal Conservation Laws, Exact Solutions, Method of Conservation Laws
National Category
Other Mathematics
Identifiers
urn:nbn:se:bth-16881 (URN)10.1166/jcsmd.2017.1123 (DOI)000437469000001 ()
Available from: 2018-08-20 Created: 2018-08-20 Last updated: 2018-08-24Bibliographically approved
Ibragimov, N. & Gainetdinova, A. A. A. (2017). Classification and integration of four-dimensional dynamical systems admitting non-linear superposition. International Journal of Non-Linear Mechanics, 90, 50-71
Open this publication in new window or tab >>Classification and integration of four-dimensional dynamical systems admitting non-linear superposition
2017 (English)In: International Journal of Non-Linear Mechanics, ISSN 0020-7462, Vol. 90, p. 50-71Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier, 2017
Keywords
Classification, Dynamical system, Integration, Non-linear superposition, Standard forms of dynamical systems, Vessiot-Guldberg-Lie algebra, Algebra, Classification (of information), Linear control systems, Integration method, Lie Algebra, Method of integration, Non linear, Dynamical systems
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-13866 (URN)10.1016/j.ijnonlinmec.2017.01.008 (DOI)000395220500006 ()2-s2.0-85009917851 (Scopus ID)
Available from: 2017-02-07 Created: 2017-02-03 Last updated: 2017-09-20Bibliographically approved
Ibragimov, N. & Gainetdinova, A. (2017). Three-dimensional dynamical systems with four-dimensional vessiot-guldberg-lie algebras. The Journal of Applied Analysis and Computation, 7(3), 872-883
Open this publication in new window or tab >>Three-dimensional dynamical systems with four-dimensional vessiot-guldberg-lie algebras
2017 (English)In: The Journal of Applied Analysis and Computation, ISSN 2156-907X, E-ISSN 2158-5644, Vol. 7, no 3, p. 872-883Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Wilmington Scientific Publisher, 2017
Keywords
Nonlinear superposition of solutions, Standard forms of L4, Time dependent dynamical system, Vessiot-guldberg-lie algebra L4
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-14676 (URN)10.11948/2017055 (DOI)000405793700006 ()
Available from: 2017-06-22 Created: 2017-06-22 Last updated: 2017-08-22Bibliographically approved
Ibragimov, R. N., Ibragimov, N. & Galiakberova, L. R. (2016). Conservation laws and invariant solutions of the non-linear governing equations associated with a thermodynamic model of a rotating detonation engines with Korobeinikov's chemical source term. International Journal of Non-Linear Mechanics, 78, 29-34
Open this publication in new window or tab >>Conservation laws and invariant solutions of the non-linear governing equations associated with a thermodynamic model of a rotating detonation engines with Korobeinikov's chemical source term
2016 (English)In: International Journal of Non-Linear Mechanics, ISSN 0020-7462, E-ISSN 1878-5638, Vol. 78, p. 29-34Article in journal (Refereed) Published
Abstract [en]

The non-linear governing gas dynamics equations that are used as a descriptor of a rotating detonation engine are investigated from the group theoretical standpoint. The equations incorporate approximation of Korobeinikov's chemical reaction model that are used to describe the two-dimensional detonation field on a surface of a two-dimensional cylindrical chamber without thickness. The transformations that leave the equations invariant are found. On the basis of these transformations, the conservation equations were constructed and the invariant solutions were obtained for specific form of the equation of state, for which the equations are non-linearly self-adjoint. The invariant solutions are given in terms of the functions that satisfy non-linear ordinary differential equations. The above reduction simplifies the analysis of the original non-linear system of partial differential equations on a surface of rotating cylinder. (C) 2015 Elsevier Ltd. All rights reserved.

Keywords
Rotating detonation engine, Invariant solutions, Conservation laws
National Category
Other Mathematics
Identifiers
urn:nbn:se:bth-11535 (URN)10.1016/j.ijnonlinmec.2015.09.015 (DOI)000366792700004 ()
Available from: 2016-02-02 Created: 2016-02-02 Last updated: 2017-11-30Bibliographically approved
Ibragimov, N., Gandarias, M., Galiakberova, L., Bruzon, M. & Avdonina, E. (2016). Group classification and conservation laws of anisotropic wave equations with a source. Journal of Mathematical Physics, 57(8), Article ID 083504.
Open this publication in new window or tab >>Group classification and conservation laws of anisotropic wave equations with a source
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2016 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 57, no 8, article id 083504Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
American Institute of Physics (AIP), 2016
National Category
Other Mathematics
Identifiers
urn:nbn:se:bth-12978 (URN)10.1063/1.4960800 (DOI)000383917300054 ()2-s2.0-84982311630 (Scopus ID)
Available from: 2016-08-31 Created: 2016-08-31 Last updated: 2017-11-21Bibliographically approved
Ibragimov, N. (2016). Integration of dynamical systems admitting nonlinear superposition. JOURNAL OF COUPLED SYSTEMS AND MULTISCALE DYNAMICS, 4(2), 91-106
Open this publication in new window or tab >>Integration of dynamical systems admitting nonlinear superposition
2016 (English)In: JOURNAL OF COUPLED SYSTEMS AND MULTISCALE DYNAMICS, ISSN 2330-152X, Vol. 4, no 2, p. 91-106Article, review/survey (Refereed) Published
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.

Place, publisher, year, edition, pages
American Scientific Publishers, 2016
Keywords
Dynamical Systems, Nonlinear Superposition, Vessiot-Guldberg-Lie Algebra, Semi-Separable Systems, Integration Method, Perturbed Dynamical Systems, Approximate Nonlinear Superposition
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-13621 (URN)10.1166/jcsmd.2016.1098 (DOI)000388585400001 ()
Available from: 2016-12-16 Created: 2016-12-16 Last updated: 2016-12-19Bibliographically approved
Anco, S., Avdonina, E. D., Gainetdinova, A., Galiakberova, L. R., Ibragimov, N. & Wolf, T. (2016). Symmetries and conservation laws of the generalized Krichever-Novikov equation. Journal of Physics A: Mathematical and Theoretical, 49(10), Article ID 105201.
Open this publication in new window or tab >>Symmetries and conservation laws of the generalized Krichever-Novikov equation
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2016 (English)In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 49, no 10, article id 105201Article in journal (Refereed) Published
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.

Keywords
integrable system, conservation law, symmetry, Krichever-Novikov equation
National Category
Mathematics
Identifiers
urn:nbn:se:bth-11660 (URN)10.1088/1751-8113/49/10/105201 (DOI)000369256700007 ()
Available from: 2016-03-02 Created: 2016-02-29 Last updated: 2017-11-30Bibliographically approved
Ibragimov, N. & Gainetdinova, A. A. (2016). Three-dimensional dynamical systems admitting nonlinear superposition with three-dimensional Vessiot-Guldberg-Lie algebras. Applied Mathematics Letters, 52, 126-131
Open this publication in new window or tab >>Three-dimensional dynamical systems admitting nonlinear superposition with three-dimensional Vessiot-Guldberg-Lie algebras
2016 (English)In: Applied Mathematics Letters, ISSN 0893-9659, E-ISSN 1873-5452, Vol. 52, p. 126-131Article in journal (Refereed) Published
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.

Keywords
Dynamical system, Nonlinear superposition, Vessiot-Guldberg-Lie algebra
National Category
Mathematics Algebra and Logic
Identifiers
urn:nbn:se:bth-11344 (URN)10.1016/j.aml.2015.08.012 (DOI)000364892900019 ()
Available from: 2016-01-04 Created: 2016-01-04 Last updated: 2017-06-22Bibliographically approved
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