| 1. |
- Avdonina, Elena D., et al.
(författare)
-
Heat conduction in anisotropic media : Nonlinear self-adjointness and conservation laws
- 2012
-
Ingår i: Discontinuity, Nonlinearity and Complexity. - : L & H Scientific Publishing. - 2164-6376. ; 1:3, s. 237-251
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
|
|
| 2. |
- Berwald, J. J., et al.
(författare)
-
Automatic recognition and tagging of topologically different regimes in dynamical systems
- 2014
-
Ingår i: Discontinuity, Nonlinearity, and Complexity. - : L and H Scientific Publishing, LLC. - 2164-6376. ; 3:4, s. 413-426
-
Tidskriftsartikel (refereegranskat)abstract
- Complex systems are commonly modeled using nonlinear dynamical systems. These models are often high-dimensional and chaotic. An important goal in studying physical systems through the lens of mathematical models is to determine when the system undergoes changes in qualitative behavior. A detailed description of the dynamics can be difficult or impossible to obtain for high-dimensional and chaotic systems. Therefore, a more sensible goal is to recognize and mark transitions of a system between qualitatively different regimes of behavior. In practice, one is interested in developing techniques for detection of such transitions from sparse observations, possibly contaminated by noise. In this paper we develop a framework to accurately tag different regimes of complex systems based on topological features. In particular, our framework works with a high degree of success in picking out a cyclically orbiting regime from a stationary equilibrium regime in high-dimensional stochastic dynamical systems.
|
|
| 3. |
- Ibragimov, Nail, 1939-2018, et al.
(författare)
-
Group Classification and Solutions of a Mathematical Model from Tumour Biology
- 2021
-
Ingår i: Discontinuity, Nonlinearity, and Complexity. - : L and H Scientific Publishing, LLC. - 2164-6376. ; 10:4, s. 605-615
-
Tidskriftsartikel (refereegranskat)abstract
- We are interested in symmetries of a mathematical model of a malignant tumour dynamics due to haptotaxis. The model is formulated as a system of two nonlinear partial differential equations with two independent variables and contains two unknown functions of the dependent variables. When the unknown functions are arbitrary, the model has only two symmetries. These symmetries allow to investigate only travelling wave solutions. The aim of the present paper is to make the group classification of the mathematical model under consideration and find the cases when the model has additional symmetries and hence additional group invariant solutions. © 2021 L&H Scientific Publishing, LLC. All Rights Reserved.
|
|
| 4. |
- Ibragimov, Nail H., et al.
(författare)
-
Conservation Laws in Thomas’s Model of Ion Exchange in a Heterogeneous Solution
- 2013
-
Ingår i: Discontinuity, Nonlinearity and Complexity. - : L&H Scientific Publishing. - 2164-6376. ; 2:2
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
|
|
| 5. |
- Ibragimov, Nail H.
(författare)
-
Method of conservation laws for constructing solutions to systems of PDEs
- 2012
-
Ingår i: Discontinuity, Nonlinearity, and Complexity. - : L&H Scientific Publishing. - 2164-6376. ; 1:4
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
|
|
