Topic
Complete group
About: Complete group is a research topic. Over the lifetime, 91 publications have been published within this topic receiving 1105 citations.
Papers published on a yearly basis
Papers
TL;DR: In this article, a complete group classification of the wave equation c 2(x)uxx−utt=0 (I) and its equivalent system vt=ux, c 2 (x)vx=ut (II) when the wave speed c(x )≠const.
Abstract: A complete group classification is given of both the wave equation c2(x)uxx−utt=0 (I) and its equivalent system vt=ux, c2(x)vx=ut (II) when the wave speed c(x)≠const. Equations (I) and (II) admit either a two‐ or four‐parameter group. For the exceptional case, c(x)=(Ax+B)2, equation (I) admits an infinite group. Equations (I) and (II) do not always admit the same group for a given c(x): The group for (I) can have more parameters or fewer parameters than that for (II); moreover, the groups can be different with the same number of parameters. Separately for (I) and (II), all possible c(x) that admit a four‐parameter group are found explicitly. The corresponding invariant (similarity) solutions are considered. Some of these wave speeds have realistic physical properties: c(x) varies monotonically from one positive constant to another positive constant as x goes from −∞ to +∞.
105 citations
TL;DR: In this article, the complete preliminary group classification of a class of nonlinear wave equations was carried out via the classification of one-dimensional Lie symmetry extensions related to a fixed finite-dimensional subalgebra of the class under consideration.
Abstract: Preliminary group classification became prominent as an approach to symmetry analysis of differential equations due to the paper by Ibragimov, Torrisi and Valenti [J. Math. Phys. 32, 2988-2995] in which partial preliminary group classification of a class of nonlinear wave equations was carried out via the classification of one-dimensional Lie symmetry extensions related to a fixed finite-dimensional subalgebra of the infinite-dimensional equivalence algebra of the class under consideration. In the present paper we implement, up to both usual and general point equivalence, the complete group classification of the same class using the algebraic method of group classification. This includes the complete preliminary group classification of the class and finding Lie symmetry extensions which are not associated with subalgebras of the equivalence algebra. The complete preliminary group classification is based on listing all inequivalent subalgebras of the whole infinite-dimensional equivalence algebra whose projections are qualified as maximal extensions of the kernel algebra. The set of admissible point transformations of the class is exhaustively described in terms of the partition of the class into normalized subclasses. A version of the algebraic method for finding the complete equivalence groups of a general class of differential equations is proposed.
87 citations
TL;DR: In this article, the authors discuss the classical statement of group classification problem and some extensions in the general case, and carry out the complete extended group classification for a class of (1+1)-dimensional nonlinear diffusion-convection equations with coefficients depending on the space variable.
Abstract: We discuss the classical statement of group classification problem and some its extensions in the general case. After that, we carry out the complete extended group classification for a class of (1+1)-dimensional nonlinear diffusion--convection equations with coefficients depending on the space variable. At first, we construct the usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements. The extended equivalence group has interesting structure since it contains a non-trivial subgroup of non-local gauge equivalence transformations. The complete group classification of the class under consideration is carried out with respect to the extended equivalence group and with respect to the set of all point transformations. Usage of extended equivalence and correct choice of gauges of arbitrary elements play the major role for simple and clear formulation of the final results. The set of admissible transformations of this class is preliminary investigated.
70 citations
TL;DR: In this article, the complete preliminary group classification of a class of nonlinear wave equations was carried out via the classification of one-dimensional Lie symmetry extensions related to a fixed finite-dimensional subalgebra of the equivalence algebra of the class under consideration.
Abstract: Preliminary group classification became a prominent tool in the symmetry analysis of differential equations due to the paper by Ibragimov, Torrisi, and Valenti [J. Math. Phys. 32, 2988–2995 (1991)10.1063/1.529042]. In this paper the partial preliminary group classification of a class of nonlinear wave equations was carried out via the classification of one-dimensional Lie symmetry extensions related to a fixed finite-dimensional subalgebra of the infinite-dimensional equivalence algebra of the class under consideration. In the present paper we implement the complete group classification of the same class up to both usual and general point equivalence using the algebraic method of group classification. This includes the complete preliminary group classification of the class and finding those Lie symmetry extensions which are not associated with subalgebras of the equivalence algebra. The complete preliminary group classification is based on listing all inequivalent subalgebras of the whole infinite-dimensi...
69 citations
TL;DR: In this paper, the authors discuss the classical statement of group classification problem and some extensions in the general case, and carry out the complete extended group classification for a class of (1 + 1)-dimensional nonlinear diffusion-convection equations with coefficients depending on the space variable.
Abstract: We discuss the classical statement of group classification problem and some its extensions in the general case. After that, we carry out the complete extended group classification for a class of (1 + 1)-dimensional nonlinear diffusion-convection equations with coefficients depending on the space variable. At first, we construct the usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements. The extended equivalence group has interesting structure since it contains a non-trivial subgroup of non-local gauge equivalence transformations. The complete group classification of the class under consideration is carried out with respect to the extended equivalence group and with respect to the set of all point transformations. Usage of extended equivalence and correct choice of gauges of arbitrary elements play the major role for simple and clear formulation of the final results. The set of admissible transformations of this class is preliminary investigated.
64 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2022 | 1 |
| 2021 | 3 |
| 2020 | 5 |
| 2019 | 8 |
| 2018 | 3 |
| 2017 | 5 |