Lie Symmetries of Differential Equations by Computer Algebra (original) (raw)
Related papers
Classical and Nonclassical Symmetries of a Generalized Boussinesq Equation
Journal of Nonlinear Mathematical Physics, 1998
We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions f (u) for which this equation admit either the classical or the nonclassical method. The reductions obtained are derived. Some new exact solutions can be derived.
Group analysis of differential equations: A new type of Lie symmetries
2018
We construct a new type of symmetries using the regular Lie symmetries as the basis, which we call Modified symmetries. The contrast is that while Lie symmetries arise from point transformations, the Modified symmetries result from the transformations of the neighborhood of that point. The similarity is that as the neighborhood contracts to the central point, the two sets of symmetries become indistinguishable from one another, meaning the Modified symmetries will cease to exist if there were no Lie symmetries in the first place. The advantage is that the group invariant solutions are not affected by all these, because they result from ratios of the symmetries, and will therefore exist in the absence of Lie symmetries, i.e,. zero symmetries. Zero symmetries lead to 0/0, and no further. With the Modified symmetries we get f (x, ω)/g(x, ω) = 0/0 as ω goes to zero, and there are numerous mathematical techniques through which this can be resolved. We develop this concept using tensors a...
1993
We give a method for using explicitly known Lie symmetries of a system of differential equations to help find more symmetries of the system. A Lie (or infinitesimal) symmetry of a system of differential equations is a transformation of its dependent and independent variables, depending on continuous parameters, which maps any solution of the system to another solution of the same systkun. Infinitesimal Lie symmetries of a system of differential equations arise as solutions of a related system of linear homogeneous partial differential equations called infinitesimal determining equations. The importance of symmetries in applications has prompted the development of many software packages to derive and attempt to integrate infinitesimal determining equations. For a. given system of differential equations we usually have a priori explicit knowledge of many symmetries of the system because of their simple form or the physical origin of the system. Current methods for finding symmetries d...
Discrete Symmetry Transformations of Third Order Ordinary Differential Equations and Applications
2020
Third order ordinary differential equations have already been classified by the Lie algebra they admit. Invariant equations corresponding to these Lie algebras are also available in the literature [17]. In this paper, list of all discrete symmetries corresponding to these invariant ordinary differential equations, are obtained. Some particular examples are given to show the significance of the work.
Applied Mathematics and Computation, 2011
The Lie-group formalism is applied to investigate the symmetries of the Benjamin-Bona-Mahony (BBM) equation with variable coefficients. We derive the infinitesimals and the admissible forms of the coefficients that admit the classical symmetry group. The ordinary differential equations deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained.
Lie symmetry analysis of the two-dimensional generalized Kuramoto-Sivashinsky equation
Mathematical Sciences, 2012
In this paper, a detailed analysis of an important nonlinear model system, the two dimensional generalized Kuramoto-Sivashinsky (2D gKS) equation, is presented by group analysis. Methods: The basic Lie symmetry method is applied in order to determine the general symmetry group of our analyzed nonlinear model. Results: The symmetry group of the equation and some results related to the algebraic structure of the Lie algebra of symmetries are obtained. Also, a complete classification of the subalgebras of the symmetry algebra is resulted. Conclusions: It is proved that the Lie algebra of symmetries admits no three dimensional subalgebra. Mainly, all the group invariant solutions and the similarity reduced equations associated to the infinitesimal symmetries are obtained.