Lie Symmetries of Differential Equations by Computer Algebra (original) (raw)
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...
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.
Equivalence Classes, Symmetries and Solutions of Linear Third-order Differential Equations
2002
The subject of this article are third-order differential equations that may be linearized by a variable change. To this end, at first the equivalence classes of linear equations are completely described. Thereafter it is shown how they combine into symmetry classes that are determined by the various symmetry types. An algorithm is presented allowing it to transform linearizable equations by hyperexponential transformations into linear form from which solutions may be obtained more easily. Several examples are worked out in detail.
Solutions of systems of ordinary differential equations using invariants of symmetry groups
AIP Conference Proceedings, 2019
We investigate the use of invariants of the admitted Lie groups of transformation in finding solutions of the systems of ordinary differential equations (ODEs). Bluman's theorem (1990) of invariant solutions of ODEs is extended for systems of ODEs. Differential invariants of a Lie group are used in reducing order of the given system. Examples are given to illustrate the methods.
Point-Symmetry Pseudogroup, Lie Reductions and Exact Solutions of Boiti–Leon–Pempinelli System
2023
We carry out extended symmetry analysis of the (1+2)-dimensional Boiti-Leon-Pempinelli system, which corrects, enhances and generalizes many results existing in the literature. The point-symmetry pseudogroup of this system is computed using an original megaideal-based version of the algebraic method. A number of meticulously selected differential constraints allow us to construct families of exact solutions of this system, which are significantly larger than all known ones. After classifying one-and two-dimensional subalgebras of the entire (infinite-dimensional) maximal Lie invariance algebra of this system, we study only its essential Lie reductions, which give solutions beyond the above solution families. Among reductions of the Boiti-Leon-Pempinelli system using differential constraints or Lie symmetries, we identify a number of famous partial and ordinary differential equations. We also show how all the constructed solution families can significantly be extended by Laplace and Darboux transformations.
Expanded Lie Group Transformations and Similarity Reductions of Differential Equations
2001
Continuous groups of transformations acting on the expanded space of variables, which in-cludes the equation parameters in addition to independent and dependent variables, areconsidered. It is shown that the use of the expanded transformations enables one to enrichthe concept of similarity reductions of PDEs. The expanded similarity reductions of diffe-rential equations may be used as a tool for finding changes of variables, which convert theoriginal PDE into another (presumably simpler) PDE. A new view on the common simi-larity reductions as the singular expanded group transformations may be used for definingreductions of a PDE to a specific target ODE.
Lie point algebraic classification of linear third order ODEs using differential invariants
PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND TECHNOLOGY 2018 (MATHTECH2018): Innovative Technologies for Mathematics & Mathematics for Technological Innovation, 2019
We obtain equivalence transformations for general linear third order ordinary differential equation (ODE) using Lie infinitesimal method. These equivalence transformations are then employed to deduce associated invariants. Derived invariants are used to reduce the linear third order ODEs with variable coefficients to simpler equations of this family with constant coefficients. It is shown that these reductions help in identifying the canonical forms of the linear third order ODEs with 4, 5, and 7-dimensional Lie point symmetry algebras. Though this algebraic classification for linear third order ODEs has already been presented, here differential invariants are shown to provide an alternate procedure to recover the same. We provide MAPLE codes used to derive the said invariants.
Similarity solutions of the Konopelchenko–Dubrovsky system using Lie group theory
Computers & Mathematics with Applications, 2016
This research deals with the similarity solutions of (2 + 1)-dimensional Konopelchenko-Dubrovsky (KD) system. Solutions so obtained are derived by using similarity transformations method based on Lie group theory. The method reduces the number of independent variables by one exploiting Lie symmetries and using invariance property. Thus, the KD system can further be reduced to a new system of ordinary differential equations. Under a suitable choice of functions and the arbitrary constants, these new equations yield the explicit solutions of the KD system which are discussed in the Similarity Solutions section of the article. Moreover, the physical analysis of the solutions is illustrated graphically in the Analysis and Discussions section based on numerical simulations in order to highlight the importance of the study.
Communications in Nonlinear Science and Numerical Simulation, 2012
A (2+1) dimensional Broer-Kaup system which is obtained from the constraints of the KP equation is of importance in mathematical physics field. In this paper, the Painlevé analysis of (2+1)-variable coefficients Broer-Kaup (VCBK) equation is performed by the Weiss-Kruskal approach to check the Painlevé property. Similarity reductions of the VCBK equation to one-dimensional partial differential equations including Burger's equation are investigated by the Lie classical method. The Lie group formalism is applied again on one of the investigated partial differential equation to derive symmetries, and the ordinary differential equations deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained.
Symmetries and Related Topics in Differential and Difference Equations
Contemporary Mathematics, 2011
Let k be a differential field and let [A] : Y ′ = A Y be a linear differential system where A ∈ Mat(n , k). We say that A is in a reduced form if A ∈ g(k) where g is the Lie algebra of [A] andk denotes the algebraic closure of k. We owe the existence of such reduced forms to a result due to Kolchin and Kovacic [Kov71]. This paper is devoted to the study of reduced forms, of (higher order) variational equations along a particular solution of a complex analytical hamiltonian system X. Using a previous result [AW], we will assume that the first order variational equation has an abelian Lie algebra so that, at first order, there are no Galoisian obstructions to Liouville integrability. We give a strategy to (partially) reduce the variational equations at order m + 1 if the variational equations at order m are already in a reduced form and their Lie algebra is abelian. Our procedure stops when we meet obstructions to the meromorphic integrability of X. We make strong use both of the lower block triangular structure of the variational equations and of the notion of associated Lie algebra of a linear differential system (based on the works of Wei and Norman in [WN63]). Obstructions to integrability appear when at some step we obtain a non-trivial commutator between a diagonal element and a nilpotent (subdiagonal) element of the associated Lie algebra. We use our method coupled with a reasoning on polylogarithms to give a new and systematic proof of the non-integrability of the Hénon-Heiles system. We conjecture that our method is not only a partial reduction procedure but a complete reduction algorithm. In the context of complex Hamiltonian systems, this would mean that our method would be an effective version of the Morales-Ramis-Simó theorem.
Group analysis of the modified generalized Vakhnenko equation
Communications in Nonlinear Science and Numerical Simulation, 2013
In this paper the Lie symmetry group, the corresponding symmetry reductions and invariant solutions of the modified generalized Vakhnenko equation are determined. Moreover a numerical algorithm that is based on a Lie symmetry group preserving scheme is applied to the ordinary differential equations obtained by symmetry reduction.
Lie symmetry analysis of (2+1)-dimensional KdV equations with variable coefficients
International Journal of Computer Mathematics, 2019
In this paper, symmetry groups are used to obtain symmetry reductions of (2+1)dimensional KdV equations with variable coefficients. Despite the fact that these equations emerge in a nonlocal form, by using suitable transformations, they can be written as systems of partial differential equations, and in potential form, as fourth-order partial differential equations. We show that the point symmetries of the potential equation involve a large number of arbitrary functions. Moreover, these symmetries are used to transform the fourth-order partial differential equations into (1+1)-dimensional fourth-order differential equations. Furthemore, we have determined all two-dimensional solvable symmetry subalgebras, under certain restrictions, which the potential equation admits. Finally, by way of example, taking into account a two-dimensional abelian subalgebra we obtain a direct reduction of the potential equation to an ordinary differential equation.
ON THE CORRESPONDENCE BETWEEN DIFFERENTIAL EQUATIONS AND SYMMETRY ALGEBRAS
Symmetry and Perturbation Theory - Proceedings of the International Conference on SPT 2007, 2008
The theory of Lie remarkable equations, i.e., differential equations characterized by their Lie point symmetries, is reviewed and applied to ordinary differential equations. In particular, we consider some relevant Lie algebras of vector fields on R k and characterize Lie remarkable equations admitted by the considered Lie algebras.