Related papers: Nonclassical symmetries as special solutions of he…
In this paper we devise a systematic procedure to obtain nonlocal symmetries of a class of scalar nonlinear ordinary differential equations (ODEs) of arbitrary order related to linear ODEs through nonlocal relations. The procedure makes use…
Similarity reductions and new exact solutions are obtained for a nonlinear diffusion equation. These are obtained by using the classical symmetry group and reducing the partial differential equation to various ordinary differential…
Lie symmetry analysis is one of the powerful tools to analyze nonlinear ordinary differential equations. We review the effectiveness of this method in terms of various symmetries. We present the method of deriving Lie point symmetries,…
We suggest the method for group classification of evolution equations admitting nonlocal symmetries which are associated with a given evolution equation possessing nontrivial Lie symmetry. We apply this method to second-order evolution…
We deal with the existence of infinitely many solutions for a class of elliptic problems with non-symmetric nonlinearities. Our result, which is motivated by a well known conjecture formulated by A. Bahri and P.L. Lions, suggests a new…
Some connections between classical and nonclassical symmetries of a partial differential equation (PDE) are given in terms of determining equations of the two symmetries. These connections provide additional information for determining…
We propose a systemic method of applying the auxiliary systems of original equations to find the high order nonlocal symmetries of nonlinear evolution equation. In order to validate the effectiveness of the method, some examples are…
We obtain the complete Lie point symmetry algebras of two sequences of odd-order evolution equations. This includes equations that are fully-nonlinear, i.e. nonlinear in the highest derivative. Two of the equations in the sequences have…
The general concept of nonlinear self-adjointness of differential equations is introduced. It includes the linear self-adjointness as a particular case. Moreover, it embraces the previous notions of self-adjoint and quasi self-adjoint…
The geometrical theory of partial differential equations in the absolute sense, without any additional structures, is developed. In particular the symmetries need not preserve the hierarchy of independent and dependent variables. The order…
The general concept of nonlinear self-adjointness of differential equations is introduced. It includes the linear self-adjointness as a particular case. Moreover, it embraces the strict self-adjointness and quasi self-adjointness introduced…
For any $n\in\mathbb{N}$ a nonlinear ordinary differential equation with Lie algebra of point symmetries isomorphic to $\frak{sl}(2,\mathbb{R})$ is given.
We give a Lie-algebraic classification of third order quasilinear equations which admit non-trivial Lie point symmetries.
We consider higher symmetries and operator symmetries of linear partial differential equations. The higher symmetries form a Lie algebra, and operator ones form an associative algebra. The relationship between these symmetries is…
New problem is considered that is to find nonlinear differential equations with special solutions. Method is presented to construct nonlinear ordinary differential equations with exact solution. Crucial step to the method is the assumption…
Symmetry groups allow to transform solutions of differential equations continuously into other solutions. This property can be used for the observability analysis of infinite-dimensional systems with input and output. In this contribution,…
We show that all Lie point symmetries of various classes of nonlinear differential equations involving critical nonlinearities are variational/divergence symmetries.
Conservation laws of a class of time-dependent damped nonlinear multidimensional wave equations are derived by Noether's theorem. For arbitrary nonzero damping coefficient and nonlinear interaction term, its infinitesimal variational…
This article complements recent results of the papers [J. Math. Phys. 41 (2000), 480; 45 (2004), 336] on the symmetry classification of second-order ordinary difference equations and meshes, as well as the Lagrangian formalism and…
In this paper we consider some new classes of integral equations that arise from Lie symmetry analysis. Specifically, we consider the task of obtaining solutions of a Cauchy problem for some classes of second order hyperbolic partial…