Related papers: $\lambda$-symmetries for discrete equations
We present the package SADE (Symmetry Analysis of Differential Equations) for the determination of symmetries and related properties of systems of differential equations. The main methods implemented are: Lie, nonclassical, Lie-B\"acklund…
We introduce the relation ${\rho}_{\lambda}$-orthogonality in the setting of normed spaces as an extension of some orthogonality relations based on norm derivatives, and present some of its essential properties. Among other things, we give…
This paper deals with the comparison of two common types of equivalence groups of differential equations, and this gives rise to a number of results presented in the form of theorems. It is shown in particular that one type can be…
We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of…
These lecture notes provide an introduction to the theory and application of symmetry methods for ordinary differential equations, building on minimal prerequisites. Their primary purpose is to enable a quick and self-contained approach for…
Two essential methods, the symmetry analysis and of the singularity analysis, for the study of the integrability of nonlinear ordinary differential equations are discussed. The main similarities and differences of these two different…
We show that $\lambda$-symmetries can be algorithmically obtained by using the Jacobi last multiplier. Several examples are provided.
In this paper, we study non-linear differential equations associated with Legendre polynomials and their applications. From our study of non- linear differential equations, we derive some new and explicit identities for Legendre…
We define discrete Hamiltonian systems in the framework of discrete embeddings. An explicit comparison with previous attempts is given. We then solve the discrete Helmholtz's inverse problem for the discrete calculus of variation in the…
The article introduces contact germs that transform solutions of some partial differential equations into solutions of other equations. Parametric symmetries of differential equations generalizing point and contact symmetries are defined.…
We state some elementary problems concerning the relation between difference calculus and differential calculus, and we try to convince the reader that, in spite of the simplicity of the statements, a solution of these problems would be a…
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…
Singular perturbation theory plays a central role in the approximate solution of nonlinear differential equations. However, applying these methods is a subtle art owing to the lack of globally applicable algorithms. Inspired by the fact…
In this paper we generalize the notion of discriminant for symmetric matrices and some results about it to the case of symmetric spaces.
A procedure for obtaining a "minimal" discretization of a partial differential equation, preserving all of its Lie point symmetries is presented. "Minimal" in this case means that the differential equation is replaced by a partial…
We show that nonlocal reductions of systems of integrable nonlinear partial differential equations are the special discrete symmetry transformations.
We define Boolean algebras in the linear context and study its symmetric powers. We give explicit formulae for products in symmetric Boolean algebras of various dimensions. We formulate symmetric forms of the inclusion-exclusion principle.
This paper studies systems of linear difference equations on the lattice $\Z^n$ that are invariant under a finite group of symmetries, and shows that there exist solutions to such systems that are also invariant under this group of…
In this paper we extend the Tanaka finiteness theorem and inequality for the number of symmetries to arbitrary distributions (differential systems) and provide several applications.
Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal…