scholarly journals Applications of Differential Form Wu’s Method to Determine Symmetries of (Partial) Differential Equations

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 378 ◽  
Author(s):  
Temuer Chaolu ◽  
Sudao Bilige
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Differential Polynomial ◽  
Decomposition Algorithm ◽  
Partial Differential ◽  
Wu’S Method ◽  
Classical Symmetry ◽  
Symmetries Of Differential Equations ◽  
Wu's Method ◽  
Nonclassical Symmetry

In this paper, we present an application of Wu’s method (differential characteristic set (dchar-set) algorithm) for computing the symmetry of (partial) differential equations (PDEs) that provides a direct and systematic procedure to obtain the classical and nonclassical symmetry of the differential equations. The fundamental theory and subalgorithms used in the proposed algorithm consist of a different version of the Lie criterion for the classical symmetry of PDEs and the zero decomposition algorithm of a differential polynomial (d-pol) system (DPS). The version of the Lie criterion yields determining equations (DTEs) of symmetries of differential equations, even those including a nonsolvable equation. The decomposition algorithm is used to solve the DTEs by decomposing the zero set of the DPS associated with the DTEs into a union of a series of zero sets of dchar-sets of the system, which leads to simplification of the computations.

scholarly journals A Potential Constraints Method of Finding Nonclassical Symmetry of PDEs Based on Wu’s Method

2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
Tonglaga Bai ◽  
Temuer Chaolu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Challenging Problem ◽  
Extended System ◽  
Partial Differential ◽  
Wu’S Method ◽  
Wu's Method ◽  
Nonclassical Symmetry ◽  
Nonclassical Symmetries ◽  
Symmetry Method

Solving nonclassical symmetry of partial differential equations (PDEs) is a challenging problem in applications of symmetry method. In this paper, an alternative method is proposed for computing the nonclassical symmetry of PDEs. The method consists of the following three steps: firstly, a relationship between the classical and nonclassical symmetries of PDEs is established; then based on the link, we give three principles to obtain additional equations (constraints) to extend the system of the determining equations of the nonclassical symmetry. The extended system is more easily solved than the original one; thirdly, we use Wu’s method to solve the extended system. Consequently, the nonclassical symmetries are determined. Due to the fact that some constraints may produce trivial results, we name the candidate constraints as “potential” ones. The method gives a new way to determine a nonclassical symmetry. Several illustrative examples are given to show the efficiency of the presented method.

scholarly journals On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations

2021 ◽  
Vol 41 (5) ◽  
pp. 685-699
Author(s):  
Ivan Tsyfra
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Partial Differential ◽  
Classical Symmetry ◽  
Kdv Equations ◽  
Classical Symmetries

We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.

scholarly journals Reductions and exact solutions of some nonlinear partial differential equations under four types of generalized conditional symmetries

1999 ◽  
Vol 41 (1) ◽  
pp. 1-40 ◽  
Author(s):  
Changzheng Qu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Dynamical Behavior ◽  
Qualitative Properties ◽  
Partial Differential ◽  
Nonclassical Symmetry ◽  
Symmetry Method

AbstractThe generalized conditional symmetry method is applied to study the reduction to finite-dimensional dynamical systems and construction of exact solutions for certain types of nonlinear partial differential equations which have many physically significant applications in physics and related sciences. The exact solutions of the resulting equations are derived via the compatibility of the generalized conditional symmetries and the considered equations, which reduces to solving some systems of ordinary differential equations. For some unsolvable systems of ordinary differential equations, the dynamical behavior and qualitative properties are also considered. To illustrate that the approach has wide application, the exact solutions of a number of nonlinear partial differential equations are also given. The method used in this paper also provides a symmetry group interpretation to some known results in the literature which cannot be obtained by the nonclassical symmetry method due to Bluman and Cole.

TOROIDAL LIE ALGEBRAS AND SOME DIFFERENTIAL EQUATIONS OF TODA TYPE

2010 ◽  
Vol 09 (06) ◽  
pp. 1015-1031
Author(s):  
YOUJUN TAN
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Hamiltonian Formalism ◽  
Differential Polynomial ◽  
Field Equations ◽  
Partial Differential ◽  
Polynomial Algebras ◽  
Recursive System

We deduce a recursive system of partial differential equations from the toroidal Lie algebra associated to a simple Lie algebra. Such a recursive system starts from the corresponding classical Toda field equations. For each truncated system we describe its Hamiltonian formalism using Gel'fand–Dikii's differential polynomial algebras, and some nontrivial local integrals are given.

Sign in / Sign up

Export Citation Format

Share Document