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 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 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.

scholarly journals The First Integrals of a Second Order Ordinary Differential Equation and Application

2019 ◽  
pp. 1-15
Author(s):  
Yanxia Hu ◽  
Xiaofei Du
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
First Integrals ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Partial Differential ◽  
Wave Solutions ◽  
Symmetry Method

The first integrals of second order ordinary differential equations are considered. The necessary conditions of the existence of analytical first integrals for the equation are presented. Then, the first integrals of the equation are obtained using Lie symmetry method. The results of the first integrals are applied to certain classes of partial differential equations, the conditions of nonexistence of the traveling wave solutions of the partial differential equations are obtained, and traveling wave solutions of the equations under the certain parametric conditions are also obtained.

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