The Methods Of Determining The Symmetry Of Partial Differential Equations And Its Applications Based On Wu’s Method

Many phenomena in mechanics are simulated by nonlinear differential equations.But the nonlinear differential equation is difficult to solve,it is the common problem of interdis-ciplinary.Therefore,the study of nonlinear differential equations has theoretical significance and application value.Symmetry theory has the advantage of optimizing nonlinear differ-ential equations.In this paper,the method of determining the symmetry of some nonlinear differential equations in mechanics and its application are studiedIn recent years,the theory of symmetry has been diversified,such as nonclassical sym-metry,high order symmetry,potential symmetry,discrete symmetry and so on.When using symmetry method to analyze partial differential equation,the precondition is to get the exact symmetric expression.It is necessary to solve the determining equations when determining symmetry.However,it is a difficult task even with the aid of a computer because the deter-mining equations of the nonclassical symmetry are nonlinear.Many nonclassical symmetry of differential equations has not been found by now.And this paper has presented an alter-native method to this problem with a satisfactory result.In addition to this,scientists have explored many effective methods to solve the exact solution of differential equation.Each of them has its own strengths and played an important role in different fields.Starting with the generalized variable separation method,this paper extends the theory of invariant subspaces and differential operatorsRelation between classical symmetry and nonclassical symmetry of partial differential equation is firstly established in this paper.Based on the study of this relation,a new method for calculating the nonclassical symmetry of partial differential equations is presented.This algorithm is used to classify the nonclassical symmetries of the generalized Burgers equa-tion(including many physics and mechanics equations)with two arbitrary terms and then derived a new symmetry.The new method is also applied to some differential equations and satisfactory results are obtained by the relatively simple method.An improved method for solving the discrete symmetry of differential equations based on Lie algebra is reviewed and the discrete symmetries of some differential equations are calculated by means of sym-bolic calculation.The properties of the Lie algebra of the approximate equations for long water waves are structurally analyzed and its optimal system and complete inner automor-phism group are calculated.Based on the above theory,the following nonlinear differential equations commonly used in mechanics are discussed:nonlinear wave equation,nonlinear heat equation,nonlinear Boussinesq equation,Chazy equation,Harry-Dym equation,ap-proximate equation for long water wave,fast diffusion equation,Hunter-Saxton equation,generalized Burgers equations etc.The paper is divided into five chapters:The first chapter is introduction of research background and brief history of differential equation research.It also includes relevant propaedeutic knowledge of basic theory of sym-metry,the Wu’s method,the basic theory of Lie algebra and the basic concept of invariant subspace.In the second chapter,firstly,the symmetry of two differential equations is calculated by using Wu’s method by which the validity of Wu’s method for calculation is illustrated.Secondly,an alternative method is proposed for computing the nonclassical symmetry of a PDEs.The method is consisted of the following three steps:firstly,a relationship between the classical and nonclassical symmetries of PDEs is established;secondly,based on the link,the sufficient conditions for non-classical symmetry are obtained from the classical symmet-ric equations,then we extend nonclassical symmetric deterministic equations and obviously the expanded system is easier to solve than the original system.Finally,we use Wu’s method to solve the extended system and get the nonclassical symmetry of the equation.This method can not only solve the incalculable problem of non-classical symmetric equations of some differential equations,but also a better understanding of the relation between symmetries.The nonclassical symmetry and potential nonclassical symmetry of Boussinesq equation are obtained by using the new algorithm as well as the rational function solution of Boussinesq equation.Applying method,the nonclassical symmetry of the generalized Burgers equation is classified as the application of this method and a new symmetry of the equation is obtained.At the beginning of the third chapter,we have researched the discrete symmetry of C-hazy equation and Harry-Dam equation by using the improved method for finding discrete symmetry.Then,the properties of Lie algebras which consist of the infinitesimal generators of approximate equation for long water wave are discussed and the one dimensional and two dimensional optimal systems have been constructed.In addition,the complete automorphis-m group of the Lie algebra of the equation is calculated and the discrete symmetry of the equation is obtained.In chapter four,two reciprocal problems about invariant subspace have been discussed.(1)Given operator,find all invariant subspace.(2)Given invariant subspace,find all differen-tial operatorF.The steps of the generalized variable separation algorithm have given and the method has been generalized.We have received the separation solutions of the second order and the third order nonlinear evolution equations.As an inverse problem,for some common invariant subspaces,the second order quadratic differential operators and the second order cubic differential operators are classified.By using the above classification,the invariant subspace of the fast diffusion equation and the Hunter-Saxton equation is identified,and some solutions of the separated variables are obtained.In chapter five,the important theorems for constructing general expressions of differen-tial operators for given linear invariant subspaces are reviewed.For the first time,this impor-tant theory is applied to a specific polynomial space and a general expression of a differential operator with a polynomial linear invariant subspace is obtained.And then,the equations are classified according to the invariant subspace.In the end,all mentioned above are summarized and the direction of future efforts is also prospected.