Semi-simple Group Classification Of A Class Of A General Third-order Evolution Equation

In this thesis,semi-simple group classification of a class of general third-order evolution equation ut?F(t,x,u,ux,uxx,uxxx)is discussed.The classification mainly consists of three steps:First of all,we construct the equivalence group,that is,the most general form of infinitesimal operator of the Lie transformation group admitted by PDEs under considera-tion.We also find the classifying equation of F about the most general symmetry group.Secondly,combining the structures of abstract semi-simple Lie algebra with the general form of Lie symmetry operators we constructed in the first step,we obtain all the realizations of Lie symmetry groups admitted by the equation under study.Thirdly,insert the symmetry operators into the classi-fying equation and solve for fuctions F.And then the corresponding invariant equations are obtained.Using this method,all inequivalent equations admitting either semi-simple or semi-direct sums of semi-simple algebras and solvable Lie algebras under study are presented.The algebraic properties of these equations can be summarized as follows.There are? two inequivalent equations admitting one-dimensional Lie algebras;? one equation admitting two-dimensional solvable Lie algebra isomorphic to A2,2;? six inequivalent equations admitting semi-simple Lie algebras isomorphic to sl(2,R);? nineteen inequivalent equations admitting symmetry algebras having non-trivial Levi factor.The outline of this dissertation is as follows:In section 1,we briefly present the methods and the significance of the group classification.In addition,the research progress and results of the methods both at home and abroad are also listed;We demonstrate the preliminary definitions,theorems and corollaries of semi-simple group classification of a class of general third-order evolution equa-tion under study in section 2;In section 3,all inequivalent twenty-eight equations of the class under study admitting semi-simple Lie symmetries are presented;Furthermore,a concluding remark is given in the last section,there we all point out some open problems.