The Construction And Classification Of The Conservation Laws Of A Class Of Differential Equations

In this dissert,the mechanical algorithm of conservation laws to differential equations is considered,especially for those nonlinear evolution equations arising from the fields of fluid mechanics,aerodynamics,plasma physics,biophysics and chemical physics.A method of the classification of the conservation laws is introduced,and it is applied to classify the conservation laws of the differential equations u_t=(A(u)u_x)_x+ B(u)Ux + C(u).The results of the classification of the conservation laws of the differential equations and the potential systems of the differential equations are obtained.Chapter 1 is devoted to introducing the related development of mathematical physics mechanization at home and abroad,the development of Lie Group,symmetry and conservation law.In addition,the development of group classification and classification of conservation laws is also presented.The constru tion of exact solutions to nonlinear partial differential equations under the instruction of the AC = BD theory is given in Chapter 2.Some examples are chosen to illustrate the principle and the application range.And some methods of constructing the operators of C and D are introduced.The mechanical algorithm for constructing the conservation laws of differential computation software Maple in Chapter 3.As applications of the algorithm and program, some higher dimensional nonlinear differential equation(s),such as Modified equation and Breaking soliton equation are considered,which is devoted to prove the effectiveness of the algorithm.In Chapter 4,a method of the classification of the conservation laws is introduced, which is applied to classify the conservation laws of the differential equations u_t=(A(u)u_x)_x+B(u)u_x+C(u).The results cover all results of the classification of the conservation laws of the differential equations u_t=(A(u)u_x)_x+B(u)u_x,which are got by Popovych,R.O.and Ivanova,N.M.And the potential systems of the differential equations are also obtained in this part.