Symmetry Reductions、Classifications And The Exact Solutions Of Two Kinds Partial Differential Equations

Nonlinear differential equations can simulate various phenomena in naturalworld, including fluid mechanics, nonlinear heat conduction, and so on. However,studies on numerous phenomena ultimately boil down to the problems of solvingnonlinear differential equations. Since1960s, many remarkable achievements havebeen made on studies of solving nonlinear differential equations. And what’s more,there also exist various solutions, such as inverse scattering method, the hyperbolicfunction method, C-K direct method, Darboux transformation method, Lie groupmethod, etc.On the one hand, studies on the exact solutions of the general nonlineardifferential equations with variable coefficients have always been extremely importantand active topics at the forefront in nonlinear scientific researches. The generalizedconditional symmetries method is an effective method to solve partial differentialequations, which is produced by Lie group method; On the other hand, people arecontinuously exploring and searching for a more convenient method to explore thesolution to partial differential equations. In1989, the C-K direct method not onlysimplifies methods of solving equation, but also gets many different valuablesolutions which put forward by Clarkson and Kruskal. In1990, Lou Senyue et al.improved this method, and it was called the modified C-K direct method.Main work of this thesis are as follows We get the relationship between the oldand the new solutions of (2+1)-dimensional dispersive long wave equations throughthe modified C-K direct method, and obtain the new exact solutions of the equations;We discuss the classifications and exact solutions of a classification (1+1)-dimensional variable coefficient nonlinear convection diffusion equation using themethod of the generalized condition symmetry.