| In this dissertation, the nonlinear partial differential algebraic equation or equations(PDAE or PDAEs) related to some nonlinear topics which origin from physics, mechanics and optics et al are studied, including exact solutions(soliton solutions, periodic solution), the projective Riccati equation method and Taylor series solutions. The computer algebra and the " AC = BD " model of Professor Zhang Hongqing are employed as the tools to deal with this problem.Chapter 1 introduces the origin and development of several subjects related to this paper, such as the soliton theory, computer algebra, mathematics mechanization. The main works and achievements that have been obtained are presented.Chapter 2 considers the construction of exact solutions of partial differential equations(PDEs) under the guidance of the theory of " AC = BD " . The basic theory pf " AC=BD" and the algorithm to construct the C-D pair are illustrated through some concrete transformations.Based on the ideas of algebraic method, algorithm realization, and mechanization for solving nonlinear evolution equations, Chapter 3 deals with the construction of exact solutions for nonlinear evolution equations by use of Wu-method and symbolic computation. The projective Riccati equation method is generalized to obtain some new exact solutions for two-dimensional generalized Burgers equation and the coupled MKdv-KdV equations.Chapter 4 is devoted to studying the Taylor series solutions. Based on the characteristic set, the case of infinity parameters is described by using of finite value and functions. The situation of linear PDAEs is extended to the nonlinear PDAEs. |
