Application Of Wu-Ritt Elimination Method In The Partial Differential Algebraic Equation

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 el al are studied, including exact solutions (soliton solution, periodic solution), compatibility of the solutions (local), Painleve Test and Taylor series solutions. The computer algebra and Wu-Ritt elimination method are employed as the tools to deal with this problem. The aim is to construct some mechanization methods.Chapter 1 introduces the origin and development of several subjects related to this paper. The main works and achievements that have been obtained are presented.Chapter 2 is devoted to constructing some solutions of PDAE or PDAEs. At first, the basic theory of C-D pair and C-D integral system are presented, and then a number of exact solutions are found by several concrete transformations. The process is come true on Maple 7.In chapter 3, the compatibility of the local solutions to partial differential algebraic equations is discussed. A sufficiency and necessity condition and a constructive method are given to decide whether the solutions to linear PDAEs are compatible or not. The case of regular solutions of nonlinear PDAEs is investigated. Meanwhile Wu-Ritt elimination theory plays important role hi the process.Chapter 4 discusses the Painleve property of PDAE by using of Wu-Ritt elimination theory. All resonance points are obtained in the case that the recursion relations are not need to be found. To use the Wu-Ritt elimination theory deals with the compatibility of the resonance equations. We implement the algorithm on computer on Maple?. The validity of the algorithm is shown by the examples, such as KdV equation and Burgers equation.Chapter 5 is devoted to studying the Taylor series solutions. Based on 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. Furthermore the dimension formula oflinear differential idea is given.