Wu Method And Its Application About Partial Differential Equations

This dissertation is changed into two parts. The first part, the theory and application about Wu-Ritt differential characteristic sequence method are discussed, which involves the theories of differential equations, abstract algebra and computer algebra etc. We apply Wu-Ritt differential characteristic sequence method(abbr.Wu-Ritt method ) to linear partial differential equations which has physics significance and give the size of solutions and formal Taylor solutions. We present // formal orderings. By using // formal orderings, we make sure Zhang's conclusion of proper solutions. The second part, with the aid of many types constructive transformation and symbolic computation(especially Wu algebraic elemination method), some topics in nonlinear evolution equation are studied, including exact solution(solitary solution, periodic solution, rational function solutions and Jacobian function solution),Backlund transformation, Cole-Hopf transformation, dromion solution and its construction etc.Charter 2 introduces AC=BD model and its application about partial differential equations. Firstly, we give a basic notion and basic theory of C-D pair and C-D integrable system and then we study their applications. How to seek for transformation u ?Cv is an important aspect in Charter 2.Chapter 3 is devoted to studying basic theory of Wu-Ritt method and its application. We apply Wu-Ritt method to linear partial differential equations and gave the size of solution arid formal Taylor solutions. Therefore we make sure Zhang's results.Chapter 4 gives hyperbolic function transformation method and its applications. Firstly we deduce hyperbolic function transformation and then apply to a class of reaction diffusion equation and Brusselator reaction diffusion model which has physics, chemistry and biology significance. Thus we obtain many new exact and explicit solutions (including solitary wave soluiton, peoiodic wave solution and rational functions solutions) to above equations. Finally we also discuss explicit exact solutions of KdV, coupled KdV and a compound KdV-Burgers equations etc. Wu algebraic elimenation method is most important basic tool during the course of solving proplem.Chapter 5 considers two mechanization algorithm of Jacobian function solutions based on sine-Gordon and sinh-Gordou equations. We call above two mechanization algorithm as Jacobian function expension method. The Jacobian function expension method is an effective method than the sine-consine method and the sn-cn method. Finally we apply Jacobian function expension method to a class of nonlinear evolution equations, RLW and a compound KdV equations and get many new Jacobian function solutions and solitary wave solutions.In Chapter 6, we study new applications of homogenous balance method and apply the method to WBK equation and obtain many new exact solutions. We apply the method to Boussinesq equation and combine Wu algebraic elimination method, many new exact solutions are obtained. Applying the method to BK equation and DLW equation, we get mulitsolition-like solutions and many dromion solutions and their constructions.In Chapter 7, we discuss exact solutions of BK, DLW, Kupershmidt equations by using expension Riccati method which Yan and Zhang developed. Also in Chapter 7, we present a mechanization algorithm of searching for soliton-like solutions and apply the algorithm to (2+1)-dimensional KD equations and we obtain many new soliton-like solutions. Therefore we extend Fan's and S.A.Elwakil's algorithms.