Uniqueness Of Difference And Explicit Solutions Of Two Class Of Equations

The fundamental challenges of science are those of description and prediction.Observing certain phenomena,we wish to know how to describe what we see now and how to determine the subsequent behavior.In many important cases,we are led to some differential or difference equations.Moreover,as a result of intense and inge-nious research,much significant information concerning geometry,physical processes,engineering,and economic sources can be derived from the analysis of some equations.Basic problems are those of existence and uniqueness of solutions of these equations.Furthermore,we now possess powerful procedures for obtaining computational solu-tions using either desk or digital computers.Despite this parts state of affairs as far as equations are concerned,we are nevertheless forced to turn to the study of more equations.And explicit solutions would be much more useful and easy to apply for applications.The search of explicit solutions poses a lot of challenges to deal with.Since the expansion to complex number can help us obtain more cognition,differential and difference equations in the complex field which develops from real field is an in-evitable results.Many efforts have been spent on getting the solutions of equations and several ways of approach to deal with complex equations have been discovered.The local theory is perhaps the most investigated of these approaches.In this thesis,from the most problems we care about,we discuss the uniqueness of a function under some conditions and give the explicit meromorphic solutions of two class of Briot-Bouquet differential equations.Chapters 1 contains the theory of Nevanlinna which develops in 1920s,and it has became a powerful tool for the study of complex equations.Then we shall give the sketch of the theory of Wiman-Valiron,which plays a very important role in the study of entire functions.Also,the study of difference and difference equation which comes from real field,is an inevitable tendency and we shall introduce some results about difference and difference equations.Finally,to make this thesis more readable,we shall discusses the Painleve test using the method of Kowalevski-Gambier and introduce the beginning of elliptic functions.In Chapter 2,we introduce our result of a difference counterpart of Briick’s con-jecture.For an transcendental function and its difference polynomial that share small function,we shall give a necessary and sufficient condition of the uniqueness.Further-more,we can get the polynomial.In Chapter 3 and 4,we investigate two class of Briot-Bouquet differential equa-tions respectively,and the aim is to obtain its explicit meromorphic solutions by the Kowalevski-Gambier method.