Approximate Functional Separation Of Variables Method And Its Nonlinear Perturbed Systems

Nonlinear Science is deeply studied as well as widely applied in the fields of natural science, technology and engineering. In particular, a lot of methods for seeking exact solutions and analysis of nonlinear partial differential equations (PDEs) have been developed. Among of which the theory of symmetry group plays an important role in the process.Another important aspect is that scientists have been pursuing research ex-tensively on nonlinear PDEs with a small parameter (or perturbed equations) based on symmetry group theory in the past few decades. Some approximate methods are developed in the combination of both symmetry group and pertur-bation theory. These methods are very beneficial to and improve on the study in the area of nonlinear science.In this dissertation, due to the combination of the perturbation theory and the functional variable separation approach, or the derivative-dependent func-tional variable separation approach, two approximate methods are proposed and they are applied to study the variable separation issue for nonlinear perturbed equations.It is organized as follows:In chapter1, we introduce the background, situations and applications of the related fundamental theory, review the symmetry group method, the variable separation approach and the approximate method, and present the main content of the dissertation.In chapter2, we define the concept of approximate functional separable so-lution (AFSS), and characterize the necessary and sufficient conditions for the (1+1)-dimensional perturbed equations that admit first-order AFSSs. As appli- cations of the new approximate functional variable separation(AFVS)approach, we discuss the following three classes of the perturbed equations ut=(A(u)uxn)x+∈F(u),A(u)F(u)≠0,(n≤1) ut=A(u,ux)Uxx+∈B(u,ux),A(u,ux)≠0,B(u,ux)≠constant, ut=A(u,ux)Uxxx+EB(u,ux)uxx+F(u,ux), A(u,ux)B(u,ux)≠0. Complete classifications of these perturbed equations that admit the AFSSs are obtained.The main Solving procedure for the AFVS approach is shown by way of examples,and the corresponding AFSSs to the resulting equations are derived.In the same way,the above AFVS approach is applied to the following two classes of the perturbed wave equations utt+∈ut=(A(u)ux).,A(u)≠0, Utt=(A(u)ux)x+∈B(u)ux+F(u),A(u)B(u)≠0. in chapter3.In chapter4.we extend the AFSS to the form with derivative of the depen-dent variable,and give the definition of approximate derivative-dependent func-tional separable solution(ADDFSS).The necessary and sufficient conditions for the(1+1)一dimensional perturbed equations that possess first-order ADDFSSs is characterized.To illustrate the new approximate derivative-dependent func-tional variable separation(ADDFVS)approach,in this chapter,we are mainly concerned with the following two classes of diffusion equations with perturbation ut=A(u)Uxx+∈B(u,ux), A(u)≠0,B(u,ux)≠constant. ut=(A(u)ux)x+∈B(u,ux), A(u)≠0,B(u,ux)≠constant Complete classifications of these perturbed equations which possess ADDFSSs are obtained.The main solving procedure for the ADDFVS approach is shown via examples. As the result, the corresponding ADDFSSs to the resulting equa-tions are constructed.The last chapter is the summary of the work and some topics to be consid-ered in the future.The novelties of the dissertation are as follows:(1) In theory, we give the definitions of the AFSS and the ADDFSS. The combination of the perturbation theory and the variable separation approach extends both the perturbation theory and the symmetry group theory to some extent.(2) We propose two kinds of approximate variable separation approaches and establish respectively their variable separation procedures. These two ap-proaches are effective and feasible to deal with nonlinear perturbed systems.(3) We have successfully applied the two approaches to several classes of nonlinear perturbed equations. Complete classifications of these perturbed equa-tions which admit approximate variable separable solutions are obtained, and then whose corresponding approximate solutions are constructed. These solu-tions generally cannot be derived by other approximate methods.