Solutions And Group Analysis Of Nonlinear Differential Equation

The nonlinear differential equation remains a major concern all the time. In the dissertation, under the guidance of mathematics mechanization and by means of symbolic-numeric computation software, the following problems around the nonlinear differential equation are discussed as follows: the exact solutions of nonlinear evolution equations; the approximation solutions of nonlinear fractional differential equations; the group analysis of nonlinear differential equations.The paper consists of the following chapters.Chapter 1 is devoted to reviewing the history and development of the mathematics mechanization, soliton theory, fractional calculous, fractional differential equation, and group analysis of differential equations. Some works and achievements on these subjects involved in this dissertation are presented at home and abroad. Finally, our main works are listed.Chapter 2 introduces the basic theories of AC = BD and pseudo-differential division with remainder, as well as the construction of exact solutions of differential equations under the instruction of the theory of AC = BD.Based on the theories in Chapter 2 and the idea of algebraic method,algorithm reality and mechanization for solving nonlinear evolution equations, Chapter 3 presents two kinds methods for obtaining the exact solutions of nonlinear evolution equations: the one is the generalized rational expansion method, which extended the known rational expansion method, the chapter takes the Burgers equation for example to illuminate the effectiveness of the method; the other is the variable coefficient Korteweg-de Vries (KdV) equation-based sub-equation method, which takes the variable coefficient KdV equation substituting ordinary differential equation as the sub-equation, and the exact solutions of the 3+1-dimensional potential-YTSF equation are obtained by the method.Chapter 4 is on the approximation solutions of nonlinear fractional differential equations.The first section is to introduce some corresponding basic definiens and properties on the fractional calculous. The rest of the chapter is to present solving methods for the approximation solutions of the nonlinear fractional differential equations and their application.The chapter applies the variational iteration method,the Adomian decomposition method and the homotopy perturbation method to obtain the rational approximation solutions of the nonlinear fractional Sharma-Tasso-Olever (STO) equation, and demonstratesthe significant features, efficiency and reliability of three methods by investigating some numerical results, absolute error and figures. The homotopy analysis which traditionally developed for differential equations of integer order are directly extended to derive approximation solutions of the nonlinear fractional Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation in the chapter.Chapter 5 is to study the group classification and the classification of conservation law for the nonlinear differential equations. Here, standard form and the package R_IFS_IMP in the symbolic computation software Maple are firstly introduced.. The package is used to reduce over-determined differential equations to standard form. The chapter applies successfullythe package R_IFS_IMP to obtain the group classification of the nonlinear variable coefficient telegraph equations f(x)u_tt = (H(u)u_x)_x + h(x)K(u)u_x, and gives the classification of conservation law with respect to the group of equivalence transformations.