Direct Reduction Method,Homogeneous Balance Method And Exact Solution Of Nonlinear Partial Differential Equations

In this thesis,direct reduction method and homogeneous balance method are applied to solve nonlinear partial differential equations.According to the basic idea and steps of direct reduction method,the idea of decomposition function is added for the first time,and the exact solutions of many nonlinear partial differential equations are successfully obtained.In combination with the latest literature ideas,several exact solutions of nonlinear partial differential equations with variable coefficients are given by homogeneous balance method.The thesis is organized the following seven chapters:In chapter 1,we make an introduction,briefly introduce the historical background,current research situation and development trend of the nonlinear partial differential equation,summarize the main methods of solving the exact solution of the nonlinear partial differential equation in recent decades.We give the research background and application process of the direct reduction method and the homogeneous balance method.Meanwhile we explain the main research content and purpose of this thesis in detail.In chapter 2,we use the direct reduction method to find the similarity solution of short pulse equation.Then we get the general form similarity reduction including traveling wave reduction,and a new form similarity reduction.In the end,we obtain the exact solutions of short pulse equation in complex form.In chapter 3,by using direct reduction method,we obtain several similarity reductions and a new similarity reduction of Rosenau equation,and we also obtain an explicit exact solution of the original equation for the new similarity reduction.In chapter 4,we deal with Thomas equation via the direct reduction method and analytical hypothesis that we find a new similarity reduction.In chapter 5,we apply the direct reduction method to obtain the new similarity reduction of Vakhnenko equation,we obtain the general form similarity reduction,and we get the exact solution of traveling wave reduction in power form of Vakhnenko equation later.In chapter 6,we discuss the exact solutions of the Boussinesq-Burgers equation with variable damping.We prove the rationality of a hypothesis,and give the exact solutions of the cylindrical Boussinesq-Burgers equation.In chapter 7,we transform the Burgers-Fisher equation with time-space variable coefficients into the classical heat equation by the homogeneous balance method.Then we give the nonlinear boundary-initial value problems for relationship between Burgers-Fisher equation with the time-space variable coefficients and the heat equation,as well as the explicit exact solution of the spherical BurgersFisher equation.