Research On Solving Exact Solutions Of Several Types Of Nonlinear Partial Differential Equations

As we all know,linear phenomena are simple and easy to understand.In fact,in the objective world,nonlinear phenomena are universal natural phenomena,such as weather changes,stock market fluctuations,water wave motion and particle motion.In order to accurately describe these non-linear phenomena,scientists get a series of non-linear models about non-linear phenomena through mathematical modeling,through in-depth study and analysis of these models,we can understand the nature and law of non-linear phenomena.From the mathematical point of view,the model based on nonlinear phenomena is a nonlinear equation.We can explore the inherent law of the phenomena described by the model by seeking the solution of the equation.Because non-linear phenomena involve many disciplines,their research and analysis have gradually formed a new discipline,that is,non-linear science.In recent years,scholars from all walks of life have made new breakthroughs in the study of non-linear science,especially in the fields of information processing and ecological environment.Nonlinear science not only promotes the integration of different disciplines,but also speeds up the development of various disciplines.The work of this paper mainly includes the following aspects:Firstly,we summarize the origin and development of non-linear science,introduce the discovery and development of solitary waves,and introduce the symbolic computing software Maple and its related commands.Secondly,we study several methods for solving nonlinear partial differential equations,which can be classified into two kinds of solutions.Based on each idea,we give some representative solutions.Then,we make two improvements to the traditional(G'/G2)-expansion method.One is to change the lower limit of the power series from zero to negative number.The other is to extend the method to three-dimensional and more than three-dimensional non-linear systems.The BBM equation and(2+1)dimensional BBM equation are solved by the(G'/G2)-expansion method.From the formal analysis of the solutions,three different types of exact solutions are obtained.When the constants in the general solution of hyperbolic functions are taken as special values,the bell and anti-bell solitary wave solutions are obtained.Finally,we use the exponential function method to solve the regular long wave equation(RLW equation)and KdV-Burgers equation,which have important physical significance,and obtain the solitary wave solution of the equation.