Studies On Several Problems Of The Solutions Of Several Types Of Integer And Fractional Differential Equations

Nonlinear problem has always been a hot research topic in mathematical physics.In recent decades,with the deepening of scientific research,nonlinear science has made great progress.It is found that many phenomena in nature can be described by establishing the mathematical model of the solution of nonlinear development equation.Although many scholars have explored a variety of effective methods to find the exact solutions of nonlinear equations,there is no one method that can be applied to all nonlinear problems,and there are still many solutions of nonlinear evolution equations to be explored.In this paper,we mainly use the Hirota bilinear method,the extended(G'/G)-expansion method and the lie symmetric analysis method to study several kinds of nonlinear partial differential equations of integer order and fractional order.In the first chapter,we introduce the research background,the research status at home and the research significance of this paper.In the second chapter,based on the bilinear system of the KP equation,we obtain the new multi soliton solutions of(3+1)-D Jimbo-Miwa equation.Similarly,many semi-rational solutions consisting of line solitons and lumps are obtained.The fusion process of line solitons and lumps and the process of splitting line solitons into line solitons and lumps are studied by drawing three-dimensional graphs.These results have never been studied before and enrich the dynamic model of Jimbo-Miwa equation,which can explain and predict the corresponding dynamic phenomena in engineering,aerospace,meteorology and other fields.In the third chapter,we first investigate the exact solutions of the generalized time-fractional foam drainage equation.The Lie-group scaling transformation method and improved(G'/G)-expansion method are adopted here.The equation describes the evo-lution of the vertical density profile of a foam under gravity.New exact solutions and maple diagrams of the generalized time-fractional foam drainage equation can help us better understand the physical phenomena.Secondly,utilizes ansatz method to find bright and dark solutions of the conformable space-time fractional modified equal width wave equation.In addition,a fractional novel(G'/G)-expansion method is used to look for periodic solutions,dark solutions,soliton solutions and soliton-like solutions of the space-time fractional modified equal width wave equation for the first time.The dynamic model of the solution is given.The results show that the two methods are applicable and more effective for solving other types of nonlinear fractional differential equations.In the fourth chapter,we study the coupling time-fractional Boussinesq-Burgers system,which is used to study the flow of fluid in the power system and describe the propagation of shallow water waves.Firstly,we consider the symmetry of Lie point and the transformation of similarity.The fractional differential operator of Erdelyi-Kober is used to transform the coupled time-fractional Boussinesq-Burgers system into a Nonlinear Fractional Ordinary differential equation.Secondly,the simplified frac-tional ordinary differential system is solved by the power series expansion method,and the convergence of the power series solution is analyzed.The conservation law of the coupled time-fractional Boussinesq-Burgers system is constructed by using the new conservation theorem.In particular,the numerical simulation of the coupled time-fractional Boussinesq-Burgers system by q-homotopy analysis is given.In the last chapter,we summarize the research results of this paper,at the same time,combined with the existing research results and the theoretical basis we have mastered,we discuss the possible research directions in the future,and give the future work prospects.