Research On Some Exact Solutions To Fractional Nonlinear Partial Differential Equations

Fractional calculus and fractional nonlinear partial differential equations are utilized to describe how the intricate physical phenomena vary with time and space in the objective world.As one of the momentous tools for mathematical modeling of complex mechan-ics and physical processes,they are extensively employed in computer graphics,fluid mechanics,oceanological dynamics,medicine,polymer physics,and other fields.In ad-dition,the motion laws and properties of these natural phenomena can be comprehended from the exact solutions of fractional nonlinear partial differential equations.Therefore,in order to analyze these phenomena better,a great many scholars have dedicated them-selves to the research in the respect of the properties,solving methods,etc.of fractional nonlinear partial differential equations.The symbolic computation system plays a crucial role in the process of obtaining the exact solutions of fractional nonlinear partial differential equations.Integrated with the symbolic computation software Maple,this thesis will streamline the solving proce-dure,carry out the complicated algebraic operations,and ultimately construct the exact solutions of fractional nonlinear partial differential equations,draw the corresponding three-dimensional diagram simultaneously to facilitate the analysis of the shape,nature and significance of these exact solutions.Regarding two space-time fractional nonlinear partial differential equations as the main research object,and with the aid of the symbolic computation software,this thesis chiefly studies the application of the exp(-ψ(ξ))-expansion method and the(G’/G)-expansion method in solving the exact solutions of fractional nonlinear partial differential equations.And on the basis of the research done by many experts and scholars,two new methods for deriving the exact solutions of fractional nonlinearity are proposed.Firstly,the definition and vital properties of several fractional calculus are intro-duced,and then the definition and properties of Jumarie’s modified Riemann-Liouville fractional derivative are given.Secondly,based on the fundamental thought of the homogeneous balance method and the exp(-ψ(ξ))-expansion method,a new method to derive the exact solutions of fractional nonlinear partial differential equations is proposed,and the proposed method is referred to as the extended exp(-ψ(ξ))-expansion method.Then we utilize this method to deal with the space-time fractional combined KdV-mKdV equation which has practical application background in physics,and obtain several exact traveling wave solutions of this equation,as well as the three-dimensional graphs of the M type,periodic type,kink type,singular kink type and dark soliton type traveling wave solutions under specific parameters.Finally,motivated by the(G’/G)-expansion method and the fractional complex trans-form,integrating with Jumarie’s modified Riemann-Liouville fractional derivative,a new method is constructed,namely the modified(G’/G)expansion method.And the feasibili-ty of this method in obtaining the exact solutions of fractional nonlinear partial differential equations is verified via the space-time fractional Cahn-Hilliard equation,while the ad-vantages and disadvantages of each can be gotten by comparing the(G’/G)-expansion method with other solving methods.