The Exact Solutions For Two Classes Of Nonlinear Space-time Fractional Partial Differential Equations

In the field of mathematical physics,due to the fractional nonlinear partial differential equations with properties of time memory and non-locality,it can describe some physical phenomena more accurately,such as Brown movement,anomalous diffusion and viscoelastic materials with memory problems,etc.As a result,the fractional theory of nonlinear partial differential equation is significant and the current researches frontier.The study of the exact solution of fractional nonlinear partial differential equation is a very important aspect.The obtained exact solution can not only reveal the regularity of mathematical physical model,but also provide a verify method for numerical calculation results.Based on the definition and properties of conformable fractional derivative,this paper constructs the rich exact solutions of the space-time fractional Drinfel'd-SokolovWilson(DSW)equations and the space-time fractional double Sine-Gordon equation by the(G'/G,1/G)-expansion method,the variable separated ODE method and the extended F-expansion method.The main contents of this paper are as follows:The chapter 1 is the introduction,mainly introduces the research background and the development status of fractional partial differential equations.Moreover,the work arrangement of this paper is provided.The chapter 2 gives the definitions and properties of several most widely used fractional differential derivatives,and introduces in detail the theoretical basis of the(G'/G,1/G)-expansion method,the ODE method of variable separation and the extended F-expansion method.In chapter 3,we introduce some existing research results and the methods of solving the space-time fractional DSW equations.Then,we use(G'/G,1/G)expansion method and the extended F-expansion method to discuss the DSW equations,and obtained lots of exact solutions,including hyperbolic,trigonometric,rational and Jacobi elliptic functional exact solutions.In chapter 4,we apply the variable separated ODE method and the extended F-expansion method to the space-time fractional double Sine-Gordon equation.As a result,we obtained abundant exact solutions in trigonometric,hyperbolic,exp-functions and Jacobi elliptic functions.Especially in the application of the extended F-expansion method,different auxiliary equations are selected to get more abundant results.The chapter 5 is the summary and prospect part.