Exact Traveling Wave Solutions For(1+1) Dimension Time-fractional Equations

Fractional order partial differential equations have a good application background and play an important role in solving practical problems.The study of fractional order partial differential equations's solution can help people to explain the natural phenomenon of the existence law of objective.In this paper,the exact traveling wave solutions of time-fractional order equations are studied by using the improved auxiliary equation method,the extended auxiliary equation mapping method,the F-expansion method and the generalized exponential rational function method,the specific timefractional order equations have the following forms:1.The(1+1)dimensional time-fractional order Klein-Gordon equation:2.The(1+1)dimensional time-fractional order BBM-Burger equation:3.The(1+1)dimensional time-fractional order Cahn-Allen equation:4.The(1+1)dimensional time-fractional order Cahn-Hilliard equation:By studying in-depth,the following research results are obtained:1.Firstly,by using traveling wave transformation,the equation(?)is reduced to an equivalent second order ordinary differential equation.With the help of planar theory and method of dynamical system,singularity analysis of planar dynamical system equivalent to ordinary differential equation,and the phase portrait of its orbit distribution is given,it is the conclusion that equation(?)exist four bell-shaped solitary wave solutions,four kink-shaped solitary wave solutions and some periodic traveling wave solutions.With the help of the improved auxiliary equation method,the exact expressions of kink-shaped solitary wave solutions and singular traveling wave solutions of the equation(?)are obtained.Finally,the dynamical behavior of partial solutions is given.2.Firstly,by applying traveling wave transformation and integral operation,the equation(?)is transformed into corresponding to second order ordinary differential equation.We carry out the theory and method of planar dynamical system,singularity analysis of planar dynamical system equivalent to ordinary differential equation,and the phase portrait of its orbit distribution is given,it is the conclusion that equation(?)exist four bell-shaped solitary wave solutions and infinite periodic traveling wave solutions.After using the extended auxiliary equation mapping method,the exact expressions of bell-shapd solitary wave and singular traveling wave solutions for the equation(?)are given.Finally,we give the dynamical behavior of these solutions by Maple software.3.Firstly,by means of traveling wave transformation,the equation(?)can be turned into corresponding to second order ordinary differential equation.By using F-expansion method,the problem of solving ordinary differential equations is transformed into the problem of solving algebraic equations,and the exact expressions of the kink-shaped solitary wave solution and bounded traveling wave solution of equation(?)are obtained.Finally,the traveling wave solutions 2D and 3D graphs of partial solutions are drawn by Maple software,and we give the dynamical behavior of these solutions.4.Firstly,by appiying of traveling wave transformation and integral operation,the equation(?)is reduced to an equivalent third order ordinary differential equation.According to generalized exponential rational function method,the exact expressions of trigonometric solution,hyperbolic solution and exponential solution of equation(?)are obtained.Finally,the 3D and 2D graphs of partial solutions are drawn by Maple software,and the dynamical behavior of the solutions are given.By studying the exact traveling wave solutions of these time-fractional partial differential equations,the exact expressions of the new bounded traveling wave solutions of these equations are obtained,which enriches the theoretical content of the traveling wave solutions of fractional partial differential equations.