Solutions And Dynamical Properties Of Time Fractional Reaction-Diffusion Equation And Fractional Pendulum Model

The mathematical model established by fractional differential equation has its own unique advantages,which are irreplaceable by integer differential model.Fractional differential equation have been widely used in signal processing,system control,abnormal diffusion and heat conduction,viscoelastic fluid mechanics,biology,magnetism and many other fields.In particular,the physical problem,chemical problems and biological population problems with memory,abnormal diffusion and viscoelastic phenomena can be modeled by fractional differential equations.Under the above background,this paper mainly studies the solutions of the time fractional reaction-diffusion equation and the fractional pendulum model,and their kinetic behaviors.The main research contents are as follows:In the first part,we discuss the solution of the time fractional reaction-diffusion equation,mainly using the new method of combining the separation of variables method with the phase diagram analysis method of dynamic system.Based on the principle of variable separation and homogeneous equilibrium,a new method is introduced to discuss the solutions of fractional nonlinear partial differential equations and their dynamic properties in combination with the plane dynamic system theory.Under some specific parameters,the exact solution of the differential equation determined by the space part is obtained by the method of orbital integration,thus the exact solution of the whole time fractional reaction-diffusion model is obtained.Then for some representative solutions,the paper studies their boundedness,periodicity and the change law of the solution with time development.Through numerical simulation,the evolution diagram of some solutions with the development of time and space is given,and then the evolution phenomenon ia analyzed.Finally,the relevant scientific conclusion are drawn,which reveals the inherent law contained in the model and successfully explains the corresponding dynamic phenomenon.The second part,by using Laplace transform method and Mittag-Leffler function,we study the exact solutions and dynamic properties of two kinds of fractional linear pendulum models defined by Caputo type fractional derivative.Under some given initial conditions,various exact solutions of the model are obtained.Then,the coordinate evolution diagram of the solution is drawn by Maple software,and the vibration of the fractional free various simple pendulum model under the influence of cycloidal friction and viscous resistance is analyzed from the image.In order to compare the dynamic phenomena and dynamic properties of fractional order and integer order models,we also study the solution and dynamic properties of the classical integer nonlinear pendulum model by using the phase diagram analysis method of dynamic system.In the case that the exact solution of the integer nonlinear classical model cannot be obtained,the time evolution of the solution of the model is shown by using the method of dynamic system numerical simulation,and the vibration of the integer nonlinear simple pendulum model is analyzed.Finally,the intrinsic law contained in the model is revealed successfully and the corresponding dynamic phenomena are explained satisfactorily.