Solution Of Time Fractional Reaction-Diffusion Equation And Camassa-Holm Equation And Its Dynamical Properties

Fractional nonlinear partial differential equations play a very important role in many natural science fields,because they can accurately depict and describe some strange nonlinear phenomena in many scientific fields,such as anomalous diffusion,viscoelastic and memory phenomena,so it is of great practical significance to deeply explore the exact solutions of such equations and their kinetic phenomena.In fact,it is rare and extremely difficult to mathematically give precise solutions to such equations.Based on the theory of fractional partial differential equation,this paper studies the exact solutions of time fractional reaction-diffusion equation and Camassa-Holm equation,the main research contents are as follows:In the first part,a new algorithm combining the separation variable method with the dynamic system is applied to solve the time fractional reaction-diffusion equation.First,by means of the method of separating variables,the equation is separated into a dynamic system in which the special function interacts with an integer-order differential equation.Secondly,the separated integer-order differential equation is transformed into a two-dimensional plane system.Then,according to the branch theory knowledge of the dynamic system,the equation includes parameter form,periodic form,hyperbolic function form and exact solution of the implicit form.Compared with other literatures,the classification basis of this part is clear,the types of exact solutions are more abundant and the expression is relatively simple.Finally,the three-dimensional coordinate diagram of the exact solutions evolving with time and space is drawn by Maple software,which intuitively reflects the dynamic phenomena of these solutions,such as the decay,boundedness and stability of the solutions with time and space.Based on the method of separating variables and the modified homogeneous equilibrium principle,the solution of fractional Camassa-Holm equation is discussed.Similarly,with the help of the separation variable method,the equation is separated into a dynamic system of interaction between a special function and an integer order differential equation.Because the order of the separated integer order differential equation is higher,it can not be transformed into a two-dimensional plane system as in the first part.This part first assumes that the integer order differential equation has some specific form of solution,then replaces it in the integer order differential equation to carry on the operation,the simplification,applies the modified homogeneous equilibrium principle to the reduced integer order differential equation,obtains the equation spatial variable part has the first function form,the exponential function form,the hyperbolic function form accurate solution.In this part,the exact solution expression is simple,and the type of solution is quite novel.Similarly,through the three-dimensional coordinate diagram of the evolution of the exact solutions with time and space drawn by Maple software,the dynamical properties such as attenuation,boundedness and stability of these solutions with time and space are reflected.