Study On Exact Solutions And Dynamical Behaviors Of The Time-Fractional F-KPP Equation And KG Equation

The time-fractional reaction-diffusion equation of the F-KPP-type has a wide application in the fields of physics,chemistry and biology.The most common application is that this kind of equation is often used as a dynamical model of biological population to study the changing law of the number of certain biological populations.And then,the anomalous diffusion phenomenon contained in this model has been paid more and more attention by people.The Klein-Gordon-type equation can be used to study the problems of complex group velocity and energy transport in the absorption medium,the propagation of the short wave in the nonlinear dispersion model and the dislocation phenomenon in the crystal,and also has a wide application in the field of mathematical physics.And the fractional Klein-Gordon equation can also better characterize the nonlinear physical phenomena with the memory characteristics.The exact solutions of these two kinds of equations can better explain and reveal the nature of the problems studied by the model,such as dynamical properties,dynamical behaviors and some strange dynamical phenomena,as well as the inherent relations and changing laws of things.Thus,in this paper two new methods are used to study the exact solutions of the time-fractional Fisher-KPPtype reaction-diffusion equation and the time-fractional Klein-Gordon-type equation and their dynamical behaviors,including the periodicity of the solution,various of dynamic properties such as bounded property,attenuating property and so forth.Our research work in this paper is quite different from the results reported in the existing literature,both in terms of method and the specific content of the study.Our research contents in this paper will be presented in the following two parts.In the first part of content,based on the combination method of variable separation method and integral bifurcation method proposed in recently,the hypothetical structure of the solution is slightly improved and extended.The exact solutions of the fractional nonlinear reaction-diffusion equation of Fisher-KPP-type are discussed by using the improved method.In this paper,four kinds of time-fractional reaction-diffusion models with Fisher-KPP-type are studied and different kinds of exact solutions of them are obtained.These exact solutions include parametric type solutions,periodic type solutions and implicit type solutions,and so on.All solutions are expressed by special functions,which are new types that have not appeared in previous literature.The dynamical properties and dynamical behaviors of these exact solutions are also discussed in this part.In order to show the dynamical property and dynamical behavior of these solutions intuitively,the dynamical profile graphs of some representative exact solutions with time evolution are given in the form of numerical simulation in this section.On the basis of the combination method of the variable separation method and the homogeneous balance principle proposed in recently,the second part of content adds a little bit of computational skills to simplify the operation.The improved method is used to study the time fractional Klein-Gordon-type equation,and the exact solutions of the equation are obtained.These solutions are all exact solutions of the explicit form.Furthermore,we discuss the dynamic properties and dynamic behaviors of these exact solutions.After detailed analysis,it is found that most of the solutions have the property of decay with the increase of time.Similarly,in order to show the dynamical property and dynamical behavior of these solutions intuitively,the dynamical profile graphs of some representative of exact solutions with time evolution are given by numerical simulation in this section.