Several Topics In Fractional Dynamics

Fractional calculus forms a new paradigm in science and engineering.It has been found that the theory of fractional calculus can more adequately describe many physical phenomena.Unlike the classical calculus,fractional calculus has characteristics of "mem-ory","long distance interaction",and "heredity" which are the main advantages to the use of fractional calculus.The most critical area in the study of fractional calculus is fractional dynamics.Nowadays,researchers and scientists have keen interest to the study of fraction-al dynamics described by the fractional derivative in two ways:one is theoretical analysis;such as,existence-uniqueness of solution,periodicity of the solution,and properties of frac-tional dynamics,whenever,on the other hand,to find the analytical and numerical solutions of fractional differential equations.However,the theoretical analysis of fractional differen-tial equations is more complicated than integer order differential equations,since fractional derivatives are non-local and have weak singular kernels.For example,we cannot find the existence of the global solution of fractional differential equations directly by using the results from local existence because,yet continuation theorems for fractional differential equations have not been derived.In the theory of ordinary differential equations,it is an evidently proved that integer order derivative(if it exists)of the periodic function is also a periodic function of same period.However,the following two natural questions remain open problems till now.(i)Can any fractional derivative of a periodic function also be a periodic function of the same period?(ii)Can any linear or nonlinear fractional differential system have periodic solutions of the same period?The solution to a fractional differential equations cannot define a dynamical system in the sense of semigroup property due to the history memory induced by the weakly singular kernel.Motivated by that work,in this dis-sertation we mainly study the local existence,continuation theorems and global existence of solution for Caputo type fractional differential system,besides the study of the fractional dynamical system with Riemann-Liouville and Hadamard derivatives and establish many results which have never been studied before.The main four contents of this dissertation are:(i)We consider fractional differential equation and system of fractional differential equa-tions with Caputo derivative and study the existence-uniqueness and global solution-s.Firstly,we prove theorems on the existence of a local solution.Then we extend the continuation theorems for ordinary differential equations to those fractional dif-ferential equations.Also,several global existence results for fractional differential equations are obtained.(?)We consider the system of fractional differential equations(FDEs)involving Riemann-Liouville and Hadamard derivatives with different types of initial condi-tions.However,these initial conditions are not completely equal to each other.We prove linearization theorems for nonlinear fractional dynamical systems in Riemann-Liouville and Hadamrd derivative sense respectively.(?)We investigate the relationship between Riesz,Riesz-Caputo,Hilfer derivatives and finite part integrals in Hadamard sense.We prove that Riesz and Hilfer fractional derivatives of a given function can be expressed by the finite part integral of a super-singular integral which does not exist.These results may provide some valuable ideas for the further exploring of the properties of finite part integrals in the future.It will be feasible to construct numerical methods for some supersingular integrals related to the Riesz and Hilfer fractional calculus.(?)In the last,a brief overview of the recent periodicity results of fractional order(Grunwald-Letnikov,Riemann-Liouville,and Caputo)derivatives and fractional d-ifferential equations are provided.At the same time,the existence or non-existence of periodicity of the long-time solution to the fractional dynamical system are also reviewed.