Numerical Solution And Dynamic Properties Of Initial Value Problems Of Some Classes Of Nonlinear Fractional Differential Equations

In recent years, many phenomena in material science, computational biology, chem-istry kinetics, control theory and other sciences have been described successfully by the mathematical models with fractional calculus, i.e. the theory of derivatives and integrals of non-integer order. Along with the development of the corresponding theories, the numerical solutions of fractional differential equations (FDEs) have been an important and hot topic in the world.In this paper, we focus on the studies of some numerical methods and approximately analytical methods for solving the initial value problems (IVPs) of nonlinear fractional differential equations. And the chaotic attractors of fractional order dynamic systems are also discussed. The main contents and results are listed as follows.In Chapter1, we introduce the background and current situations of the researches into fractional calculus, fractional differential equations and the methods for solving them. And the main research contents of this paper are proposed.In Chapter2, we introduce the basic definitions and properties. Some numerical methods for solving IVPs of FDEs are listed too.In Chapter3, we propose the cubic spline collocation method with two parameters for solving the IVPs of nonlinear FDEs. The theorem of the local truncation error of this method is given. And the results of the convergence and the stability of this cubic spline collocation method for the fractional order integral equations which is equivalent to the IVPs of nonlinear FDEs are obtained. We also obtain some results of the convergence and the stability of this method for the IVPs of nonlinear FDEs by using the relationship between the numerical solutions obtained respectively from the cubic spline method for the IVPs of nonlinear FDEs and the corresponding equivalent fractional order integral equations. Some illustrative examples successfully verify our theoretical results and show that this method is efficient.In Chapter4, we propose the cubic spline collocation method with two parameters for the IVPs of nonlinear multi-order nonlinear fractional differential equations (M-FDEs). We extend the theoretical results obtained in Chapter3. And the corresponding results about the local truncation error, convergence and stability of the cubic spline method for the IVPs of nonlinear M-FDEs are obtained. In the same way, some illustrative examples also verify our theoretical results and show that this method is efficient.In Chapter5, the generations of multi-stripe chaotic attractors of fractional order systems are considered. The original fractional order chaotir at tractors can be turned into a pattern with multiple "parallel" or " rectangular" stripes by employing certain simple periodic nonlinear functions. The relationships between the parameters in the periodic functions and the shapes of the generated attractors are analyzed. Theoretical investigations about the underlying mechanisms of the parallel stripes in the fractional order attractors are presented, with the fractional order Lorenz, Rossler and Chua systems as examples.In Chapters6and7, the variational iteration method (VIM) is applied to obtain ap-proximately analytical solutions of M-FDEs and delay integro-differential equations (DIDEs) including multi-delay integro-differential equations (MDDEs). The corresponding theorems for convergence and error estimates of the VIM for solving M-FDEs and DIDEs are given respectively. The numerical results show that our theoretical analysis are right and the VIM is a powerful method for solving these equations.