Theoretical Analysis And Numerical Computation For Fractional Differential Equations

Fractional operators have a long history, having been mentioned by Leibnitz in a letter to L'Hospital in 1695. Referring to the question of fractional differentiation, Leibnitz wrote, "It will lead to paradox, from which one day useful consequences will be drawn." Early mathematicians who contributed to fractional differential operators include Liouville, Riemann, and Holmgrem. For three centuries the theory of fractional derivatives developed mainly as a pure theoretical field of mathematics useful only for mathematicians. However, in the last few decades many authors pointed out that fractional calculus are very suitable for the description of memory and hereditary properties of various materials and processes, such effects are in fact neglected in classical models. Nowadays, fractional differential equations are increasingly used to model problems in acoustics and thermal systems, rheology and modelling of materials and mechanical systems, signal processing and systems identification, control and robotics, and other areas of application.This thesis consists of five chapters, and the body can be divided into three parts, the first part (Chapters 2-3) focuses on theoretical analysis for fractional ordinary differential equations (FODE) and the second part (Chapters 4) concentrates on numerical computation for FODE and fractional Fokker-Planck equations (it is a kind of typical fractional partial differential equations), the last chapter is of the applications of fractional differential equation, including the realization of generalized chaos synchronization and the generation of multi-scrolls by three methods.More detailed, the first chapter briefly reviews the definitions of fractional calculus and further analyzes and compares some of their properties.In Chapter 2, first attains the characteristic equation for multi-time-delayed linear FODE by using the technique of Laplace transform, then obtains the general stability criteria for multi-time-delayed linear FODE and applies the results to synchronization. Besides, some smoothness properties for the solutions of FODE are also got, and the Mittag-Leffler representation for the solutions of nonlinear FODE is presented.Chapter 3 discusses the possibilities for transforming the multi-order FODE to FODE with the same order and provides the stability results for multi-order FODE.In Chapter 4, the predictor-corrector approach for numerically solving FODE is improved, the short memory principle of fractional calculus is apprehended from a new point of view, and the idea of predictor-corrector is combined with the short memory principle for the numerical solution of FODE and the detailed error analysis is presented. Using the properties of Riemann-Liouville derivative and Caputo derivative, firstly the fractional Fokker-Planck equation is converted to a parabolic fractional partial differential equation, then combining the predictor-corrector approach with the idea of method of lines, the numerical schemes for the time-fractional Fokker-Planck equation and time and space fractional Fokker-Planck equation are designed and verified, the Chapter is the central part of this thesis.Chapter 5 numerically studies the dynamical behavior of fractional differential systems, including the realization of generalized chaos synchronization and the generation of multi-scrolls, and in fact this chapter is the further application in engineering for the numerical algorithm discussed in Chapter 4.