Theoretical Analysis And Numerical Approximate Computation For Serveral Classes Fractional Differential Equation

Two Fractional differential equations (FDE), namely Taylor method for solving the Bagley-Torvik equation and chaotic control problem of fractional order Jerk system, have been studied in this paper. Several measurements for studying Fractional differential equations are presented. For instance, study on fractional differential equations using classical calculus theory under the Riemann-Liouville definition; the research on a class fractional Jerk model by using two kinds of chaotic control methods including variable feedback control and delay feedback control; in analysis of the characteristic equation aspect, introduction of the Hopf bifurcation theory of delay differential equations to seek critical condition; approximate calculation using MATLAB in the simulation are all discussed in the paper.The main parts of this paper:Firstly, a novel method--Taylor method for solving the Bagley-Torvil equation is presented. Based on Taylor's expansion of the unknown function, the Bagley-Torvil equation can be converted to a system of linear algebraic equations for the unknown function and its derivatives up to n order. Comparing the existing literature, numerical example shows that the effectiveness of the method. Through the analysis above, sum up the method of Taylor expansion and point out the research directions.In the second part, a chaotic control problem of fractional order Jerk system with n=3.1 is considered using variable feedback control method. Firstly, we transform the Jerk system from fraction-order to approximate integral-order by Laplace transform and the method of conversion between time domain and frequency domain, and then we show the chaotic property of the model. Secondly, a feedback variable, by adjusting feedback strength, can change the speed of stable cycles, even if imposed by a small feedback coefficient can be controlled chaos, and can produce good control performance. Some numerical simulations support our analysis results.Then, a chaotic control of fractional order Jerk system is also considered by using delay feedback control method. This thesis introduces the delay feedback to control the model. Its wisely using of the system itself as part of the output signal, and the output signal makes difference with the original signal after the time delay, then bringing the difference into the original system, and the final result has also been an ideal control. Then by contrasting the linear feedback control with the delayed feedback control, we identify the advantages of these two controlling methods in such model. The structures of the two controllers are very simple, thus online adjustment of the controller and are not hard to be fulfilled in the field of engineering. The variable feedback control method applies the difference of a feedback signal and the corresponding output signals, or the direct use of the system itself chaotic output signal multiplied by the appropriate variable feedback coefficient. All of the above could be used to implement the chaos control tasks in the original nonlinear system. But in the field of delay feedback control with periodic signals using delay deviation signal to calm the periodic orbits of chaos system embedded itself, as it doesn't change the relevant attributes of the periodic orbits, it needs relative less control energy.