Fractional Ordinary Differential Equations And Anomalous Subdiffusion Equations

Fractional calculus has a long history, the application is very comprehensive, including the memory of many kinds of materials, anomalous diffusion,signal pro-cessing.control theory,vibration control of viscoelastic system and pliable structure ob-jects.fractional biological neurons, advection-diffusion in porous or fractured medium, chaotic, etc. Comparing with the classical inter-order differential equation, the new fractional order differential equation which is containing the non-integer order derivative, is more adequate to simulate practical problems. It can effectively describe the memory and transmissibility of many kinds of materials, and play an increasingly important role in engineering, physics, finance, hydrology and other fields. For inter-order differential equations, correlative numerical arithmetics are mature relatively, but for fractional differential equations in the fractional models, the investigation of numerical methods is underway, theoretical analysis is limited especially.Fractional ordinary differential equations can describe many physical phenomena, investigated widely. For example, dynamical controlled systems, chaotic model, fractional PI~λD~μcontroller simulation investigation. However they are limited in some applied fields. In recent years, researchers have proposed some numerical methods, but there are some difficulties in the error analysis. At present, the numerical methods, theoretical analysis and applications are just under exploration. Developing computationally efficient solution method and theoretical analysis of fractional ordinary differential, exploring further application of fractional ordinary differential will be very significative, which engineers are interested in.Fractional dynamic equations are very useful in describing power transmission phenomenon of complex systems, for example, a modified anomalous subdiffusion equation, etc. But it is difficult to solve such problems. For anomalous diffusion model with different situations, many investors proposed different numerical methods, and improved the error analysis of theoretical study continually. This thesis focuses on two kinds of problems: numerical methods and application of fractional ordinary equations, modified anomalous subdiffusion problem.Introduction gives some concerning fractional calculus to prepare the knowledge and present basic definitions and properties of fractional calculus. It describes numerical methods of fractional ordinary equations and anomalous subdiffusion problem comprehensively.The first kind of problems, we consider numerical methods and application of fractional ordinary equations, which are consisted of Chapters 2 to 4.In Chapter 2, we discuss the fractional Relaxation-Oscillation equation (FROE). The existence and uniqueness of solution for FROE is proven, and its analytic solution is given. A computationally effective fractional Predictor-Corrector method is proposed, and a detailed error analysis is derived. Finally, we give some numerical examples, and show the characteristic phenomena of fractional Relaxation-Oscillation equation's solution.In Chapter 3, we consider the fractional-order dynamical controlled systems. The multi-order fractional differential equation is transferred into a system of fractional-order differential equations. A new computationally effective fractional Predictor-Corrector method is proposed for simulating the fractional order systems and controllers. A detailed error analysis is derived. Finally, we give some numerical examples.In Chapter 4, we consider the application in practical physical models. We consider four chaotic models: fractional chaotic oscillator model, Chaotic "jerk" model, Chen system, Chaotic systems using state feedback controller. A computationally effective fractional Predictor-Corrector method is proposed for simulating the fractional order Chaotic systems. Finally, we give some numerical examples. The numerical results are in agreement with chaotic physical phenomena.The second kind of problems, we consider the fractional anomalous modified anomalous subdiffusion problem, which are consisted of Chapters 5 and 6.In Chapter 5, we consider a modified anomalous subdiffusion equation with nonlinear source terms for describing processes that become less anomalous as time pro- gresses by the inclusion of the second fractional time derivative acting on the diffusion term. It is an open problem. An implicit difference method is constructed. The stability and convergence are discussed using a new energy method. Finally, we give some numerical examples. The numerical results demonstrate the effectiveness of theoretical analysis.In Chapter 6, we propose a new implicit fractional Predictor-Corrector Trapezoidal method for the modified anomalous subdiffusion equation. Firstly, we give out the numerical approximation of time fractional Riemann-Liouville derivative. Using numerical techniques, the modified anomalous subdiffusion equation is transformed into a system of ordinary differential equations (ODE). The implicit fractional Predictor-Corrector Trapezoidal method for the ODE is proposed. There are some advantages: no need for iterative, high-precision, having same coefficient matrix in predictor and corrector methods. Finally, numerical results are given to demonstrate the effectiveness of this method. This technique can also be applied to solve other types of fractional partial differential equations.