High Order Approximation Method For Fractional Ordinary Differential Equations And Its Applications

In recent years, fractional derivatives and integrals play an important role in various fields of modern science,especially in biology,engineering, physics, finance, hydrology,fractional-order controllers, polymer rheology,and fractional derivative scoelastic. The most significant advantage of the fractional order models in comparison with integer-order models is that it is based on important fundamental physical considerations. However, because of the absence of appropriate mathematical methods, numerical methods and theoretical analysis of fractional calculation are still very difficult tasks.Therefore,the research of numberical method of the fractional differential equation is of important theory significance and practical value.In this paper,we consider the high order approximation method for fractional ordinary differential equations and its applications. Introduction gives some background information about fractional calculus and presents basic definitions and properties of fractional calculus. In Chapter 2,starting from the basic nonlinear fractional ordinary differential equations, a high order numerical difference scheme is constructed to solve nolinear fractional differential equations by using high order approximations of Rieman-Liouville fractional derivative. In Chapter 3, the proof of convergency and stability is given. In Chapter 4, Numerical examples are presented, which verify the efficiency of the above method.