Numerical Methods For A Class Of Initial Value Problems Of Fractional Differential Equations

In recent years, fractional integrals (FIs) and fractional derivatives (FDs) have drawn much attention due to its wide application in many science and engineering fields. They are very useful mathematic tools in electrochemical processes, colored noise, controllers theory, fluid mechanics, chaos, biology engineering etc. Modeling of systems using FDs lead in most cases, a set of fractional differntial equations (FDEs). Though some analytic solutions of FDEs can be resolved, many solutions of them are expressed by some special functions, and it is hard to express numerically. Generally, nonlinear FDEs can not be resolved analytically. Hence there has been a growing interest to develop numerical techniques.Numerical methods for a class of initial problems of FDEs are considered in this paper. Properties of Caputo derivative allow one to reduce the FDEs into a Volterra-type integral equation, therefore, the numerical schemes for Volterra-type integral equations can also be applied to the numerical solution of FDEs. In the third chapter, we apply the Adams technics developed for Volterra-type integral equation to the FDEs, and then obtain an explicit numerical schemes, error analysis is also presented, and numerical examples verify the efficiency of the numerical method. In the fourth chapter, based on the approach presented in [34], with a little improvement, an implicit numerical scheme is obtained, and error analysis is also presented. In the fifth chapter, based on the schemes presented in the third chapter and fourth chapter, we obtain a new predictor-corrector method, then present the error analysis, and numerical examples verify the efficiency of the numerical method too, which show that the method can provide higher precision when 1 <α< 2 than the one in [31]. Both in the third and the fifth chapter, fractional Relaxation-Oscillation equation was numerically resolved, which doesn't only show the efficiency of the numerical method, and also explain the situation varying from relaxation to oscillation by the picture of the numerical solution.