Types Of Fractional Differential Equation

In recent years,Fractional differential equations have been widely used in various fields,such as physics,Mechanics,Biology,and Medicine etc,and great success has been gained already.From the general review of its development,it not only shows profound research background and good application fundamentals,but also has a wide development space.Therefore,it seems very importmant to study the basic theory,nature and solution of fractional differential equations.These researches establish a solid foundation for further application of fractional differential equations.This paper has done works as follows:First,we give some concerning fractional calculus to prepare the knowledge and present basic definitions and preperties of fractional calculus.Second,the fractional regression model is considered. The analytical solution of the fractional regression model is derived by the corresponding Green's function.And we can obtain its exact solution by the operational method.Obviously,it is difficult showing its analytical solution and its exact solution numerically.Then the Taylor's expansion method is proposed for solving the fractional differentional equations of regression model for its approximate solution. Third,starting from the basic fractional ordinary differential equations, the existence and uniqueness of solution for the fractional-order differential equation is given.Then we apply a high order approximation of fractional derivative by Lubich to fractional differential equation,construct a high numerical difference scheme to solve the fractional differential equation,present error analysis of the algorithms theoretically,and prove the consistency,convergency and stability.