Research On Numerical Solutions Of Several Classes Of Fractional Differential Equations

In recent years,due to the wide applications of fractional differential equations in life sciences,engineering,physics and other disciplines,it is important for researchers to solve such equations.However,it is not an easy task to obtain the exact solutions of fractional differential equations.Thus,how to acquire numerical solutions of fractional differential equations has become a hot research domain.In general,methods of solving integer-order differential equations are no longer suitable for solving fractional differential equations.Therefore,it is necessary to construct new numerical methods for solving fractional differential equations.At present,there are some numerical methods for solving fractional differential equations including hybrid collocation method,homotopy analysis transform method,residual power series method,variational iterative method and so on.The numerical solutions of three kinds of fractional differential equations are studied in this thesis.The fractional terminal value problems are solved by the hybrid collocation method.The residual power series method and homotopy analysis transform method are applied to solve fractional partial differential equations with proportional delay and fractional predator-prey system,respectively.The specific content of the thesis is as follows:Firstly,the research background and status of fractional differential equations are collected in the first chapter.Then,the related definitions and theorems are given in the second chapter,such as fractional derivatives,fractional power series and Laplacian.In the third chapter,a hybrid collocation method for the fractional terminal value problems is given.Based on the shooting method,we convert fractional terminal value problems to initial value problem at first,then the initial value problem is transformed into the Volterra integral equation with weakly kernel.Finally,we provide a hybrid collocation method for solving it.Though numerical examples,we find that the hybrid collocation method is reliable for solving the terminal value problems of fractional differential equations.In the fourth chapter and the fifth chapter,the detailed introduction of the residual power series method and the homotopy analysis transform method for solving the time fractional partial differential equations with proportional delay as well as the fractional predator-prey system are presented respectively.By applying the theoretical methods to numerical experiments,the approximate solutions and absolute error results of the corresponding equations are obtained.The efficient of the proposed methods is illustrated by comparing the numerical results with those acquired by other existing methods.The sixth chapter summarizes the full context.