The Numerical Solutions Of Two Classes Of Fractional Differential Equations

The fractional calculus,including integrals and derivatives of arbitrary order,was first founded by Leibnitz in 1695.The theory of fractional calculus is widely used in engineering,physics and other fields of applied sciences.Due to its wide application,fractional calculus has gradually become the focus of research.Fractional differential equations are generalized,non-integer differential equations.It has been found that fractional differential equations are closer to the actual situation than integer-order differential equations when establishing system models for certain practical problems.In this paper,the numerical methods for the initial value problems of two kinds of fractional equations are studied,namely the time-fractional two-component evolutionary system of order 2 and the fractional ordinary differential equation.Firstly,the background of fractional calculus and the numerical methods of existing fractional differential equations are introduced.Some important definitions and theorems used in this paper are briefly introduced,and the definitions of the Riemann-Liuville and Caputo type fractional derivatives are given.Secondly,the coupled fractional reduced differential transform method is used to solve the numerical solution of the time-fractional two-component evolutionary system of order 2.The solution of the equation is expressed in the form of a generalized Taylor series,and the method is compared with the residual power series method.By truncating the absolute error between the series solution and the real solution,it is proved that the coupled fractional reduced differential transform method can effectively and accurately solve the time-fractional two-component evolutionary system of order 2.Finally,the hybrid collocation method is used to solve the initial value problem of fractional ordinary differential equations.The existence and uniqueness of the solution are analyzed,and the collocation scheme and convergence analysis of these problems are given.In this paper,numerical experiments are used to compare the effectiveness of the hybrid collocation method and graded collocation method insolving these problems.