A Numerical Method For Fractional Calculus Equation

With the rapid development of Computer Science, Application of fractional calculuslt has penetrated into many engineering fields.But because of the complicated mathematical form of fractional derivative,the generalized differential equation with fractional derivative to obtain analytical problem solving,So in recent years,many scholars have studied the numerical method for fractional differential equations.This article starts from the fractional calculus and the basic theory of fractional differential equations,discusses,the numerical methods for fractional differential equation, based on the fractional derivative difference scheme,with Adams-Bashforth-Moulton one step method, propose the coupling method for solving differential equations of fractional order.Mainly to do the following:chapter one, summarize the study reviewed problems,provide the fractional calculus origin,the development history and the research direction and development. The second chapter, based on the concept of Riemann—Liouville fractional calculus and Caputo’s fractional differential concept,collected and proved the basic properties of the fractional calculus, the fractional differential and integral,some operational properties and the integer order calculus.Let the fractional calculus and fractional calculus harmony,unify,systemize the theory further system of real number order calculus, easily applicated.chapter three, proofed the Riemann-Liouville fractional calculus and the fractional derivative operator in the sense of Caputo boundedness.Proofed the Riemann-Liouville fractional derivative existence and uniqueness of solutions of initial value problem in the sense of Caputo.The fourth chapter, researched the numerical algorithm for fractional differential equations, using the equivalence of Voltta integral equations, through differential and Adams-Bashforth-Moulton one step method established a coupling algorithm Caputo fractional derivative of initial value problem, and successfully used to solve Bagley-Torvik equationThrough numerical experiments, demonstrate the convergence of the algorithm and the order of error.