A new predictor-corrector method for the numerical solution of fractional differential equations

In this paper, first, a new predictor-corrector method( called the modified predictorcorrector method ) for the fractional differential equations is developed. And the new method can be formally viewed as a modification as the classical predictor-corrector method. Second, it is proved that the numerical accuracy of this new method is superior to that of the classical predictorcorrector method by two numerical examples. The comparison with the corresponding results demonstrates that the modified predictor-corrector method is more accurate than the classical predictor-corrector method when solving fractional differential equations numerically. Introduction In recent years, fractional calculus has got dramatic advances in both theory and application. For the no local property, the fractional calculus has been used to describe the basic nature of almost all sciences and engineering fields, such as fluid flow in porous materials, anomalous diffusion transport, acoustic wave propagation in viscoelastic materials, dynamics in self-similar structures, signal processing, financial theory, electric conductance of biological systems and so on[1]. However, the development process of numerical algorithm of fractional differential equations is slow at work. In 1986, Lubich[2] firstly promoted the BDF into the numerical calculation of the fractional differential equations and gained the form of the fractional BDF. And in 1993,Lubich and Ostermann[3] gained a high order approximation scheme for solving the fractional differential equations. And in 1998-2002 Diethiem[4-5]pointed an Adams-type predictor-corrector method for the numerical solution of fractional differential equations and the corresponding error analysis. In 2006, Odibat[6]presented an algorithm to numerically approximate the fractional integration and Caputo fractional differentiation. In 2011, Li, Chen and Ye[7] proposed some high-order numerical approximations for fractional integrals based on cubic Hermite interpolation and cubic spline interpolation. In 2013, Gao[8] proposed a method to approximate the Caputo fractional derivative by the quadratic interpolation. The plan of the remainder is as follows. In Section2, a new Adams-type predictor-corrector method( called the modified predictor-corrector method ) for the fractional differential equations is developed. In Section3, one test examples are used to confirm the numerical accuracy of the new method. The computational results are compared with the corresponding ones with the classical predictor-corrector method. Finally, a brief conclusion and the further work have been listed. 2. Derivation of the modified predictor-corrector method In this section, we will describe the classical predictor-corrector method and the process of the derivation of the modified predictor-corrector method in detail. Consider the fractional differential equation with initial conditions 3rd International Conference on Machinery, Materials and Information Technology Applications (ICMMITA 2015) © 2015. The authors Published by Atlantis Press 1638 ( ) ( ) ( ) ( ) ( ) ( ) * 0 , 0,1, 1 k k D y t f t y t y t y k m α  =   = = −   (2.1) Equation (2.1) is equivalent to the Volterra integral equation ( ) ( ) [ ] ( ) ( ) ( ) ( ) 1 1 0 0 0 1 , ! t k k k t y t y t f y d k α α t t t t α − −