Numerical Method For Delay Fractional Equation

Many scholars begin to pay attention to study related problems of delay differentialequations and fractional differential equations due to their more and more applicationsin various fields. Some of them begin to study problems of delay fractional differentialequations such as the existence and uniqueness of solutions, stability and numericalmethod. This paper mainly studies the numerical method of delay fractional differentialequations. In the fractional calculus, there are many definitions of the fractionalderivatives including Riemann-Liouville fractional derivatives, Caputo fractionalderivatives, etc. The differential equations studied in this paper are defined in Caputofractional derivatives. We study two classes of equations: linear fractional differentialequation with constant delay and linear fractional differential equation with proportionaldelay.In this paper, two methods, variational iteration method and Haar waveletcollocation method, are used to solve the two classes delay fractional differentialequations. These two methods have been effectively used in solving other classes ofequations. But it’s not commonly used in solving delay fractional differential equations.Fisrt, this paper gives the mathematical preliminaries of the methods. Then the formulaeof the two methods for the two classes of equations and their theoretical derivation aregiven. We use software of Mathematica and Matlab to program to realize the algorithmof the methods and solve the numerical examples. Then we give the graph of thenumerical solution and exact solution, the numerical value of error at several points, andgraph of relative error along time. In the variational iteration method, the chart of maxrelative error corresponding to the iterations is given. In the Haar wavelet collocationmethod, the graphs of the solutions corresponding to the increasing dimensions of theequation set are given. From the result of the numerical examples and the characteristicsof the methods itself, we give the evaluation of the two methods on precision ofsolutions, execution efficiency of the algorithm and the general applicability. Theadvantages and disadvantages of the two methods are summarized.