【作者】 冯光；

【导师】 肖爱国；

【作者基本信息】 湘潭大学 ， 计算数学， 2008， 硕士

【摘要】 近年来,分数阶微积分在科学工程领域的广泛应用引起了人们很大的兴趣。在电化学过程、色噪声、控制理论、流体力学、混沌、生物工程等领域,分数阶微积分是有用的数学工具。在使用分数阶导数的模型中,大部分情况下会导致出现一系列的分数阶微分方程。尽管有些方程的解析解可以求出来,但人们注意到,很多分数阶微分方程的解析解是由比较特殊的函数来表示,而要数值地表示这些特殊函数是很困难的,并且有些非线性方程是不可能求出其解析解的,于是,人们越来越关注分数阶微分方程的数值方法。本文考虑了一类分数阶微分方程初值问题的数值求解Caputo导数的性质使得该初值问题可以等价地简化为Volterra积分方程,因此,求解Volterra积分方程的数值方法同样能够应用于分数阶微分方程的数值求解。在第三章,将求解普通积分方程的Adams技巧应用于分数阶微分方程模型,得到一个求解分数阶微分方程的显式算法,给出了误差估计,并通过数值实验说明该算法是有效的。在第四章,以[34]中的方法为基础,作了微小的改动,得到一个隐式的算法,并给出了误差分析。在第五章,以第三章和第四章的方法为基础,得到一个新的预校算法,给出了误差估计,并且通过数值实验证明该方法是有效的;实验说明,该算法在1＜α＜2时比[31]所提的预校算法具有更高的数值精度。在第三章和第五章中均对分数阶松驰—震荡方程作了计算,不但说明了文中算法的有效性,还通过图形显示出解从松驰性态逐渐过渡到震荡性态的情况。更多还原

【Abstract】 In recent years, fractional integrals (FIs) and fractional derivatives (FDs) have drawn much attention due to its wide application in many science and engineering fields. They are very useful mathematic tools in electrochemical processes, colored noise, controllers theory, fluid mechanics, chaos, biology engineering etc. Modeling of systems using FDs lead in most cases, a set of fractional differntial equations (FDEs). Though some analytic solutions of FDEs can be resolved, many solutions of them are expressed by some special functions, and it is hard to express numerically. Generally, nonlinear FDEs can not be resolved analytically. Hence there has been a growing interest to develop numerical techniques.Numerical methods for a class of initial problems of FDEs are considered in this paper. Properties of Caputo derivative allow one to reduce the FDEs into a Volterra-type integral equation, therefore, the numerical schemes for Volterra-type integral equations can also be applied to the numerical solution of FDEs. In the third chapter, we apply the Adams technics developed for Volterra-type integral equation to the FDEs, and then obtain an explicit numerical schemes, error analysis is also presented, and numerical examples verify the efficiency of the numerical method. In the fourth chapter, based on the approach presented in [34], with a little improvement, an implicit numerical scheme is obtained, and error analysis is also presented. In the fifth chapter, based on the schemes presented in the third chapter and fourth chapter, we obtain a new predictor-corrector method, then present the error analysis, and numerical examples verify the efficiency of the numerical method too, which show that the method can provide higher precision when 1 <α< 2 than the one in [31]. Both in the third and the fifth chapter, fractional Relaxation-Oscillation equation was numerically resolved, which doesn’t only show the efficiency of the numerical method, and also explain the situation varying from relaxation to oscillation by the picture of the numerical solution.更多还原