| 分数阶常微分方程的改进精细积分法 |
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| 引用本文： | 鲍四元,沈峰.分数阶常微分方程的改进精细积分法[J].应用数学和力学,2019,40(12):1309-1320. |
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| 作者姓名： | 鲍四元 沈峰 |
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| 作者单位： | 1苏州科技大学 江苏省结构工程重点实验室,江苏 苏州 215011;2苏州科技大学 土木工程学院 工程力学系,江苏 苏州 215011 |
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| 基金项目： | 国家自然科学基金(11202146；51709194) |
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| 摘 要： | 基于Mittag-Leffler函数的定义式，构造Mittag-Leffler矩阵函数的精细迭代计算格式.与常规指数函数的迭代格式相比，迭代递推中多了修正项，其表达式与分数阶导数的阶次有关.对于以Caputo分数导数定义的动力学分数阶常微分方程，使用基于Mittag-Leffler函数的精细积分法可计算方程解在各时间段端点对应函数值.算例表明了所提计算方法的有效性，其精度可由所增加修正项的阶次控制.
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| 关 键 词： | Mittag-Leffler函数 精细迭代格式 修正项 分数阶常微分方程 Caputo分数阶导数 |
| 收稿时间： | 2018-12-24 |
An Improved Precise Integration Method for Fractional Ordinary Differential Equations |
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| Institution: | 1Key Laboratory of Structural Engineering of Jiangsu Province,Suzhou University of Science and Technology, Suzhou, Jiangsu 215011,P.R.China;2Department of Engineering Mechanics, School of Civil Engineering,Suzhou University of Science and Technology, Suzhou, Jiangsu 215011, P.R.China |
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| Abstract: | Based on the definition of the Mittag-Leffler function, the precise iteration computation scheme for the Mittag-Leffler matrix function was constructed. Compared with the normal iteration scheme for exponential functions, the constructed scheme has additional correction items. The expression of the correction item is related to the order of the fractional derivative. For dynamic fractional ordinary differential equation D(α)v=Hv with the Caputo fractional definition, the solution function value at the endpoint of the time phase can be obtained with the precise iteration method. The numerical examples demonstrated effectiveness and efficiency of the presented method. |
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