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计算数学  2016, Vol. 38 Issue (1): 1-24    DOI:
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分数阶微分方程的理论和数值方法研究
林世敏, 许传炬
厦门大学数学科学学院,福建省数学建模与高性能科学计算实验室, 厦门 361005
THEORETICAL AND NUMERICAL INVESTIGATION OF FRACTIONAL DIFFERENTIAL EQUATIONS
Lin Shimin, Xu Chuanju
School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientic Computing, Xiamen University, Xiamen 361005, Fujian, China
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摘要 分数阶偏微分方程的研究有很长的历史,并在最近十多年得到快速发展.相比极为有限的理论成果,数值方法的研究成果已经相当丰富,几个国际研究团队对此作出了贡献.本文旨在对分数阶微分方程的理论与数值方法研究成果做个简要的评价,聚焦总结评述与高阶方法发展密切相关的研究.主要内容为讨论最基本的三类方程:时间分数阶扩散方程、空间分数阶扩散方程、以及时空分数阶扩散方程的理论进展和数值方法研究在最近十年取得的结果.我们还有针对性地选择一些算例,用以说明几个重要方法的精度和有效性.
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关键词分数阶偏微分方程   弱解理论   数值方法   谱方法     
Abstract: The study of the fractional differential equations has a very long history, and is attracting increasing attention in recent years. As compared to the very limit theoretical work, signi cant progress has been made on numerical investigations. Several research groups have contributed to this progress. This paper has the objective to review the recent progress made in the theoretical and numerical studies of the fractional differential equations. We particularly focus on the development of high order numerical methods. The main content of the paper is to discuss the progress made in recent ten years on theoretical and numerical investigation of the three basic fractional equations: time fractional di usion equation, space fractional di usion equation, and time-space fractional di usion equation. We also provide some illustrative numerical examples to verify the accuracy and effciency of some selected numerical methods.
Key wordsfractional partial differential equation   weak solution   numerical method   spectral method   
收稿日期: 2015-09-01;
基金资助:

国家自然科学基金11471274,11421110001和91130002资助项目

引用本文:   
. 分数阶微分方程的理论和数值方法研究[J]. 计算数学, 2016, 38(1): 1-24.
. THEORETICAL AND NUMERICAL INVESTIGATION OF FRACTIONAL DIFFERENTIAL EQUATIONS[J]. Mathematica Numerica Sinica, 2016, 38(1): 1-24.
 
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