DISSIPATIVITY AND CONTRACTIVITY ANALYSIS FOR FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATIONS AND THEIR NUMERICAL APPROXIMATIONS
DISSIPATIVITY AND CONTRACTIVITY ANALYSIS FOR FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATIONS AND THEIR NUMERICAL APPROXIMATIONS
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分数阶泛函微分方程的耗散性和收缩性分析及其数值近似
DOI：
10.1137/17m1121354
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发表时间：
2019-01-01
影响因子：
2.9
通讯作者：
Zou, Jun
中科院分区：
文献类型：
--
作者：
Wang, Dongling;Zou, Jun
首先给出了一类新的时滞相关的分数阶Halanay不等式的推广，刻画了分数阶泛函微分方程的渐近性态。然后研究了具有有界吸收集的F-FDE的耗散性和具有代数压缩率的F-FDE的渐近稳定性和压缩性。基于Grunwald-Letnikov公式和Caputo分数阶导数的L1方法，结合函数项的线性插值，进一步构造了两种F-FDE的数值格式.证明了这两种格式是耗散的、压缩的，并且能保持连续方程的精确衰减率。这些结果可以直接应用于某些特殊的F-FDE情形，如分数阶延迟微分方程、分数阶积分微分方程和分数阶延迟积分微分方程。最后给出了几个数值例子来说明保结构数值方法的优点。特别是，我们将比较我们的计划和流行的预测-校正算法的F-FDES的数值性能，并证明我们的计划是更有效和强大的，特别是对一些刚性系统。
We first present a new delay-dependent fractional generalization of Halanay-like inequality to characterize the asymptotic behavior of fractional functional differential equations (F-FDEs). Then we study the dissipativity of F-FDEs with a bounded absorbing set and the asymptotic stability and the contractivity of F-FDEs with algebraically contractive rate. Two numerical schemes are further constructed for F-FDEs based on Grunwald-Letnikov formula and L1 method for Caputo fractional derivative, together with linear interpolation for the functional terms. These two schemes are proved to be dissipative and contractive and can preserve the exact decay rate as the continuous equations. These results can be directly applied to some special cases of F-FDEs, such as the fractional delay differential equations, fractional integro-differential equations, and fractional delay integro-differential equations. Finally, several numerical examples are given to illustrate the advantages of the structure-preserving numerical methods. In particular, we shall compare the numerical performance of our schemes and the popular predictor-corrector algorithms for F-FDEs and demonstrate that our schemes are more efficient and robust, especially for some stiff systems.