Complete monotonicity-preserving numerical methods for time fractional ODEs
Complete monotonicity-preserving numerical methods for time fractional ODEs
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DOI：
10.4310/cms.2021.v19.n5.a6
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发表时间：
2019-09
期刊：
影响因子：
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通讯作者：
Lei Li-;Dongling Wang
中科院分区：
文献类型：
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作者：
Lei Li-;Dongling Wang
时间分数常微分方程等价于具有弱奇异核的卷积沃尔泰拉积分方程。相应的核是典型的完全单调函数。因此，我们引入了分数常微分方程的完全单调性保持（$\mathcal{CM}$-保持）数值方法的概念，其中离散卷积核继承了连续方程的$\mathcal{CM}$属性。证明了Grunwald-Letnikov公式、基于分段插值公式的数值方法和基于$\theta$-方法的卷积求积三种具体的数值格式是$\mathcal{CM}$-保持的。这类新的数值格式，当应用于时间分数阶次扩散方程和分数阶常微分方程时，使我们能够在一个统一的框架内建立收敛性。这类格式的另一个优点是，对于标量非线性自治分数阶常微分方程，它们可以保持数值解的单调性。分析中的主要工具是由于Li和Liu（Quart.应用数学：76（1）：189-198，2018）以及由于Liu和Pego的Pick函数与完全单调序列的生成函数之间的等价性（trans.Amer.Math.Soc.368（12）：8499-8518，2016）。分数阶常微分方程的结果可以推广到具有一般完全单调核的沃尔泰拉积分方程的$\mathcal{CM}$-保持数值方法。数值例子来说明主要的理论结果。
The time fractional ODEs are equivalent to the convolutional Volterra integral equations with a weakly singular kernel. The corresponding kernel is a typical completely monotone function. We therefore introduce the concept of complete monotonicity-preserving ($\mathcal{CM}$-preserving) numerical methods for fractional ODEs, in which the discrete convolutional kernel inherits the $\mathcal{CM}$ property as the continuous equations. Three concrete numerical schemes, including the Grunwald-Letnikov formula, numerical method based on piecewise interpolation formula and convolutional quadrature based on $\theta$-method, are proved to be $\mathcal{CM}$-preserving. This class of new numerical schemes, when applied to time fractional sub-diffusion equations and fractional ODEs, allow us to establish the convergence in a unified framework. Another advantage of this kind of schemes is that, for scalar nonlinear autonomous fractional ODEs, they can preserve the monotonicity of the numerical solutions. The main tools in the analysis are convolution inverse for a completely monotone sequence due to Li and Liu (Quart. Appl. Math., 76(1):189-198, 2018) and the equivalence between Pick function and the generating function of a completely monotone sequence due to Liu and Pego (Trans. Amer. Math. Soc. 368(12):8499-8518, 2016). The results for fractional ODEs can be extended to $\mathcal{CM}$-preserving numerical methods for Volterra integral equations with general completely monotone kernels. Numerical examples are presented to illustrate the main theoretical results.