Complete monotonicity-preserving numerical methods for time fractional ODEs
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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 (CM-preserving) numerical methods for fractional ODEs, in which the discrete convolutional kernel inherits the CM property as the continuous equations. Three concrete numerical schemes, including the Grünwald-Letnikov formula, numerical method based on piecewise interpolation formula and convolutional quadrature based on θ-method, are proved to be 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 CM-preserving numerical methods for Volterra integral equations with general completely monotone kernels. Numerical examples are presented to illustrate the main theoretical results.
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