The time fractional ODEs are equivalent to convolutional Volterra integral equations with completely monotone kernels. We introduce the concept of complete monotonicity-preserving (CM-preserving) numerical methods for fractional ODEs, in which the discrete convolutional kernels inherit the CM property as the continuous equations. We prove that CM-preserving schemes are at least A(π/2) stable and can preserve the monotonicity of solutions to scalar nonlinear autonomous fractional ODEs. Significantly, by improving a result of Li and Liu (Quart. Appl. Math., 76(1):189-198, 2018), we show that the L1 scheme is CM-preserving. The good signs of the coefficients for such class of schemes ensure the discrete fractional comparison principles, and allow us to establish the convergence in a unified framework when applied to time fractional sub-diffusion equations and fractional ODEs. The main tools in the analysis are a characterization of convolution inverses for completely monotone sequences and a characterization of completely monotone sequences using Pick functions due to Liu and Pego (Trans. Amer. Math. Soc. 368(12): 8499-8518, 2016). The results for fractional ODEs are 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.

Reference: Lei Li, Dongling Wang. Complete monotonicity-preserving numerical methods for time fractional ODEs, arXiv preprint arXiv:1909.13060, 2019; to appear in Communications in Mathematical Sciences, 2021.

Prof. Dongling Wang received his PhD degree from Xiangtan University and is now an Associate Professor at School of Mathematics and Center of Nonlinear Science, Northwest University, China. Prof. Wang’ research interests include numerical methods for fractional differential equations and structure-preserving algorithms for Hamiltonian systems. Some of Prof. Wang’ research is published on journals like SIAM J. Numerical Analysis, Journal of Computational Physics and Communications in Mathematical Sciences.