Using Complete Monotonicity to Deduce Local Error Estimates for Discretisations of a Multi-Term Time-Fractional Diffusion Equation

Time-fractional initial-boundary problems of parabolic type are considered. Previously, global error bounds for computed numerical solutions to such problems have been provided by Liao et al. (SIAM J. Numer. Anal. 2018, 2019) and Stynes et al. (SIAM J. Numer. Anal. 2017). In the present work we show how the concept of complete monotonicity can be combined with these older analyses to derive local error bounds (i.e., error bounds that are sharper than global bounds when one is not close to the initial time t = 0 {t=0} ). Furthermore, we show that the error analyses of the above papers are essentially the same – their key stability parameters, which seem superficially different from each other, become identical after a simple rescaling. Our new approach is used to bound the global and local errors in the numerical solution of a multi-term time-fractional diffusion equation, using the L1 scheme for the temporal discretisation of each fractional derivative. These error bounds are α-robust. Numerical results show they are sharp.

On Some Complete Monotonicity of Functions Related to Generalized k − Gamma Function

In this paper, we presented two completely monotonic functions involving the generalized k − gamma function Γ k x and its logarithmic derivative ψ k x , and established some upper and lower bounds for Γ k x in terms of ψ k x .

Some Properties of the Kilbas-Saigo Function

We characterize the complete monotonicity of the Kilbas-Saigo function on the negative half-line. We also provide the exact asymptotics at −∞, and uniform hyperbolic bounds are derived. The same questions are addressed for the classical Le Roy function. The main ingredient for the proof is a probabilistic representation of these functions in terms of the stable subordinator.

Necessary and sufficient conditions for complete monotonicity and monotonicity of two functions defined by two derivatives of a function involving trigamma function

In the paper, by virtue of convolution theorem for the Laplace transforms, Bernstein's theorem for completely monotonic functions, some properties of a function involving exponential function, and other analytic techniques, the author finds necessary and sufficient conditions for two functions defined by two derivatives of a function involving trigamma function to be completely monotonic or monotonic. These results generalize corresponding known ones.