Analytical and numerical investigation on the tempered time-fractional operator with application to the Bloch equation and the two-layered problem

TL;DR

This paper investigates the tempered time-fractional operator, analyzing its mathematical properties, developing efficient numerical schemes, and applying it to model diffusion in MRI and layered materials, showing improved flexibility and computational efficiency.

Contribution

It introduces a systematic analysis of the tempered time-fractional operator, compares numerical methods, develops a fast computation technique, and applies the model to practical problems in MRI and layered media.

Findings

Tempered operator accelerates diffusion compared to fractional index.

The developed numerical schemes effectively handle non-smooth solutions.

Tempered model better fits MRI data and captures complex diffusion behaviors.

Abstract

In the continuous time random walk model, the time-fractional operator usually expresses an infinite waiting time probability density. Different from that usual setting, this work considers the tempered time-fractional operator, which reflects a finite waiting time probability density. Firstly, we analyse the solution of a tempered benchmark problem, which shows a weak singularity near the initial time. The L1 scheme on graded mesh and the WSGL formula with correction terms are adapted to deal with the non-smooth solution, in which we compare these two methods systematically in terms of the convergence and consumed CPU time. Furthermore, a fast calculation for the time tempered Caputo fractional derivative is developed based on a sum-of-exponentials approximation, which significantly reduces the running time. Moreover, the tempered operator is applied to the Bloch equation in nuclear magnetic resonance and a two-layered problem with composite material exhibiting distinct memory effects, for which both the analytical (or semi-analytical) and numerical solutions are derived using transform techniques and finite difference methods. Data fitting results verify that the tempered time-fractional model is much effective to describe the MRI data. An important finding is that, compared with the fractional index, the tempered operator parameter could further accelerate the diffusion. The tempered model with two parameters $\alpha$ and $\rho$ are more flexible, which can avoid choosing a too small fractional index leading to low regularity and strong heterogeneity.