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.
翻译:在连续时间随机行走模型中,时间差操作员通常表示无限的等待时间概率密度。 与通常的环境不同, 这项工作考虑到时间差操作员, 反映了一定的等待时间概率密度。 首先, 我们分析缓冲基准问题的解决方案, 这表明了接近初始时间的微弱单一性。 L1 级网格方案和WSGL 公式中含有修正术语的二层问题被调整, 以应对非超模化解决方案, 我们从趋同和消耗的 CPU 时间的角度系统地比较这两种方法。 此外, 快速计算有节制的Caputo 节奏分数衍生出的时间差, 其依据是超强的超速速速速度近似, 大大缩短运行时间。 此外, 减速操作员适用于核磁共振反应中的布洛奇方程式, 以及具有明显记忆效果的复合材料的两层问题, 分析( 半分析) 和数字性解决方案是使用变异技术和一定的差异方法。 此外,, 数据更新的结果证实, 缓冲的节制的节制的模型型模型型价制的分制的分数制的分数制, 将大大地描述MRI 的分数的分数数据。