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

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