Trapped dynamics widely appears in nature, e.g., the motion of particles in viscous cytoplasm. The famous continuous time random walk (CTRW) model with power law waiting time distribution (having diverging first moment) describes this phenomenon. Because of the finite lifetime of biological particles, sometimes it is necessary to temper the power law measure such that the waiting time measure has convergent first moment. Then the time operator of the Fokker-Planck equation corresponding to the CTRW model with tempered waiting time measure is the so-called tempered fractional derivative. This paper focus on discussing the properties of the time tempered fractional derivative, and studying the wellposedness and the Jacobi-predictor-corrector algorithm for the tempered fractional ordinary differential equation. By adjusting the parameter of the proposed algorithm, any desired convergence order can be obtained and the computational cost linearly increases with time. And the effectiveness of the algorithm is numerically confirmed.

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Incorporating subdiffusive mechanisms into the Klein-Kramers formalism leads to the fractional KleinKramers equation. Then, the equation can effectively describe subdiffusion in the presence of an external force field in the phase space. This article presents the finite difference methods for numerically solving the fractional Klein-Kramers equation and does the detailed stability and error analyses. The stability condition, mv 2 R β ≤ 16, shows the ratio between the kinetic energy of the particle and the temperature of the fluid can not be too large, which well agrees with the physical property of the subdiffusive particle, we call it "physical constraint." The numerical examples are provided to verify the theoretical results on rate of convergence. Moreover, we simulate the fractional Klein-Kramers dynamics and the simulation results further confirm the effectiveness of our numerical schemes.

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Impeller blowers are used to convey materials for various forage harvesters. As the main working component, the throwing impeller endures various static and dynamic loads while conveying the materials. This makes the throwing impeller prone to fatigue fracture, so it is very necessary to find a feasible model to estimate the fatigue life of the throwing impeller accurately. In order to obtain the accurate random cyclic load applied on the high-speed rotating impeller, the finite element analysis and the fluid-solid coupling method are adopted to calculate the stress distribution of the impeller under the combined action of the fluid-solid coupling flow field pressure, the centrifugal force, and the gravity. At the same time, the stress on the dangerous section of the impeller is measured by using the DH5909 wireless strain testing system and is compared with the calculated one. The contrast results show that the numerical calculation results are reliable. To accurately predict the fatigue life of the throwing impeller at the design stage, the two-parameter nominal stress model is deduced and combined with the linear cumulative damage model of Miner and the lognormal distribution model. Its two parameters of the average stress S m and the stress amplitude S a can be obtained through finite element analysis and do not have to be equivalent to a symmetrical cyclic load. Therefore, its precision of estimating the fatigue life is improved. By contrasting the rated and predicted fatigue lives of an impeller, it was found that the impeller's actual rated lives are closer to the predicted lives of the Goodman and Gerber twoparameter nominal stress model than those of the conventional S-N curve. In particular, they are closer to the calculation results of the Gerber-type two-parameter nominal stress model. This shows that the Gerber-type two-parameter nominal stress model is more accurate and suitable to predict the fatigue life of the throwing impeller. These achievements will play a significant role in further optimizing the impeller and improving its reliability.

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