Yanyan Yu

Yanyan Yu, Weihua Deng, Yujiang Wu et al.

Power-law probability density function (PDF) plays a key role in both subdiffusion and Lévy flights. However, sometimes because of the finite of the lifespan of the particles or the boundedness of the physical space, tempered power-law PDF seems to be a more physical choice and then the tempered fractional operators appear; in fact, the tempered fractional operators can also characterize the transitions among subdiffusion, normal diffusion, and Lévy flights. This paper focuses on the quasi-compact schemes for space tempered fractional diffusion equations, being much different from the ones for pure fractional derivatives. By using the generation function of the matrix and Weyl's theorem, the stability and convergence of the derived schemes are strictly proved. Some numerical simulations are performed to testify the effectiveness and numerical accuracy of the obtained schemes.

Yanyan Yu, Weihua Deng, Yujiang Wu

In this paper, we design a semi-implicit scheme for the scalar time fractional reaction-diffusion equation. We theoretically prove that the numerical scheme is stable without the restriction on the ratio of the time and space stepsizes, and numerically show that the convergent orders are 1 %$2-α$ in time and 2 in space. As a concrete model, the subdiffusive predator-prey system is discussed in detail. First, we prove that the analytical solution of the system is positive and bounded. Then we use the provided numerical scheme to solve the subdiffusive predator-prey system, and theoretically prove and numerically verify that the numerical scheme preserves the positivity and boundedness.

Yanyan Yu, Weihua Deng, Yujiang Wu

The continuous time random walk (CTRW) underlies many fundamental processes in non-equilibrium statistical physics. When the jump length of CTRW obeys a power-law distribution, its corresponding Fokker-Planck equation has space fractional derivative, which characterizes Lévy flights. Sometimes the infinite variance of Lévy flight discourages it as a physical approach; exponentially tempering the power-law jump length of CTRW makes it more `physical' and the tempered space fractional diffusion equation appears. This paper provides the basic strategy of deriving the high order quasi-compact discretizations for space fractional derivative and tempered space fractional derivative. The fourth order quasi-compact discretization for space fractional derivative is applied to solve space fractional diffusion equation and the unconditional stability and convergence of the scheme are theoretically proved and numerically verified. Furthermore, the tempered space fractional diffusion equation is effectively solved by its counterpart of the fourth order quasi-compact scheme; and the convergence orders are verified numerically.

Yi Zhang, Fan Wei, Jingyi Li et al.

The use of children's drawings to examining their conceptual understanding has been proven to be an effective method, but there are two major problems with previous research: 1. The content of the drawings heavily relies on the task, and the ecological validity of the conclusions is low; 2. The interpretation of drawings relies too much on the subjective feelings of the researchers. To address this issue, this study uses the Large Language Model (LLM) to identify 1420 children's scientific drawings (covering 9 scientific themes/concepts), and uses the word2vec algorithm to calculate their semantic similarity. The study explores whether there are consistent drawing representations for children on the same theme, and attempts to establish a norm for children's scientific drawings, providing a baseline reference for follow-up children's drawing research. The results show that the representation of most drawings has consistency, manifested as most semantic similarity>0.8. At the same time, it was found that the consistency of the representation is independent of the accuracy (of LLM's recognition), indicating the existence of consistency bias. In the subsequent exploration of influencing factors, we used Kendall rank correlation coefficient to investigate the effects of "sample size", "abstract degree", and "focus points" on drawings, and used word frequency statistics to explore whether children represented abstract themes/concepts by reproducing what was taught in class. It was found that accuracy (of LLM's recognition) is the most sensitive indicator, and data such as sample size and semantic similarity are related to it; The consistency between classroom experiments and teaching purpose is also an important factor, many students focus more on the experiments themselves rather than what they explain.