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In this paper, the averaging principle for stochastic Caputo fractional differential equations with the nonlinear terms satisfying the non-Lipschitz condition is considered. The work in the article is roughly divided into three parts. Firstly, we establish a generalized Gronwall inequality with singular integral kernel which is a key part in our analysis. Secondly, we discuss the existence and uniqueness of solution. And finally, the averaging principle is considered. [ABSTRACT FROM AUTHOR]

The solidification characteristics and segregation behavior of Cu-15Ni-8Sn alloy were systematically investigated in the present study. The solidification characteristics were revealed with the assistance of a solidification quenching experiment, DTA analysis and Scheil simulation. The solidification microstructure of Cu-15Ni-8Sn alloy was characterized by SEM, and the results indicated that the as-cast microstructure of Cu-15Ni-8Sn alloy mainly consists of Sn-depleted α-Cu(Ni,Sn) matrix, Sn-rich γ phase and lamellar (α + γ) structure. It has been demonstrated that the solidification process begins with the nucleation and growth of primary Sn-depleted α1 phase ( L → α 1 at 1114 °C) and terminates with the divorced eutectic reaction ( L 2 → α 2 + γ at 868 °C). During the subsequent cooling process, the discontinuous precipitation ( α 2 → α 1 + γ ) takes place in the temperature range from about 700 °C to 600 °C. In addition, the macrosegregation behavior of Sn in Cu-15Ni-8Sn alloy was investigated by adopting vertical unidirectional solidification and measuring the cooling curves at different positions. The results indicated that an inverse macrosegregation of Sn solute exists in the as-prepared ingot, which mainly segregates at the chill surface of the alloy ingot. Namely, the Sn content is higher than 8 wt pct at the chill surface and about 8 wt pct inside the ingot for Cu-15Ni-8Sn alloy. [ABSTRACT FROM AUTHOR]

We investigate the averaging principle for multivalued stochastic differential equations (MSDEs) driven by a random process under non-Lipschitz conditions. We consider the convergence of solutions in $L^{p}~(p\geq2)$ and in probability between the MSDEs and the corresponding averaged MSDEs. [ABSTRACT FROM AUTHOR]

The Caratheodory approximation for a type of Caputo fractional stochastic differential equations is considered. As is well known, under the Lipschitz and linear growth conditions, the existence and uniqueness of solutions for some type of differential equations can be established. However, this approach does not give an explicit expression for solutions; it is not applicable in practice sometimes. Therefore, it is important to seek the approximate solution. As an extending work for stochastic differential equations, in this paper, we consider Caratheodory's approximate solution for a type of Caputo fractional stochastic differential equations. [ABSTRACT FROM AUTHOR]