In this paper, we state and prove a new formula expressing explicitly the integratives of Bernstein polynomials (or B-polynomials) of any degree and for any fractional-order in terms of B-polynomials themselves. We derive the transformation matrices that map the Bernstein and Legendre forms of a degree-n polynomial on OE0, 1 into each other. By using their transformation matrices, we derive the operational matrices of integration and product of the Bernstein polynomials. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations.

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Plasmas interacting with organic liquids can lead to novel synthetic processes that are not feasible with conventional vacuum systems due to vapor pressure limitations. Of particular interest are processes that eliminate the generation of waste-streams. Here we show that He+O2 plasma can drive the epoxidation of alkenes in solution, in a process that generates epoxides without oxidant waste-streams, runs at room temperature and atmospheric pressure, and requires no catalyst. The reactions between different reactive oxygen species generated in the plasma and the target alkene, trans-stilbene in this study, have been identified and optimization of the plasma conditions within the constraints of the current experimental setup have led to yields of ~70%, which are of preparative interest.

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