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In this letter, we defined a weighted Caputo-type differential operator which was used to character relaxation and diffusion models in two different types. Then, one of the weighted Caputo-type integral operators by solving the related linear differential equation was also defined. Subsequently, we studied their main properties. Lastly, the existence of a unique solution of the related nonlinear differential equation through the Banach fixed point theorem was presented. These basic facts lay a solid foundation for the concrete practice in the future.

Fractional differential equations play an essential role in describing the shallow water wave phenomena. In this article, we discussed the coupled Burgers-type equations in the sense of the fractional derivative of Riemann-Liouville. In return, a series of new results of this considered models, were obtained. In the first place, the formulation of the time fractional coupled Burgers-type equations by applying the Euler-Lagrange variational technology, was carried out. Then, the symmetry and one-parameter group of point transformations of this researched goals through the symmetry analysis scheme, were obtained. Subsequently, the time fractional coupled Burgers-type equations can be reduced into the fractional ordinary differential equations with the help of the Erdelyi-Kober fractional differential/integral operators. Next, the approximate solution and convergence analysis, were considered. At the same time, the stability analysis of the solitary wave, was also studied. Lastly, conservation laws of this discussed equations by using a new conservation theorem and nonlinear self-adjointness, were found. The series of results obtained above can provide strong support for us to reveal the mystery of the equation.

In this article, we investigated a nonlinear higher dimensional time fractional Korteweg–de Vries-type (KdV) equation. This considered equation is usually used to describe shallow water waves phenomena in physics. Here we found firstly the symmetry of the higher dimensional time fractional KdV-type equation in the sense of the Riemann–Liouville (RL) fractional derivative with the aid of the fractional Lie symmetry method. Then, the one-parameter group of Lie point symmetry transformation and some special solutions of this considered equation, were obtained. Next, the optimal system of one-dimensional Lie subalgebra of this considered equation, was constructed. Subsequently, on the basis of the multiple-parameter Erdelyi–Kober fractional differential operator (FDO) and Erdelyi–Kober fractional integral operator (FIO), the original equation can be reduced into the lower dimensional fractional differential equation (FDE). Furthermore, it can be translated further into more lower dimensional FDE with new symmetry from the reduced equation. Finally, conservation laws of this discussed equation are also found through a new conservation theorem. These results are favorable support for us to understand this dynamic model in a deeper level.

The adsorption of Hgn (n = 1–3) on the Au-, Ag-, Cu-substituted Pd(1 1 1) surfaces as well as the PdM/γ-Al2O3(1 1 0) (M = Au, Ag, Cu) surfaces has been investigated using spin-polarized density functional theory calculations. It is found that M-substituted Pd(1 1 1) surfaces show as good Hg adsorption capacity as the perfect Pd(1 1 1) at low Hg coverage, while the Hg adsorption capacity is only slightly weakened at high Hg coverage. On the basis of stepwise adsorption energies analysis, it is concluded that M-substituted Pd(1 1 1) surfaces can contribute to the binding of Hg atom on the surfaces at high Hg coverage. The electronic properties of the second metal atoms are the main factor contributes to the Hg adsorption capacity. Gas phase Pd2 shows better Hg adsorption capacity than Pd2/γ-Al2O3, while PdM/γ-Al2O3 can adsorb Hg more efficiently than bare PdM clusters. It suggests that the γ-Al2O3 support can enhance the activity of PdM for Hg adsorption and reduces the activity of Pd2. It is also found that Pd is the main active composition responsible for the interaction of mercury with the surface for PdM/γ-Al2O3 sorbent. Taking Hg adsorption capacity and economic costs into account, Cu addition is a comparatively good candidate for Hg capture.

Spin-polarized density functional theory calculations were carried out to investigate the adsorption of Hg n ( n = 1–3) on the perfect, step and vacancy-defective Pd(1 1 1) surfaces as well as the Pd/γ-Al 2 O 3 (1 1 0) surface. It is found that Hg atoms prefer to adsorb on the hollow sites on Pd(1 1 1) surfaces. The adsorption of Hg on the step and vacancy-defective Pd(1 1 1) surfaces is stronger than on the perfect Pd(1 1 1) surface, which indicates that the existence of vacancy and step defects can enhance the mercury adsorption activity of Pd adsorbents. As indicated by the calculated adsorption energies, the mercury adsorption on γ-Al 2 O 3 is weak. The γ-Al 2 O 3 supported single Pd atom shows as good Hg adsorption activity as the perfect Pd(1 1 1) surface at low Hg coverage, while more coordination unsaturated active Pd atoms is needed to achieve high Hg adsorption capacity. In addition, it was also found that the Hg adsorption on Pd/γ-Al 2 O 3 weakens the binding of Pd to the γ-Al 2 O 3 surface.

In this paper a novel robust controller is investigated for unmanned helicopter. Based on signal compensation theory, robust controllers are designed to enhance both rotational and translational tracking performance, as well as robust performance over parameter uncertainty and external disturbance. The designed robust controllers are time invariant and easy to realize. The full controllers are implemented on onboard avionics of a small scale unmanned helicopter THeli260 developed by Tsinghua University. Real flight experiments are successfully carried out, which shows advantage of designed robust controller.