A Fractional Schrödinger Equation and Its Solution

This paper presents a fractional Schrödinger equation and its solution. The fractional Schrödinger equation may be obtained using a fractional variational principle and a fractional Klein-Gordon equation; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives to obtain the fractional Euler-Lagrange equations of motion. We present the Lagrangian for the fractional Schrödinger equation of order α. We also use a fractional Klein-Gordon equation to obtain the fractional Schrödinger equation which is the same as that obtained using the fractional variational principle. As an example, we consider the eigensolutions of a particle in an infinite potential well. The solutions are obtained in terms of the sines of the Mittag-Leffler function.     

Construction of a photon drag detector for evaluating the performance of a CO2 laser amplifier

In this article, the construction of P-type germanium (Ge) photon drag detector which is used to characterize the gain and output powers of a 10 W CO₂ laser and a CO₂ laser amplifier is described. Gain and laser amplifier output power versus laser input power measurements collected at 4.7 mbar and 12.0 mbar amplifier tube pressures are discussed. Moreover, measurements relating the CO₂ laser output power to the laser tube discharge current are provided at 6.9 mbar and 7.3 mbar tube pressures.     

Effective spectral emissivity measurements of superalloys and YSZ thermal barrier coating at high temperatures using a 1.6μm single wavelength pyrometer

A method which employs an integrating sphere and a single-wavelength (1.6 μm) pyrometer for measuring the spectral effective emissivities of superalloys in the temperature range (=650–1050°C) is described. The spectral effective emissivities for five superalloys, namely, MARM-247, MARM-509, CMSX-4, Inconel-718, N-155 and two René-N6 samples coated with YSZ thermal barrier coating were measured. Correcting the pyrometer measurements for the variations in the object emissivity would reduce the uncertainty in the temperature measurements to <±1%.     

Entropy generation in flow field subjected to a porous block in a vertical channel

Entropy generation in the flow field subjected to a porous block situated in a vertical channel is examined. The effects of channel inlet port height (vertical height between channel inlet port and the block center), porosity, and block aspect ratio on the entropy generation rate due to fluid friction and heat transfer in the fluid are examined. The governing equations of flow, heat transfer, and entropy are solved numerically using a control volume approach. Air is used as the flowing fluid in the channel. A uniform heat flux is considered in the block and natural convection is accommodated in the analysis. It is found that entropy generation rate due to fluid friction increases with increasing inlet port height, while this increase becomes gradual for entropy generation rate due to heat transfer for the inlet port height exceeding 0.03 m. The porosity lowers entropy generation rate due to fluid friction and heat transfer. The effect of block aspect ratio on entropy generation rate is notable; in which case, entropy generation rate increases for the block aspect ratio of 1:2.     

Dual input–output pairs for modeling hysteresis inspired by mem-models

We call attention to a dual-pair concept for modeling hysteresis involving instantaneous switching: Specifically, there are two input–output pairs for each hysteresis model under one specific input, namely a differential pair and an integral pair. Currently in engineering mechanics, only one pair is being recognized and utilized, not the other. Whereas this dual-pair concept is inherent in the differential and algebraic forms of memristors and memcapacitors, the concept has not been carried over to memristive system theory, nor to memcapacitive system theory. We show that the “zero-crossing” feature in memristors, memcapacitors, and memristive/memcapacitive models (i.e., the “mem-models”) is also a feature of the differential pairs of well-known non-mem-models, examples of which are Ramberg–Osgood, Bouc–Wen, bilinear hysteresis, and classical Preisach. The dual-pair concept thus connects mem-models and non-mem-models, thereby facilitating the modeling of hysteresis, and raising a set of scientific questions for further studies that might not otherwise come to awareness.     

Silver,iron, and nickel immobilized on hydroxyapatite‐core‐shell γ‐Fe2O3 MNPs catalyzed one‐pot five‐component reactions for the synthesis of tetrahydropyridines by tandem condensation of amines,aldehydes, and methyl acetoacetate

In this study, Ag, Ni²⁺, and Fe²⁺ immobilized on hydroxyapatite‐core‐shell γ‐Fe₂O₃ nanoparticles (γ‐Fe₂O₃@HAp‐Ag, γ‐Fe₂O₃@HAp‐Ni²⁺, and γ‐Fe₂O₃@HAp‐Fe²⁺) as a new and reusable Lewis acid magnetic nanocatalyst was successfully synthesized and reported for an atom economic, extremely facile, and environmentally benign procedure for the synthesis of highly functionalized tetrahydropyridines derivatives 4a‐t is described by one‐pot five‐component reaction of 2 equiv of aldehydes 1 , 2 equiv of amines 2 , and 1 equiv of methyl acetoacetate 3 in EtOH at room temperature in good to high yields and short reaction time. The presented methodology offers several advantages such as easy work‐up procedure, reusability of the magnetic nanocatalyst, operational simplicity, green synthesis avoiding toxic reagents and solvent, mild reaction conditions, and no tedious column chromatographic separation.     

Self-passivating edge reconstructions of graphene

Planar reconstruction patterns at the zigzag and armchair edges of graphene were investigated with density-functional theory. It was unexpectedly found that the zigzag edge is metastable and a planar reconstruction spontaneously takes place at room temperature. The reconstruction changes electronic structure and self-passivates the edge with respect to adsorption of atomic hydrogen from a molecular atmosphere.     

Time delay in the Einstein-Straus solution

We compute the time delay of strong lensing in the framework of the Einstein-Straus solution in presence of a cosmological constant and restrict ourselves to the case of a spatially flat universe. The calculations are done to first order in the ratio of Schwarzschild radius to peri-lens. We apply our results to the lensed quasar SDSS J1004+4112. Our predictions for the time delay between the images C and D of the quasar range from 6 to 13 years and are compatible with the experimental lower bound.     

Li‐doped nanosized TiO2 powder with enhanced photocalatylic acivity under sunlight irradiation

This work reports on the synthesis of Li‐doped TiO₂ nanoparticles using the sol–gel process and solid‐state sintering, and investigates their potential use as a photocatalyst for degradation under sunlight excitation of different organic model compounds in aqueous solution. The structure of the nanocrystals was examined by X‐ray diffraction, UV‐vis ground state diffuse reflectance absorption spectra and X‐ray photoelectron emission spectroscopy. Results showed that samples prepared by sol–gel process and calcined at 400 °C are composed of a mixture of anatase and rutile phases, in contrast to the one prepared by solid‐state sintering, which exhibits an anatase phase with Li being involved in a spinel phase. The photocatalytic degradation of aqueous solutions of different aromatic compounds was successfully achieved under sunlight excitation in presence of Li‐doped TiO₂ prepared via sol–gel process. It was shown that the calcination temperature and the preparation mode greatly affect the photocatalytic efficiency. Copyright © 2010 John Wiley & Sons, Ltd.