New exact solutions for the seventh-order time fractional Sawada–Kotera–Ito equation via various methods

ABSTRACT

In this article, we use the modified Riemann Liouville derivative and fractional complex transform for converting the time fractional equation into its corresponding ordinary differential equation. We dealt with exact solutions of the seventh-order time fractional Sawada–Kotera–Ito equation. We obtained exact solutions of the given equation via the exp-function method, the ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/twrm20/2020/twrm20.v030.i03/17455030.2018.1520407/20200617/images/twrm_a_1520407_ilm0001.gif" alt="" />ipt><img src="//:0" alt="" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/twrm20/2020/twrm20.v030.i03/17455030.2018.1520407/20200617/images/twrm_a_1520407_ilm0001.gif"}" /><img src="//:0" alt="" data-formula-source="{"type" : "mathjax"}" />iant="normal">′i> / i>Gi> ) -expansion method and the ansatz method. By using these methods, dark, bright and singular soliton solutions of this equation have been found. The physical parameters in the dark, bright and singular soliton solutions free parameters and velocity are obtained as functions of the dependent model coefficients. Finally, some graphical representations of this equation are derived. It is shown that these methods are further efficient, convenient, and can be used to establish new solutions for other kind of nonlinear fractional differential equations arising in mathematical physics.     

Momentum transport and torque scaling in Taylor-Couette flow from an analogy with turbulent convection

We generalize an analogy between rotating and stratified shear flows. This analogy is summarized in Table 1. We use this analogy in the unstable case (centrifugally unstable flow <i>vs.i> convection) to compute the torque in Taylor-Couette configuration, as a function of the Reynolds number. At low Reynolds numbers, when most of the dissipation comes from the mean flow, we predict that the non-dimensional torque <i>Gi> = <i>Ti>/ν² <i>Li>, where <i>Li> is the cylinder length, scales with Reynolds number <i>Ri> and gap width η, <i>Gi> = 1.46η^3/2(1 - η)^-7/4 <i>Ri> ^3/2. At larger Reynolds number, velocity fluctuations become non-negligible in the dissipation. In these regimes, there is no exact power law dependence the torque <i>versusi> Reynolds. Instead, we obtain logarithmic corrections to the classical ultra-hard (exponent 2) regimes: <i>Gi> = 0.50 . These predictions are found to be in excellent agreement with avail-able experimental data. Predictions for scaling of velocity fluctuations are also provided. Received 7 June 2001 and Received in final form 7 December 2001     

Multiple image encryption using an aperture-modulated optical system

A multiple image cryptosystem based on different apertures in an optical set-up under a holographic arrangement is proposed. The system is a security architecture that uses different pupil aperture mask in the encoding lens to encrypt different images. Based on this approach multiple encryption is achieved by changing the pupil aperture arrangement of the optical system among exposures. In addition to the classical speckle phase mask, the geometrical parameters characterizing the apertures are introduced to increase the system security. Even when an illegal user steals the speckle phase mask, the system cannot be broken into without the correct pupil geometrical parameters. The experimental set-up is based on a volume photorefractive BSO crystal as storing device. Information retrieval is done via a phase conjugation operation. We also have to stress that the multiple storage under this scheme, is only possible with the help of the aperture mask. Simulation and experimental results are further introduced to verify the proposed method.     

Band bending at the Ni/Si(100)-2×1 submonolayer interface

Band bending at the Ni/Si(100)-2&times;1 interface has been monitored by using Si 2p core level photoemission spectra. Two nickel-induced Si 2p components appear in the initial interaction between Ni and Si(100)-2&times;1, which is confined at the top surface and the first subsurface layers. At Ni coverage less than 0.0375 ML, Ni atoms prefer the adamantane interstitial sites on the first subsurface, but switch to the pedestal sites on Si dimer rows at higher Ni coverage. The change in the preferred occupation sites of Ni atoms on the Si(100)-2&times;1 surface strongly affects the amount of band bending shift. The shift towards higher binding energy, when Ni atoms occupy the adamantane interstitial sites, is attributed to metal-induced-gap states. While Ni atoms occupy the pedestal sites, the band bending shift is reduced which is attributed to the passivation of surface states.     

The g-values of defects in hydrogenated microcrystalline silicon

The anisotropic g-values of defects in hydrogenated microcrystalline silicon prepared by hot-wire chemical vapour deposition have been measured as a function of crystalline volume fraction at room temperature. The defect has been identified as a silicon-dangling bond existing on the surface of crystalline grain. Their anisotropic g-values are discussed in the light of theoretical calculations by Ishii et al. and Ishii and Shimizu. The defect density is also discussed as a function of crystalline volume fraction.     

Variational treatment for the dynamical pseudo Jahn-Teller system

We present a variational treatment for the <i>Ei> × <i>ei> pseudo Jahn-Teller system. Through canonical transformation the electron and phonon states are decoupled. An analytical form is obtained for the ground state energy by scaling transformation. Including both the dynamical displacement of phonon modes and the softening of phonon frequency, this approach yields fairly accurate results for the ground state energy. The energy splitting and Ham's reduction factor are calculated, which also generates fairly good results compared with other perturbation results. We argue that our variational wave function is valid for the weak and intermediate coupling range.     

Spectropolarised ray-tracing simulations in densely packed particulate medium

Light scattering from particulate medium is simulated using the Monte Carlo ray-tracing technique. The medium is modelled as a randomly packed medium of ellipsoidal grains with stochastically rough surfaces, with an optional thin coating. Optical properties are modelled using a wavelength-dependent complex refractive index and taking Fresnelian reflections and refractions from the interfaces. The size and shape of the grains are assumed to be large and smooth enough for geometric optics to apply reasonably well.The ray-tracing technique uses parallel, weighted rays for computing simultaneously over a wide wavelength spectrum and a small roughness range, and scaling to obtain a large range of sizes and absorbities simultaneously. Polarisation is fully accounted for. The multiobservation technique is effectively used at each scattering point. The scattering from thinner sample layers is also received as a subresult.Simulations are run for a set of model samples to study the effects and sensitivities regarding the values of certain parameters. It has been found that the size and composition of the grains affect the scattering in a unique and invertible way. The shape of the grain causes similar significant effects that must certainly be taken into account if any accuracy is required, although inverting for the shape is difficult without further constraints. The packing density has a small but observable effect. Polarisation can be used to study the composition of low-albedo objects.     

A model for fragment excitation and kinetic energy in nuclear fission

We present a model which allows for the calculation of fragment excitation and kinetic energy in nuclear fission. The model assumes that fission products are excited independently according to an exponential distribution function, which depends on the <i>Qi>-value of the reaction and the level density parameter in the single fragment. We develop the model description and compare the results for different fissioning systems from actinium to fermium, and for different compound reactions, namely spontaneous fission, fission following thermal neutron capture, and high-energetic reactions with experimental data.     

Demineralisation of a semianthracite char with molten salts/HCl: A study by X-ray diffraction

The effects of chemical heat treatments of a semianthracite char (AC) on the composition of the mineral fraction of the material are investigated. The starting char was first treated with a mixture of LiCl/KCl or LiCl/KCl/CaO at 743, 873 or 1173 K and the products obtained were then washed thoroughly with distilled water. A small fraction of these samples were treated with 10^&minus;3 M HCl solution. The composition changes were studied by X-ray diffraction. The predominant mineral components initially present in the starting char are quartz, mullite, muscovite and/or kaolinite and oldhamite. The treatments of AC resulted in significant changes in the mineral fraction of the material, in particular when LiCl/KCl/CaO was used. In this case, spurrite, γ-calcium orthosilicate and gehlenite were formed, which were eliminated by treatment with 10^&minus;3 M HCl solution.     

Non-perturbative approach to high-index-contrast variations in electromagnetic systems

We present a method that formally calculates exact frequency shifts of an electromagnetic field for arbitrary changes in the refractive index. The possible refractive index changes include both anisotropic changes and boundary shifts. Degenerate eigenmode frequencies pose no problems in the presented method. The approach relies on operator algebra to derive an equation for the frequency shifts, which eventually turn out in a simple and physically sound form. Numerically the equations are well-behaved, easy implementable, and can be solved very fast. Like in perturbation theory a reference system is first considered, which then subsequently is used to solve another related, but different system. For our method precision is only limited by the reference system basis functions and the error induced in frequency is of second order for first-order basis set error. As an example we apply our method to the problem of variations in the air-hole diameter in a photonic crystal fiber.