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      Volume 94, All articles

      Continuous Article Publishing mode

    • Analysis of the evolution equation of a hyperbolic curve flow via Lie symmetry method

      BEN GAO ZHANG SHI

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      In this paper, based on the classical symmetry method, the group-invariant solutions of the evolution equation of a hyperbolic curve flow are investigated. The optimal system of the obtained symmetries is found, and the reduced equations and exact solutions of the evolution equation are discussed. Then explicit solutions are obtained by the power series method. In addition, the convergence of the power series solutions is proved. Theobjective shapes of the solutions of the evolution equation are performed.

    • Analysis of ferrite nanoparticles in liquid

      SOHAIL NADEEM SHAFIQ AHMAD NOOR MUHAMMAD

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      The foremost aim of the present paper is to explore the impact of heat transport phenomenon in a ferrofluid via magnetic dipole. Three distinct ferrite nanoparticles are discussed in the present study with water as the base fluid. Magnetic dipole existing in ferrite nanoparticles plays a significant role in controlling the momentum and thermal boundary layers. The partial differential equations (PDEs) are changed into nonlinear coupled ordinary differential equation (ODEs) by utilising similar transformations. Flow occurs due to linear stretching sheet. For the evaluation of heat flux, Fourier’s law of heat conduction is employed. Effects of rising parameters on the magneto-thermomechanical coupling are examined numerically. The results indicate that the property of magneto-thermomechanical cooperation slows the motion of liquid particles, and accordingly, strengthen the heat transfer rate at the surface and skin friction coefficient. Further, Nusselt number enhances with larger solid volume fraction. A magnificent comparison with accessible results for definite cases has been made.

    • Shell model description of the core excited level structure of $^{89}\rm{Sr}$ nucleus and systematic features of the $N = 51$ odd-$A$ isotones

      YI HENG WU KE YAN MA FEI CHENG XUE YUAN CAI

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      Shell-model calculations are performed using NuShellX code in the model space $\pi( f_{5/2}, p_{3/2}, p_{1/2}, g_{9/2}) \otimes \nu(g_{9/2}, g_{7/2}, d_{5/2}, h_{11/2})$, which probe the proton core excitation from the interior of $Z = 38$ semiclosed shell and neutron core excitation from the interior of $N = 56$ semiclosed shell for the level structure of $^{89}Sr$. Our calculations show that the excitation of a single $d_{5/2}$ neutron across $N = 56$ semiclosed shell into the $h_{11/2}$ orbit should have great effects on the excited states of $^{89}Sr$. In addition, the systematic features of proton core excitation across $Z = 38$ semiclosed shell into the $g_{9/2}$ orbit and neutron core excitation across $N = 56$ semiclosed shell into the $g_{7/2}, s_{1/2}, d_{3/2}, h_{11/2}$ orbits in $N = 51$ isotones are discussed.

    • New closed form solutions of the new coupled Konno–Oono equation using the new extended direct algebraic method

      SEYED MEHDI MIRHOSSEINI-ALIZAMINI HADI REZAZADEH KUMBINARASAIAH SRINIVASA AHMET BEKIR

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      In this paper, we apply the new extended direct algebraic method (NEDAM) to solve new exact solutions of the new coupled Konno–Oono (CKO) equation, and construct exact solution expressed in terms of hyperbolic functions and trigonometric functions with arbitrary parameters. A comparison between our established results and the results obtained by the existing ones is also presented. As a newly developed mathematical tool, the proposedmethod is an effective and straightforward technique to work out new solutions of various types of nonlinear partial differential equations (NLPDEs) in applied sciences and engineering.

    • Zakharov–Kuznetsov–Burgers equation in a magnetised non-extensive electron–positron–ion plasma

      SONA BANSAL MUNISH AGGARWAL

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      In this paper, we have studied the three-dimensional (3D) electron-acoustic waves (EAWs) in a three component complex plasma containing $q$-non-extensive distributed hot electrons and positrons. The propagation characteristics of the 3D electron-acoustic (EA) shock waves under the influence of magnetic field have been studied. Our present plasma model supports the negative potential shocks. Combined action of dissipation ($\eta$), nonextensivity ($q$), concentration of positrons ($\beta$), temperature ratio of cold electrons to positrons ($\sigma$) and magnetic field ($\omega_{c}$) on the EA shock waves has been studied in detail and the findings obtained here will be beneficial in future astrophysical investigations.

    • Linear analysis of the dispersion relation of surface waves of a magnetic fluid in a square container under an external oblique magnetic field

      RABAH DJEGHIOUR BACHIR MEZIANI OUERDIA OURRAD

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      In this study, free surface evolution of a magnetic fluid in a finite size tank which is subjected to an external magnetic field was investigated. The physical problem and equations governing fluid motion and magnetic field were given with boundary conditions. Using proper selection of variables, dimensionless equation system governing magnetic fluid sloshing were written. Resolution method based on multiple scale variables was presented and solution of the linear problem was given. The dispersion relation obtained in the finite depth case was compared with that corresponding to an infinite depth calculated with the same assumptions. Direction and magnitude of the external magnetic field, magnetic permeability ratio and surface tension effects on magnetic fluid free surface stability were analysed and important results were discussed.

    • Optimised wave perturbation for the linear instability of magnetohydrodynamics in plane Poiseuille flow

      ZAKIR HUSSAIN MEHBOOB ALI MUHAMMAD SHAHZAD FAISAL SULTAN

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      In this work, linear stability of an electrically conductive fluid experiencing Poiseuille flow for minimum Reynolds value under a normal magnetic field is analysed using the Chebyshev collocation method. The neutral curves of linear instability are derived by utilising Qualitat and Zuverlassigkeit (QZ) method. Instability of the magnetohydrodynamics for plane Poiseuille flowis introduced by solving the generalised Orr–Sommerfeld equation to determine the growth rates, wave number and spatial shapes of the eigenmodes. To solve linear problems, we use numerical methods which help us at each time step of the simulation, uncoupled by physical processes, which can improve the computational performance. This article provides the stability and error analysis, presents a concise study of the Poiseuille flow, and produces computational tests to support the given theory.

    • Soret and Dufour effects in the flow of viscous fluid by a curved stretching surface

      MARIA IMTIAZ HIRA NAZAR TASAWAR HAYAT AHMED ALSAEDI

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      The main focus in this study is to study the flow of a viscous fluid through a curved stretched surface. Soret and Dufour effects along with Joule heating are incorporated. Appropriate transformations yield the nonlinear ordinary differential system. Convergent series solutions of velocity, temperature and concentration are constructed. Graphical illustrations thoroughly demonstrate the features of the involved pertinent parameters. Skin friction coefficient, Nusselt and Sherwood numbers are also obtained and discussed graphically. Current computations reveal that the radial velocity experience decline with the increase of Hartman number. Further, fluid temperature declines for higher Prandtl and Soret numbers.

    • Models for membrane curvature sensing of curvature generating proteins

      T V SACHIN KRISHNAN SOVAN L DAS P B SUNIL KUMAR

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      The curvature-sensitive localisation of proteins on membranes is vital for many cell biological processes. Coarse-grained models are routinely employed to study the curvature-sensing phenomena and membrane morphology at the length scale of a few micrometres. Two prevalent phenomenological models exist for modelling the experimental observations of curvature sensing: (1) the spontaneous curvature (SC) model and (2) the curvature mismatch (CM) model, which differ in their treatment of the change in elastic energy due to the binding of proteins on the membrane. In this work, the prediction of sensing and generation behaviour by these two models are investigated using analytical calculations as well as dynamic triangulation Monte Carlo simulations of quasispherical vesicles. While the SC model yields a monotonically decreasing sensing curve as a function of the vesicle radius, the CM model results in a non-monotonic sensing curve. We highlight the main differences in the interpretation of the protein-related parameters in the two models. We further propose that the SC model is appropriate for modelling peripheral proteins employing the hydrophobic insertion mechanism, with minimal modification of membrane rigidity, while the CM model is appropriate for modelling curvature generation using scaffolding mechanism where there is significant stiffening of the membrane due to protein binding.

    • The detection of effective atomic numbers of some potassium compounds using direct and linear differential scattering methods

      BURCU AKÇA

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      In this work, the direct method and the linear differential scattering method were used to detect the experimental effective atomic numbers of some potassium compounds ($\rm{KH_{2}PO_{4}, KNO_{3}, K_{2}S_{2}O_{8}, KOH, K_{2}HPO_{4}, K_{2}SO_{4}, KCl, KIO_{3}}$ and $\rm{KI}$). The experiment has been done by using $^{241}\rm{Am}$ radioactive source, a Si(Li) detector and an energy-dispersive X-ray fluorescence spectrometer (EDXRFS). The experimental effective atomic numbers were compared with the effective atomic numbers obtained using WinXCom, FFAST, non-relativistic theory (NRT)and relativistic theory (RT).

    • Possible effects of galactic cosmic ray flux and low-cloud amounts on global surface temperature

      M A EL-BORIE A A THABET E S EL-MALLAH M ABD EL-ZAHER A A BISHARA

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      The solar variations, solar–climate interactions, and the mechanisms controlling the response of Earth’s climate system are important to understand the effect of solar variability on climate change. The solar magnetic field is directly/indirectly disturbing the interplanetary space, the ionosphere, the magnetosphere, and even the atmosphere. To investigate the contribution of varying galactic cosmic flux, the role of sunspot number (Rz), galactic cosmic ray (GCR) rates, cloud condensation nuclei (CCN), total solar irradiance (TSI), $\rm{CO}_{2}$ concentration and the global surface temperature (GST) is examined. The variations of TSI can partially explain the global increase in temperature, and it accounts for about $0.5^{0}\rm{C}$ warming experienced from 1950 to 2016. Therefore, the future predictions of global warming should take into account the effects due to long-term changes in the galactic CRs, the low-level cloud condensation (LLC), etc. The concentrations of $\rm{CO}_{2}$ increased in the upper atmosphere by 19% during the last 65 years. A strong correlation between LLC and GST suggests a linear relationship between these parameters. These observations are suggestive of the possible role of GCRs in global climate.

    • Numerical and perturbation solutions of third-grade fluid in a porous channel: Boundary and thermal slip effects

      MUBBASHAR NAZEER NASIR ALI FAYYAZ AHMAD MADIHA LATIF

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      The steady flow of a third-grade fluid due to pressure gradient is considered between parallel plane walls which are kept at different temperatures. The space between the plane walls is assumed to be a porous medium of constant permeability. The viscosity of the fluid is taken as constant as well as a function of temperature. It is further assumed that the fluid may slip at the wall surfaces. The consequence of this assumption results in non-linear boundary conditions at the plane walls. The temperature field is also supposed to satisfy thermal slip condition at the walls. The governing equations are modelled under these assumptions and the approximate solution is obtained using the perturbation theory. The skin friction coefficient is a decreasing function of slip parameters in the case of temperature-dependent viscosity models while no variation is noted for the case of constant viscosity via boundary slip parameter. The heat transfer rate increases with the boundary slip parameter and decreases with the thermal slip parameter. The validity of the approximated solution is checked by calculating the numerical solution as well. The absolute error is calculated and listed in tabular form in the case of constant and temperature-dependent viscosity via boundary and thermal slip parameters. The influence of various emerging parameters on flow velocity and temperature profile is discussed through graphs.

    • Lump solutions with higher-order rational dispersion relations

      WEN-XIU MA LIQIN ZHANG

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      This paper aims to explore a kind of lump solutions in nonlinear dispersive waves with higher-order rational dispersion relations.We show that the second member in the commuting Kadomtsev–Petviashvili hierarchy is such an example, and construct its lump solutions, based on a Hirota trilinear form. The presented lump solutions have one peak and two valleys, where the global maximum and minimum values are achieved. A few three dimensional plots and contour plots are made for a specific example of the lumps.

    • Structural, electronic, elastic and magnetic properties of heavier $\rm{REIr}_{3}$ ($\rm{RE = Gd, Tb}$ and $\rm{Ho}$) intermetallic compounds

      PUSHPLATA SHUKLA SADHNA SINGH

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      We present results on the bonding nature, structural, electronic, magnetic and elastic properties of $\rm{REIr}_{3}$ ($\rm{RE = Gd, Tb}$ and $\rm{Ho}$) intermetallic compounds adopting simple cubic $\rm{AuCu_{3}}$-type structure obtained using the full-potential linearlised augmented plane wave (FP-LAPW) method based on density functional theory. The local spin density approximation (LSDA) with Hubbard parameter ($\rm{LSDA} +U$) has been used for exchange and correlation effects to get accurate results because of the presence of highly localised $4 f$ electrons of rare-earth $\rm{(RE) (RE = Gd, Tb}$ and $\rm{Ho}$) atoms. The calculated lattice parameter is found to be consistent with the experimental results. The calculated magnetic moments predict ferromagnetic behaviour of these compounds. The electronic and bonding properties have been solved in terms of band structure, density of states (DOS) and charge density plots. These results confirm the metallic nature of these compounds. The bonding appearances of these compounds have also been interpreted from charge density plots. The elastic constants, shear modulus and Cauchy’s pressure are computed and they reveal that $\rm{GdIr_{3}}$ and $\rm{TbIr_{3}}$ compounds are ductile while $\rm{HoIr_{3}}$ shows brittle character.

    • Joint remote state preparation of an arbitrary eight-qubit cluster-type state

      LING-YUN CAO MIN JIANG CHEN CHEN

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      In this paper, we put forward a scheme to realise joint remote state preparation (JRSP) of an arbitrary eight-qubit cluster-type state with two non-maximally entangled Greenberger–Horne–Zeilinger (GHZ) states in a recursive manner. The senders begin by helping the remote receiver to construct one intermediate state which is related to the target state closely. Then, the receiver introduces auxiliary qubits and applies appropriate local operations to obtain the target eight-qubit cluster-type state. It is shown that one new GHZ channel can be distributed among three participants with a certain probability if the initial attempt fails.Moreover, compared with the previous protocols, in our scheme both quantum resources and classical communications are considerably reduced.

    • The influence of two kinds of time delays on the vibrational resonance of a fractional Mathieu–Duffing oscillator

      LIJUAN NING WEN GUO

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      Vibrational resonance is studied in a fractional Mathieu–Duffing oscillator with two types of time delays: fixed and distributed delays. The theoretical expression of the response amplitude is obtained by utilising the methodof direct partition of slow and fast motions. Relative errors between the theoretical prediction and the numerical simulation are introduced to verify the validity of analytical approaches. The relative error of the displacement andthe relative error of the response amplitude are calculated. Small relative errors show that the theoretical analysis is statistically correct. Therefore, the effects of fractional order, linear stiffness coefficient, low-frequency signal, time delay intensity and damping coefficient on the Mathieu–Duffing oscillator with distributed delay are studied successively. In order to better illustrate the impact of distributed time delay on the model, the case of fixed time delay is analysed and compared, and it can be found that the distributed delay has more significant influence than fixed delay on the system. In addition, the influence of distributed delay on the system is more significant than that of the fixed delay.

    • Applications of three methods for obtaining optical soliton solutions for the Lakshmanan–Porsezian–Daniel model with Kerr law nonlinearity

      HADI REZAZADEH DIPANKAR KUMAR AHMAD NEIRAMEH MOSTAFA ESLAMI MOHAMMAD MIRZAZADEH

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      This paper examines new travelling wave solutions to the Lakshmanan–Porsezian–Daniel (LPD) model with Kerr nonlinearity using Bäcklund transformation method based on Riccati equation, Kudryashov method and a new auxiliary ordinary differential equation (ODE). The three methods are adequately utilised, and some new rational-type hyperbolic and trigonometric function solutions are derived in different shapes for the aforementioned model. We confirm that our methods are more efficient than the other methods and it might be used in many other such types of nonlinear equations arising in the basic fabric of communications network technology and nonlinearoptics.

    • Galerkin finite-element numerical analysis of the effects of heat generation and thermal radiation on MHD SWCNT–water nanofluid flow with a stretchable plate

      SANTOSH CHAUDHARY K M KANIKA

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      Fundamental goal of the present communication is to analyse the viscous electrically conducting nanofluid flow near a stagnation region past a stretching sheet. Investigation of single-wall carbon nanotubes (SWCNTs) are done and water is employed as the base fluid. Combinations of the effects of heat generation, thermal radiation, viscous dissipation and Joule heating are considered. Mathematical modelling and examinations are done in the presence of magnetic field. Similarity variables are introduced to convert nonlinear partial differential equations into nonlinear ordinary differential equations. Numerical solutions of the governing modelled equations are collected by applying Galerkin finite-element method. Impacts of distinct influential parameters such as velocity ratio parameter, solid volume fraction, magnetic parameter, radiation parameter, heat generation parameter and Brinkmann number on velocity, temperature, surface shear stress and surface heat flux are obtained and discussed. Furthermore, comparison of the results of the current analysis is made with the earlier published data.

    • On the exact solutions of nonlinear evolution equations by the improved tan($\varphi/2$)-expansion method

      YESIM SAGLAM ÖZKAN EMRULLAH YASAR

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      In this paper, the improved tan($\varphi/2$)-expansion method (ITEM) is proposed to obtain more general exact solutions of the nonlinear evolution equations (NLEEs). This method is applied to the generalised Hirota–Satsuma coupled KdV (HScKdV) equation and (2+1)-dimensional Nizhnik–Novikov–Veselov (NNV) system. We have obtained four types of solutions of these equations such as hyperbolic, trigonometric, exponential and rational functions as an advantage of this method. These solutions include solitons, rational, periodic and kink solutions. Moreover, modulation instability is used to establish stability of the obtained solutions.

    • Superposition behaviour between lump solutions and different forms of $N$-solitons ($N \rightarrow\infty$) for the fifth-order Korteweg–de Vries equation

      WEI TAN JUN LIU

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      A lump-type solution of the (2 + 1)-dimensional generalised fifth-order Korteweg–de Vries (KdV) equation is obtained from the two-soliton solution by applying the parametric limit method. Some theorems and corollaries about the superposition behaviour between lump solutions and different forms of $N$-soliton ($N \rightarrow\infty$) solutions are constructed, and detailed proofs are given. Besides,we give a large number of examples and spatial evolution graphics to illustrate the effectiveness of the described theorems and corollaries. Some new nonlinear phenomena and superposition behaviour, such as rational-exponential type, rational-cosh-cos type, rational-sin type, rational-logarithmic type etc., are simulated and shown for the first time. Finally, we also illustrate the superposition between high-order lump-type solutions and $N$-soliton solutions.

    • Optical soliton solutions to the Fokas–Lenells equation via sine-Gordon expansion method and $(m + (G'/G))$-expansion method

      HAJAR F ISMAEL HASAN BULUT HACI MEHMET BASKONUS

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      The purpose of this study is to find some novel soliton solutions of Fokas–Lenells (FL) equation where the perturbation terms are taken into account with nonlinearity. The sine-Gordon expansion method (SGEM) and the $(m + (G'/G))$-expansion method are used in this context. The dark, bright, dark–bright and singular optical soliton solutions are successfully obtained. Moreover, the constraint conditions for guaranteeing the existence of solutions are also given.

    • Finite-time synchronisation of uncertain delay spatiotemporal networks via unidirectional coupling technology

      SHUANG ZHOU YIXUAN HONG YIMING YANG LING LÜ CHENGREN LI

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      In this paper, the problem of finite-time synchronisation of uncertain delay spatiotemporal networks via unidirectional coupling technology is investigated. Based on Lyapunov theorem and finite-time stability theory, an effective finite-time synchronisation scheme is designed to achieve finite-time synchronisation between uncertain delay spatiotemporal networks, and adaptive estimations of coupling coefficient, unknown parameter and uncertain network topology are realised. Then, the Fisher–Kolmogorov spatiotemporal model is used as the state equation of the network node for numerical simulation. The simulation results show that the finite-time synchronisation scheme is effective.

    • Analytical study of $D$-dimensional fractional Klein–Gordon equation with a fractional vector plus a scalar potential

      TAPAS DAS UTTAM GHOSH SUSMITA SARKAR SHANTANU DAS

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      $D$-dimensional fractional Klein–Gordon equation with fractional vector and scalar potential has been studied. Both fractional potentials are taken as attractive Coulomb-type with different multiplicative parameters, namely $v$ and $s$. Jumarie-type definitions for fractional calculus have been used. We have succeeded in achieving Whittaker-type classical differential equation in fractional mode for the required eigenfunction. Fractional Whittaker equation has been manipulated using the behaviour of the eigenfunction at asymptotic distance and origin. This manipulation delivers fractional-type confluent hypergeometric equation to solve. Power series method has been employed to do the task. All the obtained results agree with the existing results in literature when fractional parameter $\alpha$ is unity. Finally, we furnish numerical results with a few eigenfunction graphs for different spatial dimensions and fractional parameters.

    • Analysis of imprecisely defined fuzzy space-fractional telegraph equations

      SMITA TAPASWINI DIPTIRANJAN BEHERA

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      Telegraph equations are very important in physics and engineering due to their importance in modelling and designing frequency or voltage transmission. Moreover, uncertainty present in the system parameters plays a vital role in the designing process. Also it is known that it is not always easy to find exact solution of fractionally ordered system. Taking these factors into consideration, here space-fractional telegraph equations with fuzzy uncertainty have been analysed. A new technique to represent fuzzy number using two different parameters in the same domain has been used along with a semianalytic approach known as Adomain decomposition method (ADM) for the solution. Gaussian and triangular shaped fuzzy numbers are considered to model the uncertainties in initial as well as boundary conditions. The obtained results are compared with the existing solution in special cases for the validation.

    • High-performance ultra-low leakage current graphene-based screen-printed field-effect transistor on paper substrate

      KAPIL BHATT SANDEEP KUMAR C C TRIPATHI

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      Exploiting the advantages of additive patterning process over complex fabrication processes, herein we report the fabrication of field-effect transistor (FET) using the screen-printing method. The graphene conductive composite dielectric ink as the channel and the dielectric layer respectively was screen printed on cellulose paper substrate. The fabricated device shows the hole and electron mobility of $\rm{135 cm^{2}/V s}$ and $\rm{98 cm^{2}/V s}$ respectively with an ultra-low leakage current of $\sim 25 \rm{nA}$. The proposed technique can be used for large-scale roll-to-roll commercial manufacturing of disposable FET-based sensors such as temperature and IR sensors, health monitoring devices etc.

    • Binding energy of excitons in an infinitely deep spherical quantum dot under intense THz laser field

      A OUADGUI J DIOURI J KHAMKHAMI

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      We study the effects of intense THz laser field on the ground-state binding energy of heavy hole excitons confined in GaAs spherical quantum dots. The calculation is performed using the variational method in the framework of the single band effective mass theory. Our results show that (i) the laser electric field lowers the binding energy for all quantum dot radii, making the exciton clustered near the centre of the dot, (ii) the binding energy is mainly due to the dressed potential making the kinetic part insensitive to the field and (iii) the behaviour of the exciton, under the approximations used, can be modelled by a unique set of plots, depending on the material only via its excitonic units.

    • Optical travelling wave solutions for the Biswas–Arshed model in Kerr and non-Kerr law media

      MUHAMMAD TAHIR AZIZ ULLAH AWAN

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      This paper scrutinises the newly proposed Biswas–Arshed model for soliton propagation through optical fibres,with small group velocity dispersion and in the absence of self-phase modulation. Spatio-temporal dispersions of higher order are considered to balance with group velocity dispersion. First integral and functional variable methods are employed to recover solitary wave, shock wave, singular wave and singular periodic wave solutions for the two nonlinear forms of the model through Kerr law and power-law nonlinearity. The constraint relations are also figured for the manifestation of these optical solutions.

    • Computational soliton solutions to (2 + 1)-dimensional Pavlov equation using Lie symmetry approach

      SACHIN KUMAR MUKESH KUMAR DHARMENDRA KUMAR

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      In this work, Lie symmetry analysis and one-dimensional optimal system for Pavlov equation are presented. All the possible vector fields, their commutative and adjoint relations are carried out under invariance property of Lie group theory. On the basis of optimal system, similarity reductions of Pavlov equation are obtained. A repeated process of similarity reductions transforms the Pavlov equation into ordinary differential equations, which generate invariant solutions. The obtained invariant solutions are supplemented by numerical simulation toanalyse the physical behaviour. Thus, their parabolic, multisoliton, nonlinear, kink and antikink wave profiles are traced in results and discussions sections.

    • Effects of the positions of scintillation detectors with fast scintillators and photomultiplier tubes on TOF–PET performance

      ELIF EBRU ERMIS CUNEYT CELIKTAS

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      The objective of this study is to improve the time resolution value of a coincidence spectrometer used in a time-of-flight–positron emission tomography (TOF–PET) system. This spectrometer is used in medical imaging systems. The coincidence spectrometer is manufactured by using a BC420-type plastic scintillator and R1828-01-type photomultiplier tube, and the time resolution value of the manufactured spectrometer is determined. The accuracy of the experimental results is determined using the FLUKA Monte Carlo simulation program. Detectors are first manufactured in this program. Experimental and simulation results are compared and are found to be in good agreement. Optimal positions of the detectors are investigated to improve the coincidence time resolution of the spectrometer. Time resolution improvement of the optimal detector positions enables higher time-of-flight (TOF) gain and spatial resolution, leading to better image quality, reduction in patient doses and detection of small lesions.

    • Bose–Einstein condensation of an imperfect Bose gas using cluster expansion

      T P SURESH K M UDAYANANDAN

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      Bose–Einstein condensations (BEC) for an ideal Bose gas and an imperfect Bose gas are presented using cluster expansion method by using a new generating function obtained by Ushcats. The saturation density is calculated from the known values of virial coefficients for both ideal and uniform hard-sphere imperfect Bose gas. The values of saturation densities are found for some experimentally observed Bose–Einstein condensates and the fractional shift in the saturation densities are also calculated using this method, which are found to be positive.

    • Anisotropic bulk viscous string cosmological models of the Universe under a time-dependent deceleration parameter

      ARCHANA DIXIT RASHID ZIA ANIRUDH PRADHAN

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      We investigate a new class of LRS Bianchi type-II cosmological models by revisiting the paper of Mishra et al (Int. J. Theor. Phys. 52, 2546 (2013)) by considering a new deceleration parameter (DP) depending on the time in string cosmology for the modified gravity theory suggested by Sáez–Ballester (Phys. Lett. 113, 467 (1986)). We have considered the energy–momentum tensor proposed by Letelier (Phys. Rev. 28, 2414 (1983)) for bulk viscous and perfect fluid under some assumptions. To make our models consistent with recent astronomical observations, we have used the scale factor (Sharma et al, Astron Astrophys. 19, 55 (2018), Garg et al, Int. J. Geo. Meth. Mod. Phys. 16, 1950007 (2019)) $a(t) = \rm{exp} [ \frac{1}{\beta}\sqrt{2\beta t + k}]$, where $β$ and $k$ are positive constants and it provides a time-varying DP. By using the recent constraints ($H_{0} = 73.8$ and $q_{0} = −0.54$) from SN Ia data in combination with BAO and CMB observations (Giostri et al, JCAP 3, 27 (2012), arXiv:1203.3213v2[astroph. CO]), we affirm $\beta = 0.0062$ and $k = 0.000016$. For these constraints, we have substantiated a new class of cosmological transit models for which the expansion takes place from the early decelerated phase to the current accelerated phase. Also, we have studied some physical, kinematic and geometric behaviour of the models, and have found them consistent with observations and well-established theoretical results. We have also compared our present results with those of Mishra et al (Int. J. Theor. Phys. 52, 2546 (2013)) and observed that the results in this paper are much better, stable under perturbation and in good agreement with cosmological reflections.

    • Magnetohydrodynamic creeping flow around a weakly permeable spherical particle in cell models

      M KRISHNA PRASAD TINA BUCHA

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      The present paper studies the impact of applied uniform transverse magnetic field on the flow of incompressible conducting fluid around a weakly permeable spherical particle bounded by a spherical container. Analytical solution of the problem is obtained using Happel and Kuwabara cell models. The concerned flow is parted in two regions, bounded fluid region and internal porous region, to be governed by Stokes and Darcy’s law respectively. At the interface between the fluid and the permeable region, the boundary conditions used are continuity of normal component of velocity, Saffman’s boundary condition and continuity of pressure. For the cellsurface, Happel and Kuwabara models together with continuity in radial component of the velocity has been used. Expressions for drag force, hydrodynamic permeability and Kozeny constant acting on the spherical particle under magnetic effect are presented. Representation of hydrodynamic permeability for varying permeability parameters, particle volume fraction, slip parameter and Hartmann numbers are represented graphically. Also, the magnitude of Kozeny constant for weakly permeable and semipermeable sphere under a magnetic effect has been presented. In limiting cases many important results are obtained.

    • Lie symmetries and invariant solutions of (2 + 1)-dimensional breaking soliton equation

      MUKESH KUMAR DIG VIJAY TANWAR

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      The present article deals with the symmetry reductions and invariant solutions of breaking soliton equation by virtue of similarity transformation method. The equation represents the collision of a Riemann wave propagating along the $y$-axis with a long wave along the $x$-axis. The infinitesimal transformations under one parameter for the governing system have been derived by exploiting the invariance property of Lie group theory. Consequently, the number of independent variables is reduced by one and the system remains invariant. A repeated application transforms the governing system into systems of ordinary differential equations. These systems degenerate well-known soliton solutions under some limiting conditions. The obtained solutions are extended with numerical simulation resulting in dark solitons, lumps, compactons, multisolitons, stationary and parabolic profiles and are shown graphically.

    • Correction to: Vibrational resonance in a higher-order nonlinear damped oscillator with rough potential

      J A LAOYE T O ROY-LAYINDE K A OMOTESO O O POPOOLA U E VINCENT

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      In our recently published paper [Pramana – J. Phys.93: 102 (2019), https://doi.org/10.1007/s12043-019- 1865-5] one of the author’s affiliation (O O Popoola) was wrongly indicated. The correct affiliation is Depart- ment of Physics, University of Ibadan, Ibadan, Nigeria.

    • Exact solitary wave solutions to the (2 + 1)-dimensional generalised Camassa–Holm–Kadomtsev–Petviashvili equation

      BEHZAD GHANBARI JIAN-GUO LIU

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      In this paper, a (2+1)-dimensional nonlinear evolution equation (NLEE), namely the generalised Camassa–Holm–Kadomtsev–Petviashvili equation (gCHKP) or Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation (KP-BBM), is examined. After applying the newly developed generalised exponential rational function method (GERFM), 14 travelling wave solutions are formally generated. It is worth mentioning that by specifying values to free parameters some previously obtained solutions can be recovered. The simplest equation method (SEM) is used to prove that the solutions obtained by GERFM are good. With the aid of a symbolic computation system, we prove that GERFM is more efficient and faster.

    • Development of a zero-cost multichannel analyser based on digital signal processing for $\gamma$ -ray spectroscopy using the PC sound card

      A JANA S K SINGH A GUPTA S DAS K BASU S SAMANTA R RAUT S S GHUGRE A K SINHA

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      A zero-cost multichannel analyser (MCA) system based on the digital signal (pulse) processing (DSP) convenient for $\gamma$ -ray spectroscopy with conventional detectors such as scintillators and high-purity germanium (HPGe) has been implemented. The in-built high-performance analog-to-digital converter (ADC) in the sound card, an integral component of the present day personal computers, was used to digitise the signals from the radiation detectors. These pulses were then shaped using the established digital signal processing recursive algorithms. The filtered data were then displayed as histograms which then could be subjected to the traditional analysis to obtain peak parameters and the associated quantities were deduced. The developed system combines the performance of the sound card hardware with the flexibility allowed by the DSP to achieve a versatile MCA.

    • Dynamic analysis and synchronisation control of a novel chaotic system with coexisting attractors

      CHENGQUN ZHOU CHUNHUA YANG DEGANG XU CHAOYANG CHEN

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      In this paper, a novel four-dimensional continuous chaotic system is constructed from the simplified Lorenz-like system. The novel system has three equilibria, strange attractors, coexisting attractors, and performs Hopf bifurcation with the variation of system parameters. The coexisting attractors, which are the most remarkable dynamic features of the system, are numerically studied. The coexisting attractors show that the system coexists as a pair of point, periodic, and chaotic attractors. Some basic dynamic behaviours are studied as well. The synchronisation control problem of the system is analysed. The theoretical and numerical analyses demonstrate that the system can easily achieve synchronisation by using the passive control technique.

    • Exact solution of perturbed nonlinear Schrödinger equation using $(G'/G, 1/G)$-expansion method

      YANJIE WEN YONGAN XIE

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      By constructing auxiliary equations and combining the expansion method of $(G'/G, 1/G)$, we study a class of nonlinear Schrödinger equation with perturbation terms which describes the propagation of the waves in optical metamaterials. More types of exact solutions, particularly solitary wave solutions, are obtained for the first time.

    • Propagation of nonlinear waves with a weak dispersion via coupled (2 + 1)-dimensional Konopelchenko–Dubrovsky dynamical equation

      ALY R SEADAWY DAVID YARO DIANCHEN LU

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      This work applies the modified extended direct algebraic method to construct some novel exact travelling wave solutions for the coupled (2 + 1)-dimensional Konopelchenko–Dubrovsky (KD) equation. Soliton, periodic, solitary wave, Jacobi elliptic function, new elliptic, Weierstrass elliptic function solutions and so on are obtained, which have several implementations in the field of applied sciences and engineering. In addition, we discuss the dynamics of some solutions like periodic, soliton and dark-singular combo soliton by their evolutionary shapes.

    • Physical origins of the ideality factor of the current equation in Schottky junctions

      LING-FENG MAO

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      After the carrier drift velocity at the semiconductor/metal interface is considered, current transport in Schottky diodes under a forward electric field is physically modelled. This model reveals that the ideality factor can be physically originated from the drift velocity and the drift velocity can also reduce the effective Schottky barrier height. This proposed model predicts that both the ideality factor and the Schottky barrier height depend on temperature, voltage and doping density, which agree well with the experimental results reported in the literature. The proposed diode current model also predicts a linear dependent relation between the reciprocal of the ideality factor and the effective Schottky barrier height, which is validated by experimental results. Such a model is useful to better understand the thermionic emission current physically in semiconductor/metal contact. It is also useful to characterise the material properties by using the ideality factor.

    • Influence of magnetic field and Coulomb field on the Rashba effect in a triangular quantum well

      SHU-PING SHAN SHI-HUA CHEN

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      The influence of magnetic field and Coulomb field on the Rashba spin–orbit interaction in a triangular quantum well was studied using Pekar variational method. We theoretically derived the expression of the boundmagnetopolaron ground-state energy. The energy of the bound magnetopolaron splits under the influence of the Rashba effect. From this phenomenon, it is concluded that the effects of orbital and spin interactions on the polaron energy in different directions must be considered. Because of the contribution of the magnetic field cyclotron resonance frequency to the Rashba spin–orbit splitting, the energy spacing becomes larger as the magnetic field cyclotron resonance frequency increases. Compared to the bare electron, the bound polaron is more stable, and the energy of bound polaron split is more stable.

    • Mixed-mode oscillations and the bifurcation mechanism for a Filippov-type dynamical system

      MIAO PENG ZHENGDI ZHANG ZIFANG QU QINSHENG BI

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      In this paper, mixed-mode oscillations and bifurcation mechanism for a Filippov-type system including two time-scales in the frequency domain are demonstrated. According to classic Chua’s system, we investigate a non-smooth dynamical system including two time-scales. As there exists an order gap between the exciting frequency and the natural one, the whole external excitation term can be considered as a slow-changing parameter, which results in two smooth subsystems divided by the non-smooth boundary. In addition, the critical condition about fold bifurcation (FB) is studied, and by applying the Hopf bifurcation (HB) theorem, specific formulas for determining the existence of HBs are presented. By introducing an auxiliary parameter via differential inclusions theory, the non-smoothbifurcations on the boundary are discussed. Then, the equilibrium branches and the bifurcations are derived, and two typical cases associated with different bifurcations are considered. In light of the superposition between the bifurcation curve and the transformed phase portrait, the dynamical behaviours of the mixed-mode oscillations as well as sliding movement along the non-smooth boundary are obtained, which reveal the corresponding dynamical mechanism.

    • Ion-acoustic waves in magnetised plasma with nonthermal electrons and positrons

      J K CHAWLA P C SINGHADIYA R S TIWARI

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      Zakharov–Kuznetsov (ZK) equation for ion-acoustic waves (IAWs) is derived using the reductive perturbation method (RPM) in magnetised plasma consisting of ions, positrons and nonthermal electrons in small but finite amplitude limit. Propagation characteristics of ion-acoustic solitary waves (IASWs) in three-dimensional space are analysed to determine their region of existence. Investigations reveal that ion-acoustic solitary pulses (IASPs) may exist in such plasmas and presence of nonthermal electrons significantly affects the amplitude and width of solitary pulses. Dependence of velocity, amplitude and width of solitary pulses on plasma parameters arepresented graphically. The amplitude of soliton increases with increase in ion temperature ratio ($\sigma$) and positron concentration ($\alpha$). However, it decreases with increase in nonthermal electron parameter ($\beta$) keeping other plasma parameters constant. Width of the soliton increases with increase in $\beta$, $\sigma$ and $\alpha$. Phase velocity of ion-acoustic wave ($\lambda$) increases with increase in nonthermal $\beta$ and $\sigma$. In our analysis, we found that magnetisation of plasma affect the width of the soliton but not the amplitude.

    • Numerical and microcontroller simulations, and electronic circuit realisation of Minorsky’s equation

      NGOUABO ULRICH GAËL NOUBISSIE SAMUEL FOTSIN HILAIRE BERTRAND WOAFO PAUL

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      This work deals with the mathematical analysis, numerical and microcontroller simulations and electronic circuit realisation of the dynamics of Minorsky’s equation.We consider the model including the nonlinear derivative feedback with delay. The study of stability is done by linearising the equation. An alternation between the zones of stability and instability as a function of the values of the delay is found. The bifurcation diagrams allowed us to validate the analytical predictions. These bifurcation diagrams show Hopf bifurcations and complex dynamics of the system. The analog and microcontroller simulations together with the experimental analysis were carried out in order to validate the theoretical analysis.

    • Investigation of the radiation shielding capability of $\rm{xPbO–(50 − x)BaO–50B_{2}O_{3}}$ glass system using Geant4, Fluka, WinXCOM and comparison of data with the experimental data

      A ASKIN C MUTUWONG T NUTARO M DAL

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      In this study, mass attenuation coefficient ($\mu_{\rm{m}}$), transmission fractions ($T$), effective atomic numbers ($\rm{Z_{eff}}$) and half-value layer (HVL) of the $\rm{xPbO–(50 − x)BaO–50B_{2}O_{3}}$ (where x = 10, 20, 30, 40 mol%) glass system have been determined from the Monte Carlo simulations carried out with Geant4 and Fluka simulation toolkits and WinXCOM database software. The calculated results were compared with the experimentally obtained $\mu_{m}$ values of the selected glass in order to validate the Geant4 model of HPGe detector and Fluka model of NaI(Tl) detectors. $T$, $\rm{Z_{eff}}$ and $\rm{HVL}$ shielding parameters of the studied glass system indicate that increase of PbO content from 10 to 40% results in a better shielding behaviour thanks to the high atomic number of lead.

    • Rogue wave solutions of the chiral nonlinear Schrödinger equation with modulated coefficients

      L FERNAND MOUASSOM ALAIN MVOGO C BIOUELE MBANE

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      The research of rogue wave solutions of the nonlinear Schrödinger (NLS) equations is still an open topic. NLS equations have received particular attention for describing nonlinear waves in optical fibres, photonics, plasmas, Bose–Einstein condensates and deep ocean. This work deals with rogue wave solutions of the chiral NLS equation. We introduce an inhomogeneous one-dimensional version, and using the similarity transformation and direct ansatz, we solve the equation in the presence of dispersive and nonlinear coupling which are modulated in time and space. As a result, we show how a simple choice of some free functions can display a lot of interesting rogue wave structures and the interaction of quantum rogue waves. The results obtained may give the possibility of conducting relevant experiments in quantum mechanics and achieving potential applications.

    • On: New optical soliton solutions for nonlinear complex fractional Schrödinger equation via new auxiliary equation method and novel (G'/G)-expansion method

      MARWAN AL-RAEEI MOUSTAFA SAYEM EL-DAHER

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      A comment on an article by Khater et al published in Pramana – J. Phys. 90: 59 (2018) is presented here. We represent two quotes on the article, the two quotes about the space-dependent fractional Schrödinger equation type and about one of the constants used by the authors.

    • Specific criteria for BCS-type cuprate superconductivity and peculiar isotope effects on the critical superconducting transition temperature

      S DZHUMANOV B L OKSENGENDLER SH S DJUMANOV

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      So far, many researchers have been misled to believe that the Bardeen–Cooper–Schrieffer (BCS)-like ($s-$ or $d-$wave) pairing theory is adequate for explaining high-$T_{c}$ superconductivity in doped cuprates from underdoped to overdoped regime.We show that the doped cuprates, depending on the Fermi energy ($\varepsilon_{F}$) and the energy ($\varepsilon_{A}$) of the effective attraction between pairing carriers, might be either unconventional (non-BCS-type) superconductors (at intermediate doping) or BCS-type superconductors (at higher doping). We argue that specific criteria for BCS-type superconductivity formulated in terms of two ratios $\varepsilon_{A}/\varepsilon_{F}$ and $\Delta/\varepsilon_{F}$ (where $\Delta$ is the BCS-like gap) must be met in these systems. We demonstrate that these criteria are satisfied only in overdoped cuprates but not in underdoped and optimally doped cuprates, where the origin of high-$T_{c}$ superconductivity is quite different from the BCS-type ($s-$ or $d-$wave) superconductivity. The BCS-like pairing theory is then used to calculate the critical superconducting transition temperature ($T_{c}$) and the peculiar oxygen and copper isotope effects on $T_{c}$ in overdoped cuprates.

    • Exact solitary wave solutions for a system of some nonlinear space–time fractional differential equations

      MUHAMMAD HANIF M A HABIB

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      We have enumerated new and exact general wave solutions, along with multiple exact soliton solutions of space–time nonlinear fractional differential equations (FDE), namely Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM), foam drainage and symmetric regularised long-wave (SRLW) equations, by employing a relatively new technique called (G'/G, 1/G)-expansion method. Also, based on fractional complex transformation and the properties of the modified Riemann–Liouville fractional-order operator, the fractional partial differential equations transform into a form of ordinary differential equation (ODE). This method is a recollection of the commutation of the well-appointed (G'/G)-expansion method introduced by Wang et al, Phys. Lett. A 372, 417 (2008) In this paper, it is mentioned that the two-variable (G'/G, 1/G)-expansion method is more legitimate, modest, sturdy and effective in the sense of theoretical and pragmatical point of view. Lastly, the peculiarities of these analytic solutions are illustrated graphically by utilising the computer symbolic programming Wolfram Mathematica.

    • Theoretical investigation of structural, electronic and thermoelectric properties of $p−n$ type $\rm{Mg_{2}Si_{1−x}Sn_{x}}$ system

      BRAHIM MARFOUA BRAHIM LAGOUN HAMZA LIDJICI ALI BENGHIA AHMED GUEDDOUH

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      Based on the density functional theory and the Boltzmann transport theory, the thermoelectric properties of $\rm{Mg_{2}Si_{1−x}Sn_{x}}$ solid solution with $x = 0.25, 0.5$ and $075$ were investigated. The calculated structural parameters were in good agreement with the previous work and the mechanical and dynamical stabilities were confirmed. The electronic band structure computed using the Tran-Blaha-modified Becke and Johnson (TB-mBJ) exchange potential indicated that the band gap can be tuned by the alloy effect. We combined first-principles calculations and the semiclassical Boltzmann transport theory by considering the electronic transport in the $\rm{Mg_{2}Si_{1−x}Sn_{x}}$ solid solution to determine the effect of varying the Sn composition on the thermoelectric performance. Our results have shown exceptionally high electrical conductivity for $\rm{Mg_{2}Sn}$ and higher Seebeck coefficient for $\rm{Mg_{2}Si}$. The highest figure of merit (ZT) was predicted for $\rm{Mg_{2}Si_{1−x}Sn_{x}}$ solid solution with x = 0.5 where ZT has reached 0.55 with carrier concentration charge $n = 10^{20} \rm{cm}^{−3}$ (p-type doping) at intermediate temperatures. Consequently, the alloying system with p-type doping may improve the thermoelectric properties compared to the $\rm{Mg_{2}Si}$ and $\rm{Mg_{2}Sn}$ pristine compounds.

    • Improvement of transconductance and cut-off frequency in $\rm{In_{0.1}Ga_{0.9}N}$ back-barrier-based double-channel $\rm{Al_{0.3}Ga_{0.7}N/GaN}$ high electron mobility transistor by enhancing the drain source contact length ratio

      RACHITA MOHAPATRA PRADIPTA DUTTA

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      An aluminium gallium nitride/gallium nitride ($\rm{Al_{0.3}Ga_{0.7}N/GaN}$) high electron mobility transistor (HEMT) is designed at a gate length ($L_{G}$) of 0.1 $\mu$m, drain-to-source spacing ($L_{SD}$) of 3 $\mu$m and drain length to source length ratio ($L_{D}:L_{S}$) of 1. The HEMT is investigated by considering four different heterostructures, namely single channel, single channel with back-barrier, double channel and double channel with back-barrier. A two-dimensional electron gas (2DEG) is formed at the interface of AlGaN/GaN HEMT (DC HEMT). The physical importance of indium gallium nitride (InGaN) as back-barrier is to increase carrier confinement by raising the conduction band of GaN buffer. The double-channel HEMT (DC HEMT) with back-barrier shows the highest current drive. There is an improvement of 3.16% in drain current and an improvement of 4.58% in cut-off frequency at a gate-to-source voltage of −0.5 V for the DC HEMT with back-barrier compared to the DC HEMT without back-barrier. For further improvement in transconductance and cut-off frequency, the structure of DC HEMT with back-barrier is modified by increasing the drain contact length and decreasing the source contact length, that is $L_{D}:L_{S} = 3$, keeping the drain-to-source spacing unchanged, i.e. $L_{SD} = 3 \mu m$. There is 32.55% improvement in transconductance and 14.03% improvement in cut-off frequency at a gate-to-source voltage of −0.5 V for the DC HEMT with back-barrier at $L_{D}:L_{S} = 3$ compared to the DC HEMT with back-barrier at $L_{D}:L_{S} = 1$.

    • Convective heat transfer and double diffusive convection in ionic nanofluids flow driven by peristalsis and electromagnetohydrodynamics

      J PRAKASH ASHISH SHARMA D TRIPATHI

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      An analytical study to investigate the double diffusive convection in peristaltic pumping of ionic nanofluids through asymmetric microchannel under the influence of electromagnetohydrodynamics (EMHD) is presented. Thermal radiation effect is also considered. Velocity slip and convective boundary conditions are employed at the permeable channel walls. Debye Hückel linearisation is considered to simplify the Poisson–Boltzmann equation. The normalised two-dimensional conservation equations formass, momentum, energy, solutal concentration and nanoparticle fraction are reduced when long wavelength and low Reynolds number are assumed. Analytical solutions are computationally illustrated with MATLAB software to describe the pumping, flow and thermal characteristics under the effects of relevant parameters like Biot number, slip parameters, thermal radiation, Debye length, Hartmann number and Grashof numbers. Furthermore, solutal concentration, nanoparticle fraction and heat transfer coefficient are also analysed to see the influences of pertinent parameters. Such observations may be applicable to develop electro-osmotically actuated bio-microfluidic systems for smart drug delivery andmicrolevel physiological transport.

    • Chirped solitons in optical monomode fibres modelled with Chen–Lee–Liu equation

      BUSHRA YOUNAS MUHAMMAD YOUNIS

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      The paper studies the extraction of chirped soliton to Chen–Lee–Liu equation (CLLE) with the group velocity dispersion (GVD) and self-steeping coefficients that describe pulse transmission through optical monomode fibres. The chirped bright, dark and singular optical solitons are obtained and the results show that nonlinear chirp parameters strongly vary on self-steeping, GVD and spreading effects. The constraint conditions for the existence of solitons are also derived during the derivation. The results are helpful and important for understanding the propagation of optical pulses.

    • Single and multiband THz metamaterial polarisers

      BAGVANTH REDDY SANGALA ARVIND NAGARAJAN PRATHMESH DESHMUKH HARSHAD SURDI GOUTAM RANA VENU GOPAL ACHANTA S S PRABHU

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      We report single and multiband linear polarisers for terahertz (THz) frequencies using cut-wire metamaterials (MM). The MMs were designed by finite-element method (FEM), fabricated by electron beam lithography, and characterised by THz time-domain spectroscopy. The MM unit cells consist of single or multiple length cut-wire pads of gold on semi-insulating gallium arsenide (GaAs) for single or multiple band polarisers. For example, a MM with a square unit cell of 50 $\mu$m size on 1 mm GaAs substrate with a gold cut wire of 65 $\mu$m length, 2 $\mu$m width, and 150 nm height gives a resonance around 1.05 THz. The dependence of the resonance frequency of the single-band polariser on the length of the cut-wires was explained based on transmission line model.

    • Quantisation of particle motion in dissipative harmonic environment

      EQAB M RABEI AHMED AL-JAMEL

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      In this work, the quantisation of particle propagating in a dissipative harmonic medium will be investigated using the creation and annihilation operator formalism, which is more appropriate in some fields of physics. Modelling the problem as damped harmonic oscillator, the equations of motion are then written in terms of Poisson brackets, and the Heisenberg equations are written in terms of the quantum counterpart of the Poisson bracket, known as commutators. The creation and annihilation operators are introduced and used to obtain the energy and eigenstates. Our results are in exact agreement with different quantisation approaches as in Serhan et al, J. Math. Phys. 59, 082105 (2018). The normalisable coherent states are obtained as eigenstates of the annihilation operator, which overcome the non-normalisability of these states that appeared via the dual coordinate method.

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