• Volume 90, Issue 6

June 2018

• A systematic literature review of Burgers’ equation with recent advances

Even if numerical simulation of the Burgers’ equation is well documented in the literature, a detailed literature survey indicates that gaps still exist for comparative discussion regarding the physical and mathematical significance of the Burgers’ equation. Recently, an increasing interest has been developed within the scientific community, for studying non-linear convective–diffusive partial differential equations partly due to the tremendous improvement in computational capacity. Burgers’ equation whose exact solution is well known, is one of the famous non-linear partial differential equations which is suitable for the analysis of various important areas. A brief historical review of not only the mathematical, but also the physical significance of the solution of Burgers’ equation is presented, emphasising current research strategies, and the challenges that remain regarding the accuracy, stability and convergence of various schemes are discussed. One of the objectives of this paper is to discuss the recent developments in mathematical modelling of Burgers’ equation and thus open doors for improvement. No claim is made that the content of the paper is new. However, it is a sincere effort to outline the physical and mathematical importance of Burgers’ equation in the most simplified ways. We throw some light on the plethora of challenges which need to be overcome in the research areas and give motivation for the next breakthrough to take place in a numerical simulation of ordinary/partial differential equations.

• A new 4D chaotic system with hidden attractor and its engineering applications: Analog circuit design and field programmable gate array implementation

Nowadays, designing chaotic systems with hidden attractor is one of the most interesting topics in nonlinear dynamics and chaos. In this paper, a new 4D chaotic system is proposed. This new chaotic system has no equilibria, and so it belongs to the category of systems with hidden attractors. Dynamical features of this systemare investigated with the help of its state-space portraits, bifurcation diagram, Lyapunov exponents diagram, and basin of attraction. Also a hardware realisation of this system is proposed by using field programmable gate arrays(FPGA). In addition, an electronic circuit design for the chaotic system is introduced.

• Multiple periodic-soliton solutions of the (3 + 1)-dimensional generalised shallow water equation

Based on the extended variable-coefficient homogeneous balance method and two newansätz functions, we construct auto-Bäcklund transformation and multiple periodic-soliton solutions of (3+1)-dimensional generalised shallowwater equations.Completely newperiodic-soliton solutions including periodic cross-kinkwave, periodic two-solitary wave and breather type of two-solitary wave are obtained. In addition, cross-kink three-soliton and cross-kink four-soliton solutions are derived. Furthermore, propagation characteristics and interactions of the obtained solutions are discussed and illustrated in figures.

• Effective atomic numbers in some food materials and medicines for $\gamma$ -ray attenuation using $^{137}\rm{Cs}$ $\gamma$ -ray

A light-weight multichannel analyser (MCA)-based $\gamma$ -ray spectrometer, developed earlier at the Inter University Accelerator Centre, New Delhi, has been used as part of the PG curriculum, to determine the effective atomic numbers for $\gamma$ attenuation of $^{137}\rm{Cs}$ $\gamma$ -ray in different types of samples. The samples used are mixtures of graphite, aluminum and selenium powders in different proportions, commercial and home-made edible powders, fruit and vegetable juices as well as certain allopathic and ayurvedic medications. A narrow beam good geometry set-up has been used in the experiments. The measured attenuation coefficients have been used to extract effective atomic numbers in the samples. The results are consistent with XCOM values wherever available. The present results suggest that the $\gamma$ attenuation technique can be used as an effective non-destructive method for finding adulteration of food materials.

• Multiswitching combination synchronisation of non-identical fractional-order chaotic systems

In this paper, multiswitching combination synchronisation (MSCS) scheme has been investigated in a class of three non-identical fractional-order chaotic systems. The fractional-order Lorenz and Chen systems are taken as the drive systems. The combination of multidrive systems is then synchronised with the fractional-orderLü chaotic system. In MSCS, the state variables of the two drive systems synchronise with different state variables of the response system, simultaneously. Based on the stability of fractional-order chaotic systems, the MSCS of three fractional-order non-identical systems has been investigated. For the synchronisation of three non-identical fractional-order chaotic systems, suitable controllers have been designed. Theoretical analysis and numerical results are presented to demonstrate the validity and feasibility of the applied method.

• Fractional Klein–Gordon equation composed of Jumarie fractional derivative and its interpretation by a smoothness parameter

Klein–Gordon equation is one of the basic steps towards relativistic quantum mechanics. In this paper, we have formulated fractional Klein–Gordon equation via Jumarie fractional derivative and found two types of solutions. Zero-mass solution satisfies photon criteria and non-zero mass satisfies general theory of relativity. Further, we have developed rest mass condition which leads us to the concept of hidden wave. Classical Klein–Gordon equation fails to explain a chargeless system as well as a single-particle system. Using the fractional Klein–Gordon equation, we can overcome the problem. The fractional Klein–Gordon equation also leads to the smoothness parameter which is the measurement of the bumpiness of space. Here, by using this smoothness parameter, we have defined and interpreted the various cases.

• A comparative study of graphene and graphite-based field effect transistor on flexible substrate

In the present era, there has been a great demand of cost-effective, biodegradable, flexible and wearable electronics which may open the gate to many applications like flexible displays, RFID tags, health monitoring devices, etc. Due to the versatile nature of plastic substrates, they have been extensively used in packaging, printing, etc. However, the fabrication of electronic devices requires specially prepared substrates with high quality surfaces, chemical compositions and solutions to the related fabrication issues along with its non-biodegradable nature. Therefore, in this report, a cost-effective, biodegradable cellulose paper as an alternative dielectric substrate material for the fabrication of flexible field effect transistor (FET) is presented. The graphite and liquid phase exfoliated graphene have been used as the material for the realisation of source, drain and channel on cellulose paper substrate for its comparative analysis.The mobility of fabricated FETs was calculated to be $\rm{83 cm^{2}/V s}$ (holes) and $\rm{33 cm^{2}/V s}$ (electrons) for graphite FET and $\rm{100 cm^{2}/V s$ (holes) and $\rm{52 cm^{2}/V s}$ (electrons) for graphene FET, respectively. The output characteristic of the device demonstrates the linear behaviour and a comprehensive increase in conductance as a function of gate voltages. The fabricated FETs may be used for strain sensing, health care monitoring devices, human motion detection, etc.

• On synchronisation of a class of complex chaotic systems with complex unknown parameters via integral sliding mode control

Chaotic systems demonstrate complex behaviour in their state variables and their parameters, which generate some challenges and consequences. This paper presents a new synchronisation scheme based on integral sliding mode control (ISMC) method on a class of complex chaotic systems with complex unknown parameters. Synchronisation between corresponding states of a class of complex chaotic systems and also convergence of the errors of the system parameters to zero point are studied. The designed feedback control vector and complex unknown parameter vector are analytically achieved based on the Lyapunov stability theory. Moreover, the effectiveness of the proposed methodology is verified by synchronisation of the Chen complex system and the Lorenz complex systems as the leader and the follower chaotic systems, respectively. In conclusion, some numerical simulations related to the synchronisation methodology is given to illustrate the effectiveness of the theoretical discussions.

• Apodization of two-dimensional pupils with aberrations

The technique proposed to enhance the resolution of the point spread function (PSF) of an optical system underneath defocussing and spherical aberrations. The method of approach is based on the amplitude and phase masking in a ring aperture for modifying the light intensity distribution in the Gaussian focal plane ($Y_{D} = 0$) and in the defocussed planes ($Y_{D} = \pi$ and $Y_{D} = 2\pi$). The width of the annulus modifies the distribution of the light intensity in the side lobes of the resultant PSF. In the presence of an asymmetry in the phase of the annulus, the Hanning amplitude apodizer [cos($\pi\beta\rho$)] employed in the pupil function can modify the spatial distribution of light in the maximum defocussed plane ($Y_{D} = 2\pi$), results in PSF with improved resolution.

• Propagation of nonlinear shock waves for the generalised Oskolkov equation and its dynamic motions in the presence of an external periodic perturbation

Propagation of nonlinear shock waves for the generalised Oskolkov equation and dynamic motions of the perturbed Oskolkov equation are investigated. Employing the unified method, a collection of exact shock wave solutions for the generalised Oskolkov equations is presented. Collocation finite element method is applied to the generalised Oskolkov equation for checking the accuracy of the proposed method by two test problems including the motion of shock wave and evolution of waves with Gaussian and undular bore initial conditions. Consideringan external periodic perturbation, the dynamic motions of the perturbed generalised Oskolkov equation are studied depending on the system parameters with the help of phase portrait and time series plot. The perturbed generalised Oskolkov equation exhibits period-3, quasiperiodic and chaotic motions for some special values of the systemparameters, whereas the generalised Oskolkov equation presents shock waves in the absence of external periodic perturbation.

• A new linear plasma device for the study of plasma waves in the electron magnetohydrodynamics regime

A new, user-friendly, linear plasma device has been developed in our laboratory where a quiescent ($\Delta n/n \approx 1%$), low temperature (1–10 eV), pulsed (3–10 ms) plasma can be produced over a large uniform region of 30–40 cm diameter and 40 cm length. Salient features of the device include the flexibility of tuning the plasma density in the range of $10^{10}$ to $10^{12} \rm{cm}^{−3}$ and capability of scanning the plasma and field parameters in two dimensions with a precision of < 1 mm. The plasma is produced by a multifilamentary cathode and external magnetic field by Helmholtz coils, both designed and constructed in-house. The plasma parameters can be measured by Langmuir probes and electromagnetic field parameters by miniature magnetic probes and Rogowski coils. The plasma produced is uniform and essentially unbounded for performing experiments on waves and turbulence. The whole device can be operated single-handedly by undergraduate or graduate students. The device can be opened, serviced, new antennas/probes installed and ready for operation in a matter of hours. Some results on the excitation of electromagnetic structures in the context of electron magnetohydrodynamics (EMHD) are also presented to demonstrate the suitability of the device for carrying out such experiments.

• Shape, size and temperature dependency of thermal expansion, lattice parameter and bulk modulus in nanomaterials

A theoretical model is described here for studying the effect of temperature on nanomaterials. The thermodynamic equation of state (EoS) proposed by Goyal and Gupta in High Temp.-High Press. 45, 163 (2016); Oriental J. Chem. 32(4), 2193 (2016), is extended in the present study using Qi and Wang model [Mater. Chem. Phys. 88, 280 (2004)]. The thermal expansion coefficient is expressed in terms of shape and size and used to obtain the isobaric EoS of nanomaterials for the change in volume $V/V_{0}$. The variation in $V/V_{0}$ with temperature is estimatedfor spherical nanoparticles, nanowires and nanofilms. It is found that the volume thermal expansivity decreases as size of the nanomaterial increases, whereas $V/V_{0}$ increases with temperature across nanomaterials of different sizes. The lattice parameter variation with temperature is studied in Zn nanowires, Se and Ag nanoparticles. It is found that lattice constant increases with increase in temperature. Also, bulk modulus is found to increase with temperature in nanomaterials. The results obtained from the present model are compared with the available experimental data. A good consistency between the compared results confirms the suitability of the present model for studying thermalproperties of the nanomaterials.

• Monte Carlo simulation of secondary electron images for gold nanorods on the silicon substrate

Recently, gold nanorods (Au NRs) have attracted much attention because at a particular photoelectricity the gold nanorods present a characteristic which is different from other types of Au nanomaterials with variousshapes. Accurate measurement of aspect ratios does provide very high value of optical property for Au NRs.Monte Carlo (MC) simulation is thought of as the most accurate tool to perform size measurement through extracting structure parameters from the simulated scanning electron microscopy (SEM) image which best matches the experimental one. In this article, a series of MC-simulated secondary electron (SE) images have been taken for Au NRs on a silicon substrate. However, it has already been observed that the two ends of Au NRs in the experimental SEM image is brighter than that of the middle part. It seriously affects the accuracy of size measurement for Au NRs. The purpose of this work is to understand the mechanism underlying this phenomenon through a series of systematical analysis. It was found that the cetyltrimethylammonium bromide (CTAB) which covers the Au NRs indeed can alter the contrast of Au NRs compared to that without CTAB covering. However, SEs emitting fromCTAB are not the reason for the abnormal brightness at the two ends of NRs. This work reveals that the charging effect might be the leading cause for this phenomenon.

• Pramana – Journal of Physics

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