Simple Elementary Functions Research Articles

Течение маслосодержащего материала в канале шнека маслопресса

The paper presents an analysis of the basic operational concepts for the screw channel of the oil press. The channel is chosen by the authors as the object of research. The subject of the study is the the flow velocities distribution for the oil-containing material in the channel of the local auger of the second section of the oil press and the influence of the channel parameters on the performance of the press. The mathematical modeling of the flow of oil-containing material in the channel of the oil press screw is used as a research method. Of the proposed simplifying assumptions for the flow in the auger channel, three assumptions are considered and further developed. The cross section of the channel is assumed to be rectangular. The developed hydrodynamic model, in contrast to previously published works, takes into account the presence of a gap δ between the edge of the screw spiral and the inner surface of the pressing cage, as well as the non-uniformity of the movement velocity for points on the screw. A fundamentally new point in the formulation of the problem is that the velocity is not assumed to be constant on the side walls of the channel. The coordinate dependence of the velocity vz on the rectangular boundaries of the screw is presented. The flow of viscoplastic material is obtained in the form of four components, which are determined by the pressure gradient, the flow of material along the axis of the screw channel, the movement of the side walls of the channel and the viscoplastic material through the gap between the screw spiral and the pressing cage. Simple analytical expressions containing only simple elementary functions are derived for the flow velocity of a viscoplastic material. The found solutions of hydrodynamics use the Newtonian model of fluid flow but can be easily generalized to non-Newtonian Bingham viscoplastic rheology. The results demonstrated can be employed to optimize the parameters of the oil press augers, increase its productivity and oil yield.

Electrically nonneutral ground states of stars

To approximately compute the non-relativistic ground state of an electrically non-neutral star, an exactly solvable model was recently introduced, and partly solved, by Krivoruchenko, Nadyozhin, and Yudin. The model generalizes the well-known Lane--Emden equation of a polytropic gas ball of index $n=1$ to a two-fluid setting. Here its complete solution is presented in terms of simple elementary functions; it is also generalized to a more-than-two-fluid setting where it remains exactly solvable. It is shown that, given the number of nuclei, a maximal negatively and a maximal positively charged solution exists, plus a continuous family of solutions which interpolates between these extremes. Numerical comparisons show that this exactly solvable model captures the qualitative behavior of the more physical model it is supposed to approximate. Furthermore, it correctly answers the question: how non-neutral can the star be? The answer is independent of the speed of light $c$ and the Planck quantum $h$. It supports Penrose's weak cosmic censorship hypothesis, in the sense that the bounds on the excess charge are compatible with the bound on the charge of a Reissner--Weyl--Nordstr\"om black hole.

On new approximations for the modified Bessel function of the second kind \(K_0(x)\)

A new series representation of the modified Bessel function of the second kind \(K_0(x)\) in terms of simple elementary functions (Kummer's function) is obtained. The accuracy of different orders in this expansion is analysed and has been shown not to be so good as those of different approximations found in the literature. In the sequel, new polynomial approximations for \(K_0(x)\), in the limits \(0< x\leq 2\) and \(2\leq x < \infty\), are obtained. They are shown to be much more accurate than the two best classical approximations given by the Abramowitz and Stegun's Handbook, for those intervals.

Unified Performance Analysis of Multi-Hop FSO Systems Over Double Generalized Gamma Turbulence Channels With Pointing Errors

Free space optical (FSO) communication systems provide high bandwidth in unregulated spectrum and act as a powerful line-of-sight wireless connectivity solution. The performance of FSO systems can be seriously impaired by fading as a result of atmospheric turbulence and/or pointing errors due to misalignment. In the context of FSO systems, relaying was proposed as an effective fading mitigation technique due to the fact that the variance is distance-dependent in turbulence channels. In this article, we present a unified performance analysis of multi-hop FSO systems over Double Generalized Gamma (DGG) turbulence channels with pointing error impairments. We assume amplify-and-forward relaying and consider both heterodyne detection and intensity modulation with direct detection. We derive tight closed-form expressions for the outage probability and bit error probability of both fixed-gain and channel state information (CSI)-assisted relaying in terms of the bivariate Fox-H functions and Fox-H functions, respectively. We further analyze asymptotic behavior of the outage probability in terms of simple elementary functions and obtain the achievable diversity orders. Diversity gain is found to be a function of atmospheric turbulence parameters, pointing error, detection type and the number of hops. Monte Carlo simulation results are further provided to verify the accuracy of the derived expressions.

Performance analysis of modulating retro-reflector FSO communication systems over Málaga turbulence channels

In this work we present a unified performance analysis of bistatic modulating retro-reflector (MRR) communication systems for all turbulence regimes where the free space optical (FSO) link is modeled by the generalized Málaga distribution. In particular, we obtain closed-form expressions for probability density function (PDF) of the system fading channel as well as PDF, cumulative distribution function (CDF), moment generating function (MGF), and moments of the system received signal to noise ratio (SNR) using Meijer’s G function. Based on the conducted analysis, we derive exact closed form expressions for the system average bit error rate (BER), outage probability, ergodic capacity, and amount of fading (AF). The closed form expressions are obtained in terms of Meijer’s G function. Furthermore, asymptotic expressions of the system average BER and outage probability at high SNR values are presented in terms of simple elementary functions.

Motion of a simple pendulum: a digital technology approach

In differential equations textbooks, the motion of a simple pendulum for small-amplitude oscillations is analyzed. This is due to the impossibility of expressing, in terms of simple elementary functions, the solutions of the nonlinear differential equation (NLDE) that models the pendulum, which is why the authors usually choose the linearized differential equation arising from the sin θ ≈ θ approximation for small angles θ. In this article, we use the computational power of Mathematica software and take advantage of its capacity to graphically show the solutions of the NLDE. In this manner, we determine approximations of the periods that result from varying the initial amplitude θ 0, without assuming that it takes small values. From a comparison of the solutions of both types of modelling of a simple pendulum, a criterion for deciding how small θ 0 must be for the linear equation to be adequate for modelling is obtained; this comparison includes an analysis of the pendulum′s motion when the cable length and the gravitational acceleration constant are varied.

Thermal effects of magnetohydrodynamic micropolar fluid embedded in porous medium with Fourier sine transform technique

This paper investigates the thermal effects of magnetohydrodynamic micropolar fluid with hidden phenomenon of heat and mass transfer via Caputo–Fabrizio fractional derivative. Analytical solutions are obtained for velocity field, mass concentration, microrotation and temperature distribution by implementing Fourier Sine and Laplace transform. The general solutions have been expressed in terms of simple elementary functions involving the convolution theorem for the Laplace transform. The graphical illustration is depicted in order to explore the influence of rheological parameters, i.e., Grashof, Prandtl, Schmidt numbers, transverse magnetic field, microrotation parameter, porosity and few other parameters on micropolar fluid flow.

Poroelastic response of spherical indentation into a half space with a drained surface via step displacement

Indentation of a poroelastic solid by a spherical-tip tool is analyzed within the framework of Biot’ s theory. We seek the response of the indentation force and the field quantities as functions of time when a rigid pervious indenter is loaded instantaneously to a fixed depth. The particular case where the surface of the semi-infinite domain is permeable and under a drained condition is considered. Compressibility of both the fluid and solid phases is taken into account. The solution procedure based on the McNamee–Gibson displacement function method is adopted in this work. One of the difficulties in solving poroelastic contact problems theoretically is in evaluating integrals with kernels that oscillate rapidly. We show that such issues can be overcome by using alternative integral representations with exponentially decaying functions in the kernels. Special functions, such as the modified Struve functions and the modified Bessel functions, and the method of contour integration can be utilized to aid the removal of the oscillation. Problem formulation and the solution procedure are first introduced. Implications of the poroelastic solution for incipient failure in form of tensile crack initiation and onset of plastic deformation are then discussed. An interesting outcome from this analysis is that the transient response of the dimensionless indentation force shows only weak dependence on one derived material constant and can in fact be fitted by a simple elementary function, which can then be conveniently used for material characterization in the laboratory.

Numerical evaluation of the interpolation accuracy of simple elementary functions

The paper contains a comparison of the accuracy of the restoration of elementary functions by the values in the nodes for algorithms of low-degree piecewise-polynomial interpolation. The test results demonstrate in graphical form the advantages and disadvantages of the widely used cubic interpolation splines. The comparison revealed that, contrary to popular belief, the smoothness of the interpolant is not directly related to the accuracy of the approximation. In the 20 different examples considered, the piecewise quadratic interpolation is rarely and only slightly inferior in the form of the used classical cubic splines, often by orders of magnitude better than many of them. In several examples, the high interpolation error of simple functions on a fixed grid appears to be almost independent of the degree of the algorithm and the smoothness of the interpolant. The piecewise-linear interpolation unexpectedly appeared the most accurate in one of the examples. A new problem arises: to find a local interpolation algorithm, accurately restoring any rational functions of the second order.

Charged isotropic model with conformal symmetry

We investigate the behaviour of a charged isotropic model with conformal symmetry. The relationship between the gravitational potentials arising from the conformal condition is used to generate a new class of exact solutions to the Einstein-Maxwell equations. A specific form of the electric field intensity and the metric potential is required to avoid a singularity at the centre. We can find simple elementary functions for the matter variables and the potentials with realistic profiles. The causality conditions, stability conditions and energy conditions are satisfied. Masses, radii, central densities and surface redshifts are generated, and the values are consistent with the compact stars 4U 1538-52 and PSR J1614-2230.

Thermal analysis in Stokes’ second problem of nanofluid: Applications in thermal engineering

Present study is prepared to analyze the heat transfer for the Stokes’ second problem of nanofluid. Water is taken as base fluid and two types of nanoparticles namely copper (Cu) and silver (Ag) are suspended in it. Exact solutions for velocity field and temperature distribution have been investigated by utilizing the Laplace transform method and presented in the form simple elementary functions. The results lead to the few facts regarding the effects of rheological and pertinent parameters on the graphical illustrations. Heat transfer is decreased with increasing nanoparticles volume fraction. Hartman number and porosity have opposite effects on fluid motion. This study has several applications in thermal engineering.

A comparative mathematical analysis of RL and RC electrical circuits via Atangana-Baleanu and Caputo-Fabrizio fractional derivatives

This research article is analyzed for the comparative study of RL and RC electrical circuits by employing newly presented Atangana-Baleanu and Caputo-Fabrizio fractional derivatives. The governing ordinary differential equations of RL and RC electrical circuits have been fractionalized in terms of fractional operators in the range of $0 \leq \xi \leq 1$ and $0 \leq \eta \leq 1$ . The analytic solutions of fractional differential equations for RL and RC electrical circuits have been solved by using the Laplace transform with its inversions. General solutions have been investigated for periodic and exponential sources by implementing the Atangana-Baleanu and Caputo-Fabrizio fractional operators separately. The investigated solutions have been expressed in terms of simple elementary functions with convolution product. On the basis of newly fractional derivatives with and without singular kernel, the voltage and current have interesting behavior with several similarities and differences for the periodic and exponential sources.

On approximation of ultraspherical polynomials in the oscillatory region

For k≥2 even, and α≥−(2k+1)∕4, we provide a uniform approximation of the ultraspherical polynomials Pk(α,α)(x) in the oscillatory region with a very explicit error term. In fact, our result covers all α for which the expression “oscillatory region” makes sense. To that end, we construct the almost equioscillating function g(x)=cb(x)(1−x2)(α+1)∕2Pk(α,α)(x)=cosB(x)+r(x). Here the constant c=c(k,α) is defined by the normalization of Pk(α,α)(x), B(x)=∫0xb(x)dx, and the functions b(x) and B(x), as well as bounds on the error term r(x), are given by some rather simple elementary functions.

Analytical results for a parity-time-symmetric two-level system under synchronous combined modulations

We propose a simple method of combined synchronous modulations to generate the analytically exact solutions for a parity-time symmetric two-level system. Such exact solutions are expressible in terms of simple elementary functions and helpful for illuminating some generalizations of appealing concepts originating in the Hermitian system. Some intriguing physical phenomena, such as stabilization of a non-Hermitian system by periodic driving, non-Hermitian analogs of coherent destruction of tunneling (CDT) and complete population inversion (CPI), are demonstrated analytically and confirmed numerically. In addition, by using these exact solutions we derive a pulse area theorem for such non-Hermitian CPI in the parity-time symmetric two-level system. Our results may provide an additional possibility for pulse manipulation and coherent control of the parity-time symmetric two-level system.

BER Performance of an Optically Pre-amplified FSO System Under Turbulence and Pointing Errors With ASE Noise

The performance of a free space optical (FSO) communication system is significantly affected by various atmospheric turbulence conditions and pointing errors (PEs) apart from the additive noise, which is assumed to be Gaussian. Optical pre-amplifiers are an essential component of FSO systems for improving the receiver sensitivity. However, optical pre-amplification results in amplified spontaneous emission (ASE), which dominates the receiver thermal and shot noises. The square law photodetection process at the receiver in a FSO system necessitates the consideration of chi-square statistics for the decision variable contrary to the Gaussian approximation that is widely used in the literature. In this paper, we evaluate the bit error rate (BER) performance of a FSO system assuming non-return-to-zero on–off keying modulation in the presence of ASE noise under weak, moderate to strong, and very strong atmospheric turbulence regimes and PEs. We also derive asymptotic BER expressions for the considered FSO system for large values of the signal-to-noise ratio in terms of simple elementary functions. Further insight into the system is provided by performing a diversity analysis.

A family of Finch and Skea relativistic stars

Several new families of exact solution to the Einstein–Maxwell system of differential equations are found for anisotropic charged matter. The spacetime geometry is that of Finch and Skea which satisfies all criteria for physical acceptability. The exact solutions can be expressed in terms of elementary functions, Bessel functions and modified Bessel functions. When a parameter is restricted to be an integer then the special functions reduce to simple elementary functions. The uncharged model of Finch and Skea [R. Finch and J. E. F. Skea, Class. Quantum Grav. 6 (1989) 467.] and the charged model of Hansraj and Maharaj [S. Hansraj and S. D. Maharaj, Int. J. Mod. Phys. D 15 (2006) 1311.] are regained as special cases. The solutions found admit a barotropic equation of state. A graphical analysis indicates that the matter and electric quantities are well behaved.

Dual-Hop FSO Transmission Systems Over Gamma–Gamma Turbulence With Pointing Errors

In this paper, we analyze the end-to-end performance of dual-hop free-space optical (FSO) fixed gain relaying systems under heterodyne detection and intensity modulation with direct detection techniques in the presence of atmospheric turbulence as well as pointing errors. In particular, we derive the cumulative distribution function (cdf) of the end-to-end signal-to-noise ratio (SNR) in exact closed form in terms of the bivariate Fox’s H function. Capitalizing on this cdf expression, novel closed-form expressions for the outage probability, the average bit-error rate (BER) for different modulation schemes, and the ergodic capacity of dual-hop FSO transmission systems are presented. Moreover, we present very tight asymptotic results for the outage probability and the average BER at high SNR regime in terms of simple elementary functions, and we derive the diversity order of the considered system. By using dual-hop FSO relaying, we demonstrate a better system performance as compared with the single FSO link. Numerical and Monte Carlo simulation results are provided to verify the accuracy of the newly proposed results, and a perfect agreement is observed.

A Novel Equivalent Definition of Modified Bessel Functions for Performance Analysis of Multi-Hop Wireless Communication Systems

A statistical model is derived for the equivalent signal-to-noise ratio of the Source-to-Relay-to-Destination (S-R-D) link for Amplify-and-Forward (AF) relaying systems that are subject to block Rayleigh-fading. The probability density function and the cumulated density function of the S-R-D link SNR involve modified Bessel functions of the second kind. Using fractional-calculus mathematics, a novel approach is introduced to rewrite those Bessel functions (and the statistical model of the S-R-D link SNR) in series form using simple elementary functions. Moreover, a statistical characterization of the total receive-SNR at the destination, corresponding to the S-R-D and the S-D link SNR, is provided for a more general relaying scenario in which the destination receives signals from both the relay and the source and processes them using maximum ratio combining (MRC). Using the novel statistical model for the total receive SNR at the destination, accurate and simple analytical expressions for the outage probability, the bit error probability, and the ergodic capacity are obtained. The analytical results presented in this paper provide a theoretical framework to analyze the performance of the AF cooperative systems with an MRC receiver.

Performance Analysis of Free-Space Optical Links Over Málaga ($\mathcal{M} $) Turbulence Channels With Pointing Errors

In this work, we present a unified performance analysis of a free-space optical (FSO) link that accounts for pointing errors and both types of detection techniques [i.e., intensity modulation/direct detection (IM/DD) and heterodyne detection]. More specifically, we present unified exact closed-form expressions for the cumulative distribution function, the probability density function, the moment generating function, and the moments of the end-to-end signal-to-noise ratio (SNR) of a single link FSO transmission system, all in terms of the Meijer's G function except for the moments that is in terms of simple elementary functions. We then capitalize on these unified results to offer unified exact closed-form expressions for various performance metrics of FSO link transmission systems, such as the outage probability, the scintillation index (SI), the average error rate for binary and $M$-ary modulation schemes, and the ergodic capacity (except for IM/DD technique, where we present closed-form lower bound results), all in terms of Meijer's G functions except for the SI that is in terms of simple elementary functions. Additionally, we derive the asymptotic results for all the expressions derived earlier in terms of Meijer's G function in the high SNR regime in terms of simple elementary functions via an asymptotic expansion of the Meijer's G function. We also derive new asymptotic expressions for the ergodic capacity in the low as well as high SNR regimes in terms of simple elementary functions via utilizing moments. All the presented results are verified via computer-based Monte-Carlo simulations.

Mixed mmWave RF/FSO Relaying Systems over Generalized Fading Channels with Pointing Errors

This paper studies the performance of mixed millimeter-wave radio-frequency (mmWave RF), free-space optics (FSO) systems in a highly scalable and cost-effective solution for fifth-generation (5G) mobile backhaul networks. The mmWave RF and FSO fading channels are, respectively, modeled by the Rician and the generalized Malaga ( $\mathcal {M}$ ) distributions. The effect of pointing errors due to the misalignment between the transmitter and the receiver in the FSO link is also included. Novel accurate closed-form expressions for the cumulative distribution function, the probability density function, and the moment generating function (MGF) in terms of Meijer's G functions are derived. Capitalizing on these new results, we analytically derive precise closed-form expressions for various performance metrics of the proposed system, including the outage probability, the average bit error rate (ABER), and the average capacity. Additionally, new asymptotic results are provided for the outage probability, the MGF, and the ABER in terms of simple elementary functions by applying the asymptotic expansion of the Meijer's G function at high signal-to-noise ratios (SNRs). Numerical results further validate the mathematical analysis by Monte-Carlo simulations.