Trigonometric Cosine Research Articles

Ramanujan-Berndt-type integrals of order four and series associated with Jacobi elliptic functions

In this paper, we evaluate in closed forms two families of infinite integrals containing hyperbolic and trigonometric cosine functions in their integrands. We call them Ramanujan-Berndt-type integrals of order four since they initiated the study of similar integrals. We establish explicit evaluations of twelve classes of hyperbolic sums by special values of the Gamma function by two completely different approaches, which extend those sums considered by Xu, Yang, Zhao and the second author previously. We discover the first by refining two results of Ramanujan concerning some q-series. For the second we compare both the Fourier series expansions and the Maclaurin series expansions of a few Jacobi elliptic functions. Furthermore, by contour integrations we convert two families of Ramanujan-Berndt-type integrals of order four to the above hyperbolic sums, all of which can be evaluated in closed forms. We then discover explicit formulas for one of the two families. Moreover, we provide many examples which enable us to formulate a conjectural explicit formula for the other family of the Ramanujan-Berndt-type integrals at the end, and obtain some results on Barnes' multiple zeta function.

On Coefficient Inequalities of Starlike Functions Related to the q-Analog of Cosine Functions Defined by the Fractional q-Differential Operator

This article extends the study of q-versions of analytic functions by introducing and studying the association of starlike functions with trigonometric cosine functions, both defined in their q-versions. Certain coefficient inequalities like coefficient bounds, Zalcman inequalities, and both Hankel and Toeplitz determinants for the new version of starlike functions are investigated. It is worth mentioning that most of the determined inequalities are sharp with the support of relevant extremal functions.

Thermal Instability in Nanoliquid Under Four Types of Magnetic-Field Modulation Within Hele-Shaw Cell

Abstract The influence of trigonometric cosine, square, sawtooth, and triangular wave types of magnetic-field modulation in nanoliquid within Hele-Shaw cell is studied in this paper utilizing linear/nonlinear explorations. The solvability condition to the third-order solution of the referred model equation has been imposed to get the cubic Ginzburg–Landau equation (GBL-equation) which is utilized to measure the rate of heat (or mass) transfer. In the sequel, the influence of the nondimensional parameters is discussed graphically in detail. It is demonstrated that Prandtl number (Pr)/magnetic Prandtl number (Prm)/Lewis-number (Le)/redefined diffusivity-ratio (NA)/concentration Rayleigh-number (RS1) and magnitude of the magnetic-modulation (δ) destabilize the system, that is, the heat/mass transfer increases. On the other hand, nanoliquid magnetic-number (Q), Hele–Shaw number (Hs), and modulating-frequency (ω) stabilize the system. The outcomes demonstrate that the magnetic-field modulation can be imposed significantly to increase or decrease the heat/mass transfer.

Trigonometric Cosine, Square, Sawtooth and Triangular Waveforms of Internal Heating Modulations for Three-Component Convection in a Couple Stress Liquid: A Detailed Analysis

Trigonometric Cosine, Square, Sawtooth and Triangular Waveforms of Internal Heating Modulations for Three-Component Convection in a Couple Stress Liquid: A Detailed Analysis

Pipeline Implementation of the Unified CORDIC Algorithm in FPGA

This work presents a pipeline implementation of the COordinate Rotation DIgital Computer (CORDIC) algorithm in VHDL. This implementation computes both the trigonometric (sine, cosine, and arctangent of two parameters) and hyperbolic (hyperbolic sine, hyperbolic cosine, and hyperbolic arctangent) fixed-point functions. The implementation was synthesized on a Xilinx ® Zynq UltraScale + ZCU102 FPGA evaluation board, reaching an operating frequency of 297.89 MHz. The results show the proper operation of the proposed architecture, achieving relative errors of less than 1.4% for a 16-bit fixed-point representation. The VHDL implementation is easily adjustable to meet the specific requirements of each system and is available as an open-source code under a Creative Commons license.

Heat and mass transfer of triple diffusive convection in viscoelastic liquids under internal heat source modulations

AbstractThe influence of sinusoidal (trigonometric cosine [TC]) and nonsinusoidal waveforms (square, sawtooth, and triangular) of internal heat source modulation on triple diffusive convection in viscoelastic liquids is investigated. An Oldroyd‐B type model is taken into account for viscoelastic liquids. Nonlinear analysis is carried out using a truncated representation of the Fourier series. To analyze the heat and mass transfer over a triply diffusive liquid layer, expressions for average Nusselt and average Sherwood numbers are derived using 8‐mode generalized Lorenz equations. The transient behavior of Nusselt and Sherwood numbers is analyzed on different parameters of the problem. From the results, it is found that the internal heat source enhances the heat transfer and diminishes the mass transfer while the heat sink diminishes the heat transfer and enhances the mass transfer. The results for respective waveforms are obtained for each parameter and it is found that the maximum heat and mass transfer occurs due to TC waveform. The limiting cases of viscoelastic liquids (Newtonian, Oldroyd‐B, Maxwell, and Rivlin–Ericksen) have been tabulated and corresponding results for each of the waveforms on heat and mass transfer have been shown.

On trigonometric cosine, square, sawtooth, and triangular wave‐type rotational modulations on triple‐diffusive convection in salted water

AbstractThe effect of sinusoidal (trigonometric cosine) and nonsinusoidal (square, sawtooth, and triangular) rotational modulations on triple‐diffusive convection in a Newtonian fluid is studied in this paper. The derived Ginzburg–Landau equation for the stationary mode of convection is used to quantify heat and mass transport. The enhancement of heat and mass transport in water due to the addition of different salts is explained in terms of the thermophysical properties of the fluid and the aqueous solutes. The flow equations are reduced to a Lorenz‐type scheme of nonlinear evolution equations using a truncated Fourier series. The equations describing the growth of convection amplitudes are solved using the multidomain spectral collocation method. With the aid of thermophysical properties of the fluid, numerical computations are performed for different values of the revised Rayleigh number (), Taylor number (), the amplitude of modulation , and modulation frequency . A comparison of heat and mass transport in different combinations of aqueous solutions is made. The study reveals that all modes of rotational modulation can control the rate of heat and mass transport. It is noted that amongst all other modulations, the square wave modulation is the most destabilizing one.

Oscillation rogue waves for the Kraenkel–Manna–Merle system in ferrites

In this work, new analytical rogue wave solutions are first derived for the Kraenkel–Manna–Merle (KMM) system in ferrites by the truncated Painlevé method. Two free functions, which are respectively space and time variables, are involved in the solutions, and can be utilized to generate certain perturbations to the rogue waves. As the two functions are taken as trigonometric cosine functions, the novel periodic oscillation rogue waves can be observed. Further, it is found that the amplitude and frequency parameters of these cosine functions play crucial roles during the oscillation. The rogue waves can be separated into two symmetric ones by adjusting the amplitude parameter, while the rogue waves can be divided into many peaks and troughs by changing frequency parameter. Besides, both the amplitude and frequency parameters are also highly sensitive to the amplitude of the oscillation rogue waves. A series of figures and data are illustrated these new features.

Wave propagation of resonance multi-stripes, complexitons, and lump and its variety interaction solutions to the (2+1)-dimensional pKP equation

This study deals with the (2+1)-dimensional potential Kadomtsev-Petviashvili (pKP) equation, which is used to describe the dynamics of a wave of small but finite amplitude in two dimensions in diverse areas of physics and applied mathematics. Through symbolic computations with Maple, the resonance multi-stripe solutions in real fields, and multi-stripe complexiton solutions in complex fields are derived via the simplified linear superposition principle in conjunction with Hirota's bilinear method (HBM). Furthermore, lump, lump-stripe, and lump-triangular periodic wave solutions are derived by employing the HBM. A positive quadratic function with exponential, hyperbolic cosine and trigonometric cosine functions are considered to reach such aims. For lump solutions, it is found that the shape and amplitude of a lump wave remain unchanged during its propagation. On the other hand, lump-stripe interaction solitons present the fission and fusion interaction phenomena between the lump and stripe solitons. To illustrate the dynamical characteristics of the attained solutions, the three-dimensional (3D) and contour plots of some of the representative attained solutions are displayed with the particular choice of the free parameters. The obtained solutions and their physical features might be helpful to understand the propagation of small but finite amplitude waves in shallow water.

Linear and nonlinear triple diffusive convection in the presence of sinusoidal/non-sinusoidal gravity modulation: A comparative study

The non-uniform vertical vibrations (gravity modulation) or g-jitter or time-periodic body force, can be realized by oscillating the system vertically. The time periodic variations are considered to vary sinusoidally (trigonometric cosine) or non-sinusoidally (square, sawthooth and triangular) to study three-component convection in a Newtonian fluid using linear and non-linear analyses. In the linear theory, the expressions for the Rayleigh number and the correction Rayleigh number are obtained by using a regular perturbation method. The eigen value is obtained by adopting the classical Venezian approach. The generalized Lorenz model is derived using a Fourier series with additional modes and under the assumption of the Boussinesq approximation and small-scale convection motion. The resulting non-autonomous Lorenz model is solved numerically to quantify the heat and mass transports. The results are considered against the background of the results of no modulation. Study reveals that all the four types of gravity modulation delay the onset of convection and thereby lead to the diminished heat transfer situation. It is also found that in the case of trigonometric type of gravity modulation heat and mass transports are between those of the cases of triangular and square types. The results in respect of saw-tooth and triangular wave modulations are identical.

Effect of internal heat source modulations on the onset of triple diffusive convection in viscoelastic liquids

The paper aims to study the dynamic behavior of a triple diffusive system subjected to sinusoidal (trigonometric cosine) and non-sinusoidal wave forms (square, sawtooth and triangular) of internal heat source modulation. The configuration of the system is such that a layer of viscoelastic liquid is heated and salted with two solutes from below. An Oldroyd-B type model is made use for viscoelastic liquids. In order to regulate the convection onset, internal heat source modulation is applied. This investigation is modelled using a linear stability analysis where a stationary convection is preferred. Venezian approach facilitates a solution by finding the eigen values of the problem. The influence of pertinent parameters which are varied for a wide range of values have been reported. It is captured via graphs that for small values of frequency of modulation, square wave form is more stable while sawtooth wave form is more stable for an increment in the values of frequency of modulation. Further, liquids such as Newtonian, Maxwell and Rivlin-Ericksen are analysed as the limiting cases of the problem. It seems worthwhile to discuss the results of the present study as it is the first work on linear theory of different wave forms of internal heat source modulation and thus paves a way for new theoretical and experimental endeavors.

Determination of the calorific value of natural gas using predictive modelling

The analysis of the measured data on the calorific value of natural gas in different regions of Ukraine for 2014–2019, which are in the public domain, has been carried out. Since 2020, such data has not been published. This predetermines the need to use calculation methods for determining this physical quantity for subsequent years in different regions of Ukraine. It is proved that during 2018–2019 there was a trend for the stability of calorific value of natural gas, and fluctuations of calorific value had a smaller amplitude (within the range of 9.31–9.80 kW·h/m3), the spread in the values of the calorific value almost halved: from 0.88 to 0.49 kW·h/m3 compared to 2014–2017. Therefore, the measured data for the calorific value of natural gas for the period 2018–2019 were taken as a basis for predictive modeling of this physical quantity. It has been established that for most regions of Ukraine it is possible to use a single average value of the calorific value of natural gas for the subsequent determination of the energy of this energy carrier. The exceptions are Donetsk, Ivano-Frankovsk, Lugansk and Chernivtsi regions, in which the measurement data of the calorific value of natural gas are described with sufficient accuracy by the trigonometric cosine function. By the method of predictive modeling, a mathematical model has been developed to determine the calorific value of natural gas for a specific month of the year in Ukraine. The adequacy of the developed model has been verified by the example of measuring data on the calorific value of natural gas along route 406 (Ivano-Frankivsk region). It was found that the relative error in calculating the combustion heat of natural gas on this route does not exceed plus 1, minus 3%, which is comparable with the accuracy of measuring the volume of natural gas with household gas meters (the volume of gas and its heat of combustion are parameters for determining the energy of natural gas). Thus, a predictive mathematical model has been developed with sufficient accuracy to describe the change in the calorific value of natural gas and can serve as a basis for calculating this gas parameter in the absence of measurement data.

Loop-like kink breather and its transition phenomena for the Vakhnenko equation arising from high-frequency wave propagation in electromagnetic physics

We report a novel type of breather by studying the Vakhnenko equation (VE) describing high-frequency wave (HFW) propagation in electromagnetic physics. By extending the bilinear function into a mixed exponential and trigonometric cosine function in Hirota bilinear method, an analytical multiple-valued function solution is constructed, which is a verified loop-like kink breather. The propagation control and evolution based on the parameter are investigated for the breather. Several interesting transition phenomena are revealed, such as, the transitions from the soliton to breather, from the single loop to the double loops, and from the leaping background waves to the flat ones. The results are helpful to understand the compressed mechanism to pulses in ultrafast optics.

Three-Dimensional Vibration Analysis of Laminated Composite Rectangular Plate with Cutouts.

The main purpose of this paper is to establish an analysis model of laminated composite rectangular plate with/without cutouts on the basis of three-dimensional elasticity theory and provide the exact three-dimensional solution. In the present work, the effect of the cutout is considered by subtracting the energies of cutouts from the total energy of the entire plate. The standard three-dimensional trigonometric cosine Fourier series and auxiliary Fourier series are chosen as admissible functions, and the Hamilton’s principle and Rayleigh-Ritz procedure are used to obtain the exact solution. In order to verify the effectiveness and reliability of the proposed method, some numerical results are obtained, and the results are compared with the ones available in the literature or finite element analysis. Finally, the effects of some key parameters which will affect the vibration characteristics are analyzed, and the non-dimensional frequency parameters are obtained, which can serve as benchmark data for the future research.

Role of Some Integral Transforms in Cryptograph

There are so many methods for the process of cryptography in literature. In this paper we present encryption and decryption method by using Laplace transform &Sumudu transform and their inverses. The purpose of using this method is for more security in communication as compared to other methods because cipher text obtained by this method could not be cracked by other persons easily. In the first part we apply Laplace transform to trigonometric cosine function for Sumudu transform for the same purpose.Fiinally we conclude by comparing these two methods.

A Solution to Remove Railway Track Discontinuities at Spiral Junctions

To design railway track layouts, engineers use tangent and circle segments. To maintain curvature and bank angle continuities between flat straight lines, with zero curvature, and banked circles, they use spiral segments where curvature and bank angle vary linearly as function of distance. This method meets the behavior of track engineers for which curvature and super-elevation are the main data used to lay the groundwork. However, it demonstrates dynamical drawbacks at both ends of spirals where bank angle and curvature derivatives would be discontinuous. Several unsuccessful attempts have been made to change spirals definition to alleviate this issue. In reality, track engineers smoothen the theoretical layout at the junctions, but there is no analytical definition for that although it is necessary track data at least for simulation codes. Smoothing portions are called ‘doucines’ but the maps of railway lines do not mention these unknown data and railway numerical codes ignore them, thus having to deal with subsequent dynamical perturbations which are avoided by doucines in reality. This paper proposes an analytical definition of practical doucines which does not modify actual theoretical method for designing lines. It also describes a method for programming it in railway codes. It uses trigonometric functions, sine and cosine which are simple to integrate to get at first the track orientation and then the X-Y coordinates of the layout line. In addition to doucine definitions, usable as well in practice as for all multibody codes, the paper shows how it can be implemented into the specific French calculation method, which uses an imaginary space, developed at first for calculating the dynamics of TGV. After modifying the author’s Ocrecym code according to the proposed doucine method, dynamical simulations of a light trailer coach, running over curved lines with and without doucines are presented to demonstrate the effectiveness of proposed solution.

SECURITY IN COMMUNICATION BY USING SUMUDU TRANSFORMS AND CRYPTOGRAPHY

In this paper we will show that how to increase the security in communication between two persons by using Sumudu transform & cryptography .In the first part of the paper we will consider one plain text (original message) and convert it to cipher text(text in hidden form) by applying Sumudu transform to trigonometric cosine function. In second part we will convert this cipher text to plain text by applying inverse Sumudu transform.

Simulation of number sets based on probability distributions

A three-parameter probability distribution is presented, modeled on the basis of the negative binomial distribution. This distribution corresponds structurally to the author’s distribution of the hyperbolic cosine type by the form of the characteristic function. The difference lies in the use of standard trigonometric cosine and sine for the characteristic function instead of the corresponding hyperbolic functions. The moment-forming polynomials of two arguments are found, derived from the recurrent differential relation to calculate the moments of distribution. A recurrent algebraic formula to produce the integer coefficients of these polynomials is introduced. The set of the coefficients depending on three arguments is ordered and geometrically interpreted as a numerical prism. Numerical prism sections are numerical triangles and numerical sequences. Among the sections of the prism there are well-known and new numerical sets, for example, Stirling numerical triangle, Bessel numerical triangle with alternating signs of elements, alternating tangential numbers, etc. A connection with the numerical prism derived from the hyperbolic cosine type distribution is indicated.

Modeling and Simulation of Transverse Free Vibration Analysis of a Rectangular Plate with Cutouts Using Energy Principles

A modeling method is proposed for the vibration characteristics of rectangular plates with cutouts having variable size. Different from the existing modeling method by considering the cutout as an extremely thin part of the plate, the energy principles in conjunction with Rayleigh‐Ritz solution technique are employed for the modeling of the structure. Under this theoretical framework, the effect of the cutout is taken into account by subtracting the energies of the cutout domains from the total energies of the whole plate with arbitrary boundary conditions. The displacement of the rectangular plate with nonuniform physic parameters is expressed as the combination of a two‐dimensional trigonometric cosine series and supplementary terms introduced to ensure the uniform convergence of the solution over the entire solution domain including the cutouts boundary. The effectiveness and reliability of the eigenmodes of the rectangular plate with cutouts are checked against the results obtained by the finite element method (FEM). The cutout number, position, and size are varied to illustrate the effect of the cutouts on the vibration characteristics of the rectangular plate with cutouts.

Geometric Properties of Cesàro Averaging Operators

Using positivity of trigonometric cosine and sine sums whose coefficients are generalization of Vietoris numbers, we find the conditions on coefficient {ak} to characterize the geometric properties of the corresponding analytic function f(z)=z+∑k=2∞akzk in the unit disc D. As an application, we also find geometric properties of generalized Cesàro-type polynomials.