Transcendental Equation Research Articles

Алгоритм определения намагниченностей подрешеток двухподрешеточного ферримагнетика с точкой компенсации. Часть 2. Степенное приближение

The calculation algorithm of sublattice magnetizations in two-sublattice ferrimagnet with compensation point is proposed. On the consideration of magnetic as the combination of spin with two orientations it is found the dependence of magnetization from the temperature in the firm of transcendent equation. The calculated decision of this equation make the coincidence the calculated dependence with experiment in precision about 10%. For the analytic presentation of magnetization temperature dependence in obviously view it is proposed the power index approximation which consist of relation magnetic temperature to curie temperature in six power index. On the basis of power index approximation the phase diagram of two-sublattice ferrimagnet with compensation point is constructed. It is shown the scheme of sublattice magnetization determination on the basis of using meanings by temperature character points of phase diagram lower branch. It is found the equation system which connects the character points parameters with magnetizations in initial point and in the point of maximum of diagram in the temperature more than compensation. By solving this equation system it is found the enough simple analytical expressions which determine the saturation magnetizations both sublattices through parameters of character points on lower brunch of phase diagram. The proposed methods is applied to determination of sublattice magnetizations in experiments by measuring summary magnetization which are curried out on the films of two different compositions using the vibration magnetometer. For the examination of determined meanings the comparison with constructed on its basis phase diagram with the diagram determined in experiment was carried out. The examination shows the coincidence calculated data with experiment in limits from 8 to 40%. The analysis of correspondence determined diagram data with experimental data. It is found the good (to part of percent) coincidence of whole magnetization meaning in the beginning of diagram and also the coincidence of compensation temperature and Curie temperature. For the interpretation of visible deflections it is proposed the hypothesis about influence of third sublattice which changes the magnetizations of main sublattices. It is proposed some recommendations for further development of investigations.

Effect of equilibrium contact angle on water equilibrium film thickness for the carbon dioxide–brine–mineral system based on surface force theory

The thickness of the thin wetting film depends on disjoining pressure forces, and it evolves with pH evolution due to brine acidification at the physical and chemical conditions of geological carbon dioxide storage becoming thinner in response to dewetting. In the literature, molecular dynamic simulation (MDS) studies have been employed to understand the effect of pressure/capillary pressure on the thin wetting film evolution. In this paper, a theoretical approach based on the Frumkin–Derjaguin Equation (FDE), models of electric double layer repulsion, and van der Waals forces have been used for the calculation of the wetting film thickness. The approach excluded hydration forces contribution to disjoining pressure forces due partly to its poorly understood nature, and partly to the high salinity conditions encountered in geological carbon storage. Due to its promising global storage capacity compared to other lithologies, the carbon dioxide–brine–silica systems was chosen to simulate sandstone saline aquifers. The validation of the model benefited much from literature resources on data and a universal model of carbon dioxide–brine interfacial tension. Calculated results confirm pH-induced dewetting and they follow trends controlled by pH and pressure as found in the literature. The novelty of the paper can be seen from the fact that it has demonstrated a theoretical supplement to MDS studies in addition to justifying the fundamental utility and versatility of the FDE. Moreover, the paper links for the first time, a transcendental equation to the thin wetting film theory encountered in the carbon dioxide–solid–brine system found in geological carbon storage.

A form-grinding wheel optimization method for helical gears based on a PSO-SVM model

A gear form-grinding optimization method is proposed to obtain an optimal profile of a form-grinding wheel by solving a transcendental equation of the contact line between a form-grinding wheel and a helical gear workpiece. Since the equation of the contact line is transcendental, the relationship between the installation angle of the form-grinding wheel and the shape of the contact line cannot be represented by explicit functions, which makes it difficult to obtain the optimal profile of the form-grinding wheel. An optimization method of the contact line between the form-grinding wheel and the gear is proposed using three evaluation parameters that are the overrun, the shift, and the offset. Some gear form-grinding performances, such as machining stroke, tooth deviation, and grinding chatter, can be quantitatively described using those evaluation parameters. Further, a method for optimizing evaluation parameters is proposed, which uses a particle swarm optimization-support vector machine (PSO-SVM) model with the advantage of small-sample robustness to solve evaluation parameters. The PSO-SVM model is trained under the condition of different installation angles of the form-grinding wheel as the input and the evaluation function as the output. The R-squared value of the overrun is 0.986 that vilidate the high accuracy of the PSO-SVM model. Gear form-grinding test results show the proposed method can effectively improve grinding accuracy.

Modified Bracketing Iterative Method for Solving Nonlinear Equations

Non-linear equations, depicted as curves in numerous everyday situations, have long piqued the curiosity of researchers and engineers due to their frequent occurrence in practical problems. Despite attempts to tackle these equations both analytically and numerically, analytical methods often fall short when the equation's degree exceeds five, prompting the adoption of numerical approaches to yield approximate solutions. Consequently, this study places emphasis on segmenting intervals into smaller sub-intervals, with a particular focus on employing the Regula-Falsi method to integrate these segmented intervals, thereby enhancing its convergence rate. Furthermore, by utilizing the Regula-Falsi formula for interval segmentation, the number of iterations and computational time required are minimized. Additionally, the effectiveness of the proposed method is verified through numerical experiments involving various equation types, including algebraic, trigonometric, exponential, logarithmic, and transcendental equations, comparing the outcomes with established methods. The findings demonstrate that the proposed algorithm not only efficiently segments intervals but also enhances accuracy and reduces errors when these segmented intervals are utilized in conventional bracketing methods.

ASSESSMENT OF THE EFFICIENCY OF FIRE EXTINGUISHING SYSTEMS

As previously shown, many indicators characterise the quality of fire extinguishing systems. The most common quality indicator is fire extinguishing time. For fire extinguishing systems focused on extinguishing Class B fires using sprayed water, the authors have constructed a transcendental equation, the root of which is the fire extinguishing time. We derived the equation provided that the intensity of the sprayed water supply to the combustion centre to be described by the Heaviside function. The dynamic properties of fire are defined using the integral Laplace transform. We obtained the functional dependence of class B fire extinguishing time on fire parameters and showed that this time does not exceed a tenth of the constant fire time. In the first approximation, the inertial properties of the fire extinguishing system can be taken into account by the transfer function of an aperiodic link with a generalised time constant. For this case, we constructed a transcendental equation that determines the fire extinguishing time by the inertial fire extinguishing system. We note that, according to inertial properties, fire extinguishing systems fall into three classes. For each of the three classes, estimates of the extinguishing time of a class B fire were obtained, depending on the dynamic properties of this fire. The results show that this time can change several times. The authors have introduced the criterion for the efficiency of fire extinguishing systems, which characterises the degree of approximation of the speed of the fire extinguishing system to the maximum possible value. This criterion also indicates how much it is necessary to increase the extinguishing agent mass supplied per unit area of the fire relative to the extinguishing agent mass supplied in leaps per unit area of the fire, provided that the extinguishing times are the same. We obtained estimates of the effectiveness of fire extinguishing systems using this criterion. Furthermore, we provided recommendations on the use of the efficiency criterion. Keywords: fire extinguishing system, extinguishing time, efficiency criterion.

A new optimal iterative algorithm for solving nonlinear equations

The aim of this paper is to propose a new iterative algorithm (scheme or method) for solving algebraic and transcendental equations considering fixed point and an initial guess value on the x-axis. The concepts of slope of a line and Taylor series are used in the derivation. The algorithm has second-order convergence and requires two function evaluations in each step, which shows that it is optimal with computational efficiency index 1.414 and informational efficiency 1. The validity of the algorithm is examined by solving some examples and their comparisons with the Newton’s method.

Fractional periodic boundary value and Cauchy problems with Hilfer–Prabhakar operator

We introduce a successive approximations method to study one fractional periodic boundary value problem of the Hilfer-Prabhakar type. The problem is associated to the corresponding Cauchy problem, whose solution depends on an unknown initial value. To find this value we numerically solve the so-called ’determining system’ of algebraic or transcendental equations. As a result, we determine an approximate solution of the studied problem, written in a closed form. Finally, we evaluate efficiency of our method on a nonlinear numerical example.

Dynamical complexity of a fractional‐order neural network with nonidentical delays: Stability and bifurcation curves

Recently, many scholars have discovered that fractional calculus possess infinite memory and can better reflect the memory characteristics of neurons. Therefore, this paper studies the Hopf bifurcation of a fractional‐order network with short‐cut connections structure and self‐delay feedback. Firstly, we use the Laplace transform to obtain the characteristic equation of the model, which is the transcendental equation containing four times transcendental item. Secondly, by selecting the communication delay as the bifurcation parameter and the other delay as the constant in its stability interval, the conditions for the occurrence of Hopf bifurcation are established; the bifurcation diagrams are provided to ensure that the derived bifurcation findings are accurate. Thirdly, in the case of identical neurons, the crossing curves method is exploited to the fractional‐order functional function equation to extract the Hopf bifurcation curve. Finally, two numerical examples are employed to confirm the efficiency of the developed theoretical outcomes.

An explicit approximation of the central angle for the curved interface in double-circle model for horizontal two-phase stratified flow

Stratified flow in horizontal tubes is frequently observed in gas-liquid two-phase flow system. In the two-fluid modeling, it is important to define the interface shape in solving the balance equations to determine the key parameters such as the interfacial transfer terms, void fraction, and pressure drop. A double-circle model is usually introduced to depict the concave-down interface in a horizontal circular tube under the stratified-wavy flow condition. However, calculation of the central angle in the double-circle model, which represents the interfacial curvature, requires an appropriate iterative numerical root-finding scheme to solve the implicit transcendental equation. In this study, an explicit approximate equation has been proposed without requirement of the iterative scheme and numerical instability, which is expected to improve the coding process and computation efficiency in the analysis code with the two-fluid model.

Hole subband dispersions in a cylindrical Ge nanowire: exact results based on the axial Luttinger–Kohn Hamiltonian

Based on the Luttinger–Kohn Hamiltonian in the axial approximation, the transcendental equations determining the hole subband dispersions in a cylindrical Ge nanowire are analytically derived. These equations are more general than that derived using the spherical approximation, and are suitable to study the growth direction dependence of the subband dispersions. The axial approximation almost gives rise to the accurate low-energy subband dispersions for high-symmetry nanowire growth directions [001] and [111]. The perturbation correction from the non-axial term is negligible for these two directions. The lowest two subband dispersions can be regarded as two shifted parabolic curves with an energy gap at kz=0 for both growth directions [001] and [111]. At the position of the energy gap, the eigenstates for growth direction [111] are inverted in comparison with the normal eigenstates for growth direction [001]. A nanowire growth direction where the energy gap closes at kz=0 is predicted to exist between directions [001] and [111].

Selective harmonic elimination in PUC‐5 multilevel inverter using hybrid IGWO‐DE algorithm

AbstractThe use of a multilevel inverter has received significant attention in recent years due to its numerous advantages. Because of the widespread use of multilevel inverters in industries and applications that require a wide range of voltages, achieving high‐quality voltage has presented a number of challenges. Many studies have been carried out to address the problem of unwanted harmonics in multilevel inverters. The inverter switching can be done at a low frequency using the Selective Harmonic Elimination (SHE) technique, and the unwanted harmonics can be significantly reduced; however, the issue with SHE is solving the transcendental equations and determining the optimum switching angle. To address this problem, a new and improved hybrid algorithm is proposed that combines two evolutionary algorithms, gray wolf optimization (GWO) with an improved and new convergence factor and Differential evolution (DE) with a dynamic scaling factor using a crossover operator. In this paper, the optimum switching angles for a packed U cell 5‐level inverter are estimated using the proposed algorithm for distinct modulation index values, and simulation results are compared with different algorithms. The simulation result of the proposed algorithm, IGWO‐DE is confirmed through experiment.

STUDY OF THE EQUILIBRIUM POSITION OF THE MECHANISMS
 OF THE WORKING CHAMBER OF A CHAIN UNIT

The object of study in the article is the working chamber of the chain unit. We consider its previously proposed threerod
 model in the equilibrium position under the influence of gravity, the shape of which was determined using the angular
 parameters αл and αп borrowed from experience. Here, in a new study, a variant of the analytical determination of these
 parameters is proposed. To achieve the goal, transcendental equations are obtained and an algorithm for the successive
 approximation method is developed to solve them. An algorithm used for numerical studies of the nature of changes in angles
 αл, αп depending on the ratio of gravity forces applied in kinematic pairs and the crank rotation angle. The corresponding
 graphs have been built. It has been established that the inclination of the plane of movement of the working parts of the
 chamber does not affect the angular parameters obtained for the equilibrium position in the vertical plane.

Controlled random initialization of search vectors for application of gravitational search algorithm to selective harmonic elimination PWM

The prime objective of SHE PWM is to eliminate selected lower order harmonic by solving a set of non-linear transcendental equation while maintaining the fundamental component at a desired satisfactory value. In order to solve these non-linear equations different metaheuristic optimization techniques has been used in recent past. Due to faster convergence rate increased precision and higher probability to calculus based mathematical approximation techniques. In this manuscript an improved initialized Gravitational search algorithm has been proposed for a uni-polar 3-phase inverter. The algorithm is improved upon through a controlled random initialization of search agent, specifically for the purpose of solving the set of non-linear transcendental equation in SHEPWM for 3- switching angles. The simulation of the gravitational search algorithm and the proposed improved initialized GSA has been performed in MATLAB software and comparison of different performance parameter like Harmonic distortion, convergence rate etc. has been studied. The simulation results proves that a considerable improvement has been achieved by applying improved initialized GSA algorithm in terms of harmonic mitigation, convergence rate and accuracy in finding out the appropriate switching angles. Finally, the results obtained in simulation are verified through an appropriate experimental setup.

Analytical Study of the Behavioral Trend of Klein-Gordon Equation in Different Potentials

In this work, we present the analysis of behavioral trend of Klein-Gordon Equation involving potential as regards when it comes to the study of particle, it has been observed that in every case of handling of KGE with potential of any type, it is made clear here that the equation has to first off all be transformed into a particular standard differential equation with a well-known solution which appears in form of implicitly defined transcendental equation. The equation on the other hand is to be solved analytically since the exact solution is not easily attainable without the use of mathematical tool especially when it comes to the consideration of the energy eigenvalue and the corresponding wave function because the solution is also always accompanied with a normalization constant often coupled with a condition that requires an arbitrarily chosen quantum number that come up when (l=0) and so on. In general, the analysis reveals the fact that the of trend of KGE involving potential gives a good understanding in the study of inter-molecular structure, diatomic crystals, and such case that involves inter-atomic interaction which is gives very nice idea in the study of bound state in atom.

On the roots of systems of transcendental equations

On the roots of systems of transcendental equations

Cables of a prescribed length under concentrated or distributed loads

A general analysis of inextensible perfectly flexible cables of negligible weight and a prescribed length, loaded by an arbitrary number of concentrated loads, or by a uniformly distributed load per unit horizontal length, is presented. A catenary cable under its own weight, with or without a concentrated force, is also considered. The sags under loads and tensions in all segments of the cable are determined. The results are derived in closed-form, apart from a numerical solution of a single nonlinear algebraic equation for a suitably introduced parameter in the last step of the analysis. The study complements the treatment of cables under concentrated forces available in undergraduate textbooks, which omit the consideration of cables whose length is prescribed in advance. The general analysis may be found appealing for a senior or graduate structural mechanics course, while its specialization to cables with two or three loads only is well suited for an introductory mechanics course. Expressions for a catenary cable subjected to a concentrated load at its midpoint are also derived, with only one nonlinear transcendental equation for the introduced load parameter needing to be solved numerically. Three student projects are suggested which address the existence and uniqueness of the cable configuration under a concentrated force with a specified sag and length of the cable, the lifting of two symmetrically hung loads by an upward force, and the determination of the deformed shape and tension in a catenary cable subjected to a concentrated force at an arbitrary point of the cable.

Study of Algorithms to Calculate Real Root of Transcendental Equations

This study deals with a comparative analysis of algorithms for solving transcendental equations. It includes algorithms for RF-EXP, RF-Halley, Arc-sine, Tanh and RF-ArcTanh methods to check the accuracy, number of iterations, and errors for the solution of the trigonometric, logarithmic, and exponential equations. Several numerical examples are presented to illustrate the algorithms’ efficacy which are programmed in MATLAB. The findings indicate that Arc-sine and RF-ArcTanh, RF-Halley, and Tanh are in favour of exponential equations, trigonometric equations, and logarithmic equations respectively.

Generalized One-Dimensional Periodic Potential Wells Tending to the Dirac Delta Potential

The solution of a quantum periodic potential in solid state physics is relevant because one can understand how electrons behave in a corresponding crystal. In this paper, the analytical solution of the energy formulation for a one-dimensional periodic potential that meets certain boundary conditions is developed in a didactic and detailed way. In turn, the group speed and effective mass are also calculated from the transcendental energy equation of a potential V=V(x). From this, a comparison is made with periodic potentials with known analytical solutions, such as the Dirac delta, as well as rectangular and triangular potentials. Finally, some limits are proposed in which these periodic potentials of the form V=V(x) approach the periodic Dirac delta potential of positive intensity. Therefore, the calculations described in this paper can be used as the basis for more-complex one-dimensional potentials and related simulations.

Novel Exact Solution of Elastic Catenary and Applications on Floating Wind Turbine Mooring Systems

Abstract This research addresses the mathematical solution of the elastic catenary, a fundamental problem in offshore mooring engineering. A novel exact solution in a non-Lagrangian form is developed through rigorous mathematical derivation, distinguishing it from classical Lagrangian solutions. The procedure is reported in detail, and the resulting expressions are applied to analyze the mooring system of a reference floating turbine. A general approach is introduced to solve the transcendental equations associated with catenary mooring problems. The newly derived formulae exhibit greater applicability to geometry-to-force problems compared to existing Lagrangian expressions, making them particularly useful for conceptual designs and front-end engineering. In summary, this work provides valuable new insights into the exact solution of the elastic catenary, enhancing understanding and enabling practical applications in the field of floating wind turbines.

Wigner functions of the finite square-well bound states

The bound states of a particle confined in a one-dimensional finite square well cannot be solved analytically, since the eigen-energies are determined by transcendental equations. Here, we numerically calculate the bound states and show their non-classical properties, using Wigner's quasi-probability distribution (also called the Wigner functions) in the phase space (x, p). In contrast to the infinite-well case, we find that the Wigner functions spread over the space dimension x, get squeezed along the momentum dimension p, and show negativity outside the well. Negativity in a Wigner function indicates non-classical properties of the bound states.