Transcendental Equation Research Articles

Solutions of an advance-delay differential equation and their asymptotic behaviour

The paper considers a scalar differential equation of an advance-delay type \begin{equation*} \dot{y}(t)= -\left(a_0+\frac{a_1}{t}\right)y(t-\tau )+\left(b_0+\frac{b_1}{t}\right)y(t+\sigma )\,, \end{equation*} where constants $a_0$, $b_0$, $\tau $ and $\sigma $ are positive, and $a_1$ and $b_1$ are arbitrary. The behavior of its solutions for $t\rightarrow \infty $ is analyzed provided that the transcendental equation \begin{equation*} \lambda = -a_0\mathrm{e}^{-\lambda \tau }+b_0\mathrm{e}^{\lambda \sigma } \end{equation*} has a positive real root. An exponential-type function approximating the solution is searched for to be used in proving the existence of a semi-global solution. Moreover, the lower and upper estimates are given for such a solution.

Optimum regulation of THD profile in multilevel inverter using parameter-less AI technique for electrical vehicle application

This article proposes investigation on harmonic profile improvement using a novel fitness function in the multiphase multilevel inverter. For the proposed study in this paper, three-phase, seven-level cascaded H-bridge (CHB) multilevel inverter (MLI) is considered. Modulation of the stepped waveform output of the MLI is done using selective harmonic elimination (SHE) method. Many algorithms are proposed for solving the set of nonlinear transcendental trigonometric equations for SHE methods. Teaching learning-based optimisation (TLBO) algorithm is a parameter less optimisation technique. Due to the lack of controlling parameters, the proposed algorithm is most robust among the family of artificial intelligence (AI) techniques. In this paper, an investigation is carried out on a novel fitness function proposed for controlling total harmonic distortion (THD) for the proposed inverter. It is observed that the proposed fitness function improves THD profile below IEEE standards. It is also observed that THD profile obtained through this method is far better than THD profile obtained through various proposed methods so far. The results of THD profile from the MATLAB Simulink simulations are verified experimentally using seven-level three-phase hardware controlled by Arduino MEGA 2560 low-cost controller.

Dynamics of an HIV infection model with two time delays

In this paper, a mathematical model for HIV-1 infection and immune response is considered, involving two discrete time delays in the intracellular as well as in activation of immune response. Using a recently developed geometric method for studying a class of transcendental equation with two time delays and delay dependent coefficients, we obtain the stability and bifurcation results at the non-trivial equilibrium. In particular, the crossing curves on the two-delays parameter plane can be completely characterized, on which Hopf and double-Hopf bifurcation will take place. In the case of Hopf bifurcation, there exist stability switches, and the direction and stability of delay induced Hopf-bifurcation can be determined using normal form theory and center manifold theorem. These results imply the model will exhibit complex temporal dynamics, such as period oscillations, quasi-periodic solutions, etc. Numerical examples are also carried out to verify these results.

Robustness analysis of exponential synchronization in complex dynamic networks with random perturbations

<abstract><p>This article discusses the robustness of exponential synchronization (ESy) of complex dynamic networks (CDNs) with random perturbations. Using the Gronwall-Bellman lemma and partial inequality techniques, by solving the transcendental equation, the maximum perturbation intensity of the CDN is estimated. This implies that the disturbed system achieves ESy if the disturbance intensity is within the range of our estimation. We illustrate the theoretical results with two numerical examples.</p></abstract>

Robustness analysis of stability of Takagi-Sugeno type fuzzy neural network

<abstract><p>In this paper, inequality techniques, stochastic analysis and algebraic methods are used to analyze the robustness of the stability of recurrent neural networks containing Takagi-Sugeno fuzzy rules. By solving the transcendental equations, the upper bounds of time delay and noise intensity are given, and the dynamic relationship between the two disturbance factors is derived. Finally, numerical examples are given to verify the results of this paper.</p></abstract>

АЛГОРИТМЫ ВЫЧИСЛЕНИЯ СОБСТВЕННЫХ ЗНАЧЕНИЙ ДИСКРЕТНЫХ ПОЛУОГРАНИЧЕННЫХ ОПЕРАТОРОВ, ЗАДАННЫХ НА КВАНТОВЫХ ГРАФАХ

Spectral problems for differential operators defined on quantum graphs are of great scientific interest related to problems in quantum mechanics, computer network modeling, image processing, ranking algorithms, modeling of electrical, and mechanical and acoustic processes, in networks of a diverse nature, in designing nano systems with prescribed properties and in other areas. Theoretical solutions of direct and inverse spectral problems on quantum graphs have been developed, but computational algorithms based on these methods are computationally inefficient. We have not seen any published works that consider examples of numerical solutions of spectral problems on finite connected graphs with a large number of vertices and edges. Therefore, the development of new computationally effective algorithms for numerical solution of spectral problems given on finite connected graphs is urgent. This paper develops a technique for finding the eigenvalues of boundary value problems on finite connected graphs with a required number of vertices and edges. To use this technique, it is necessary to know the eigenvalues and vectors of the eigenfunctions of corresponding unperturbed vector operators which are usually self-adjoint. Finding them manually, if the graph has a large number of vertices and edges, is difficult. This led to writing a package of programs in the mathematical environment Maple to find transcendental equations in the symbolic mode to calculate eigenvalues and find the eigenfunctions of unperturbed boundary value problems. Examples of calculating eigenvalues for a quantum graph which models an anthracene aromatic compound molecule are presented.

New Trigonometrically Method for Solving Non-Linear Transcendental Equations

Abstract: The transcendental equations' non-zero positive real roots can be found using a new approach presented in this research. The proposed approach is based on the union of the Newton-Raphson method and the inverse tan(x) function. To ensure the method's applicability, it is implemented in MATLAB and applied to various issues. The suggested approach is evaluated on a variety of numerical instances, the results show that our approaches are superior and more efficient than widely used methods. For both the new proposed method and the currently accessible existing methods, error calculations have been made. When compared to well-known procedures, the mistakes were quickly minimised, and the genuine root was discovered in fewer repetitions. The proposed method's convergence is studied, and it is demonstrated that it reduces to the quadratic convergent Newton-Raphson method. This method will also assist in the production of a non-zero real root of a specified nonlinear equation in the commercial software (transcendental, algebraic, and exponential).

A Novel Two Point Optimal Derivative free Method for Numerical Solution of Nonlinear Algebraic, Transcendental Equations and Application Problems using Weight Function

It’s a big challenge for researchers to locate the root of nonlinear equations with minimum cost, lot of methods are already exist in literature to find root but their cost are very high In this regard we introduce a two-step fourth order method by using weight function. And proposed method is optimal and derivative free for solution of nonlinear algebraic and transcendental and application problems. MATLAB, Mathematica and Maple software are used to solve the convergence and numerical problems of proposed and their counterpart methods.

A Step by Step Analytical Solution to the Single Diode Model of a Solar Cell

Making use of previous results where the series resistance, Rs, and the light-generated current, IL, of a solar cell are determined through the knowledge of the open-circuit voltage, Voc, the short-circuit current, Isc, the voltage and current at the maximum power point, Vmp and Imp, respectively, a simple and analytical step-by-step approach has been developed to determine the shunt resistance, Rsh, the reverse saturation current, Is and the ideality factor, A, of a solar cell. Making use of these results and with the knowledge of the operating temperature, T, this work demonstrates that a single I-V curve is enough to fully solve the transcendental equation governing the behavior of a solar cell in the Single-Diode Model.

Developing a method for determining the time parameters of a mobile fire extinguisher operator

The object of this study is the process of functioning of the "man-machine" system on the example of the operator of a mobile fire installation. One of the issue when building models of such systems is to determine the parameters a priori of a given model of the human operator – the delay time and a time constant. For one of the promising means of fire extinguishing such as a mobile installation based on Segway, a method for determining the time parameters of the operator has been devised. A feature of the method is the use of approximation of partial derivatives from the phase-frequency characteristics of the operator in frequency, determined at two frequencies. This approach makes it possible to get rid of the need to use transcendental equations to determine time parameters and move on to an algebraic equation. To substantiate the values of frequencies at which partial derivatives are approximated, tolerance accuracy criteria are used. It is shown that working range of the operator of the mobile fire installation is in the infra-frequency region. Therefore, it is advisable to determine the phase-frequency characteristics of the operator numerically using an array of data on the transition function of the operator. An array of such data is formed using the Kotelnikov-Nyquist-Shannon theorem. A list of sequential procedures for the implementation of the method for determining the time parameters of the operator of a mobile fire installation is provided. The method for determining the time parameters of the operator of a mobile fire installation was verified by solving a test problem. It is shown that with permissible errors in the time parameters of the operator at the level of 5.0 %, the errors in their determining do not exceed 2.0 %. The reported results can be used for determining the dynamic parameters of the model of the operator of the fire installation, provided that the tolerance criterion for accuracy is set.

Analysis and Experimental Validation of Selective Harmonic Elimination in Single-Phase Five-Level Inverter using Particle Swarm Optimization Algorithm

This paper presents the analysis and experimental validation of selective harmonic elimination in single-phase multilevel inverter using one of the most effective optimization algorithms known as particle swarm optimization. The presented inverter is one of the popular emerging topologies composed of a few switches compared to conventional topologies in order to produce a five-level output voltage. The optimization algorithm is applied to find the optimal commutation angles to implement the selective harmonic elimination by solving a system of transcendental non-linear equations. The results of the computational simulations are in good agreement with the theoretical analysis. The method is simple compared with the analytical methods, effective for driving the inverters and inexpensive to apply in real-time using a low-price DSP–Delfino-C2000-TMS320F28379D Chip. The theoretical analysis is verified by simulation using MATLAB/ Simulink environment and an experimental prototype composed of efficient electronic components, respectively. The obtained results of simulations and experiments confirm the effectiveness of the presented modulation method.

Asymmetric Thermal Stability in GPL Reinforced Composite Circular Plates on Partial Winkler Foundation

Buckling of a circular graphene-platelet-reinforced composite plate resting on an elastic foundation is investigated in this research for the first time. The equations governing the thermal buckling of the circular plate were derived based on Hamilton’s principle, classical theory, and the von Kármán strain field. The effective material properties were determined by the Halpin-Tsai model and the rule of mixture. The plate is divided into two sections where solution of stability equation for each section is determined exactly. Applying the boundary and continuity conditions, a transcendental equation is established which may be used to obtain the critical buckling temperature and number of nodal diameters at the onset of buckling. In the end, and after validating the results, the effects of the laminated graphene-platelet-reinforced plate configuration, elastic foundation properties and dimensions, and the graphene platelet weight fraction on the critical thermal buckling temperature were investigated.

A review on technological aspects of different PWM techniques and its comparison based on different performance parameters

AbstractUse of Pulse Width Modulation (PWM) techniques has enabled the converters to be used in low‐frequency high‐power applications. The main objectives of PWM are to reduce the line current harmonic, switching energy loss, and torque pulsation and motor acoustic noise (for motor drive applications). This paper mainly deals with selective harmonic elimination PWM (SHEPWM), Hysteresis current controlled PWM (HCPWM), space vector PWM (SVPWM), bus‐clamping PWM (BCPWM), and the most advanced wavelet PWM technique (WPWM). Different conventional as well as advanced soft computing techniques have been discussed for SHEPWM in order to solve the complex non‐linear transcendental equations and obtain the optimized switching angles. The paper also deals with different conventional and advanced bus clamping sequences that have shown superior result in mitigating the switching loss. In order to judge the true performance of SHEPWM and CSVPWM in terms of accuracy of obtaining the switching angles, convergence rate, THD, torque ripple, switching energy loss, etc., a common experimental test bench consisting of three‐phase, two‐level, six switch inverter coupled with a 2 hp, 50 Hz, three‐phase induction machine has been formed. The selected BIA‐based SHEPWM and CSVPWM have been implemented in the MATLAB simulation software, and the same has also been applied to the experimental test bench through dSPACE 1104 DAC card. Performance of HCPWM and different BC‐PWM sequences has also been verified using the same experimental test bench. This manuscript will help the researches and practitioners working in this field, to gain a clear idea about the type of PWM to be used for their specific application in order to achieve superior performance.

Two Practical Methods for the Forward Kinematics of 3-3 Type Spatial and 3-RRR Planar Parallel Manipulators

The forward kinematics in parallel manipulators is a mathematically challenging issue, unlike serial manipulators. Kinematic constraint equations are non-linear transcendental equations that can be reduced to algebraic equations with appropriate transformations. For this reason, sophisticated and time-consuming methods such as the Bezout method, the Groebner bases method, and the like, are used. In this paper, we demonstrate that these equations can be solved by non-complicated mathematical methods for some special types of manipulators such as the 3-3 and 6-3 types of Stewart platforms, and the 3-RRR planar parallel manipulator. Our first method is an analytical approach that exploits the special structure of kinematic constraint equations and yields polynomials of 32nd and 16th order, as mentioned in the previous works. In the second method, an error function is defined. This error function is employed to find the most appropriate initial values for the non-linear equation solver which is used for solving kinematic constraint equations. Determining the initial values in this manner saves computation time and guarantees fast convergence to real solutions.

On one Approach to the Solution of the Lambert Problem using the Decompositional Method of Modal Control

A new approach to the solution of the Lambert’s problem in spaceflight mechanics is proposed for elliptical orbits. The system of four transcendental algebraic equations is solved using the method of modal synthesis which is based on multilevel decomposition of discrete dynamic system and applied to solve the problem of identification of parameters of discrete system by a state observer. The solution algorithm is as follows: conditional and identification discrete models (systems) are built for the specified system of equations; initial values of estimates are given; initial conditions in the equations of residuals are formed. Using the method of modal synthesis, the problem of search for control of the auxiliary system is solved, as a result of which the matrix of state observer feedback coefficients is calculated. This matrix is used to predict the state vector and to obtain refined estimates --- parameters of the planar orbit. A numerical example of the Lambert’s problem solution using the proposed algorithm is given. In essence, an approach to the solution of nonlinear algebraic systems of the fourth order, which can be extended to systems of any observable order, is proposed. The peculiarity of the proposed algorithm is that the convergence of the iterative process of finding a solution can have a different "adjustable" speed using the control law

Bound states of a one-dimensional Dirac equation with multiple delta-potentials

Two approaches are developed for the study of the bound states of a one-dimensional Dirac equation with the potential consisting of N δ-function centers. One of these uses Green’s function method. This method is applicable to a finite number N of δ-point centers, reducing the bound state problem to finding the energy eigenvalues from the determinant of a 2 N × 2 N matrix. The second approach starts with the matrix for a single delta-center that connects the two-sided boundary conditions for this center. This connection matrix is obtained from the squeezing limit of a piecewise constant approximation of the delta-function. Having then the connection matrices for each center, the transmission matrix for the whole system is obtained by multiplying the one-center connection matrices and the free transfer matrices between neighbor centers. An equation for bound state energies is derived in terms of the elements of the total transfer matrix. Within both approaches, the transcendental equations for bound state energies are derived, the solutions to which depend on the strength of delta-centers and the distance between them, and this dependence is illustrated by numerical calculations. The bound state energies for the potentials composed of one, two, and three delta-centers (N = 1, 2, 3) are computed explicitly. The principle of strength additivity is analyzed in the limits as the delta-centers merge at a single point or diverge to infinity.

Discussion of “Contact Unloading Behaviors of Elastic-Power-Law Strain Hardening Material Considering Indenter Elasticity Effect” (Chen, C., Wang, Q., Wang, H., Ding, H., Hu, W., Xie, W., Weng, P., Jiang, L., and Yin, X., 2022, ASME J. Tribol., 144(12), p. 121501)

Discussion of “Contact Unloading Behaviors of Elastic-Power-Law Strain Hardening Material Considering Indenter Elasticity Effect” (Chen, C., Wang, Q., Wang, H., Ding, H., Hu, W., Xie, W., Weng, P., Jiang, L., and Yin, X., 2022, ASME J. Tribol., 144(12), p. 121501)

Power Quality Improvement of a Reduced Switch MLI Using a Modified ANN-Based SHE Technique

A cross-connected source (CCS) based multilevel inverter (MLI) is proposed for the generation of “9” and 13-levels using asymmetrical switching. The reduced switch CCS-MLI is controlled with a low frequency selective-harmonic-elimination pulse-width-modulation (SHE-PWM) strategy. The switching angles are synthesized by solving SHE equations utilizing the proposed modified artificial-neural-network (ANN) like architecture. The ANN can solve multiple transcendental equations starting from random initial guess points and without any initial data. This adds a significant improvement to the existing techniques that suffer severely with an increased number of SHE equations to be solved. To verify these unique features of the modified ANN, a CCS-MLI is realized with a higher number of levels. The enhanced performance and reasonable convergence rate of the ANN validate the suitability of the proposed control. Furthermore, CCS-MLI is simulated for resistive-inductive load considering THD as a performance parameter. The power quality improvement of the proposed inverter is shown for different modulation index (MI) and significantly low THD is reported for both “9” and 13-level case. Experimental results of a hardware prototype are also illustrated. Furthermore, the proposed system is implemented as a central inverter of standalone PV system to test its practical feasibility.

Exact Solutions of Fractional Order Oscillation Equation with Two Fractional Derivative Terms

The fractional oscillation equation with two fractional derivative terms in the sense of Caputo, where the orders alpha and beta satisfy 1<alpha le 2 and 0<beta le 1, is investigated, and the unit step response and two initial value responses are obtained in two forms by using different methods of inverse Laplace transform. The first method yields series solutions with the nonnegative powers of t, which converge fast for small t. The second method is our emphasis, where the complex path integral formula of the inverse Laplace transform is used. In order to determine singularities of integrand we first seek for the roots of the characteristic equation, which is a transcendental equation with four parameters, two coefficients and two noninteger power exponents. The existence conditions and properties of the roots on the principal Riemann surface are given. Based on the results on the characteristic equation, we derive these responses as a sum of a classical exponentially damped oscillation, which vanishes in an indicated case, and an infinite integral of the Laplace type, which converges fast for large t, with a steady component in the unit step response. Asymptotic behaviors of solutions for large t are derived as algebraic decays in negative power laws characterized by the orders alpha and beta. The fractional system exhibits a magical transition from oscillation to monotonic decay in negative power law.

Phenomenological rate-independent uniaxial hysteretic models: A mini-review

A great variety of phenomenological models has been proposed over the years to model rate-independent hysteretic forces in structural mechanics. The classification of such models is usually based on the type of equation that needs to be solved to evaluate the output variable. In particular, we distinguish among algebraic, transcendental, differential and integral models. For algebraic (transcendental) models, an algebraic (a transcendental) equation needs to be solved to compute the output variable; conversely, differential equations are employed for differential models, whereas equations expressed in integral form characterize integral models. This paper provides a mini-review of the most adopted phenomenological rate-independent uniaxial hysteretic models. Such models are selected in order to provide a complete overview of the four types of previously mentioned models, currently available in the literature. In particular, we illustrate the fundamental characteristics of each model and discuss their peculiarities in terms of 1) number of adopted parameters and variables, 2) physical interpretation of parameters and related calibration procedures, 3) type of hysteresis loop shapes that can be simulated.