Trigonometric Method Research Articles

Innovative Bipolar Fuzzy Sine Trigonometric Aggregation Operators and SIR Method for Medical Tourism Supply Chain

Bipolar fuzzy sets (BFSs) are effective tool for dealing with bipolarity and fuzziness. The sine trigonometric functions having two significant features, namely, periodicity and symmetry about the origin, are helping in decision analysis and information analysis. Taking the advantage of sine trigonometric functions and significance of BFSs, innovative sine trigonometric operational laws (STOLs) are proposed. New aggregation operators (AOs) are developed based on proposed operational laws to aggregate bipolar fuzzy information. Certain characteristics of these operators are also discussed, such as boundedness, monotonicity, and idempotency. Moreover, a modified superiority and inferiority ranking (SIR) method is proposed to cope with multicriteria group decision-making (MCGDM) with bipolar fuzzy (BF) information. To exhibit the relevance and feasibility of this methodology, a robust application of best medical tourism supply chain is presented. Finally, a comprehensive comparative and sensitivity analysis is evaluated to validate the efficiency of suggested methodology.

Novel hybrid algorithms for root determining using advantages of open methods and bracketing methods

There are two numerical approaches for root determining methods: open methods and bracketing methods. Open methods usually converge much more quickly than the bracketing methods, but they may diverge in some cases. Bracketing methods always converge but they are slow compared to the open methods. In this paper, two novel blended algorithms are proposed. These algorithms have advantages of the bracketing methods and the open methods. In particular, the first hybrid algorithm consists of false-position method with modified secant method (FP-MSe) and the second blended algorithm consists of false-position method with trigonometric secant method (FP-TMSe). The numerical results show that the proposed algorithms overcome bisection (Bi) and false-position (FP) methods. On the other hand, the presented algorithms overcome the trisection (Tri), secant (Se) and Newton- Raphson (NR) algorithms according to the iteration number and the average of running time. Finally, the implementation results show the superiority of the proposed algorithms on other related algorithms.

Robust optimization of composite cylindrical shell by a trigonometric mixed response surface method

AbstractThe response surface method (RSM) as a popular fitting method has been applied to numerous engineering issues. However, the existing RSM models cannot satisfy the analysis of complex nonconvex structure problems, especially in composite cylindrical shells. Thus a trigonometric mixed response surface method (TMRSM) is suggested to improve the accuracy and the fitting performance combing optimal Latin hypercube design (LHD), the moving least square method and the trigonometric functions. An improved robust design optimization (RDO) process is performed based on the weighted norm method and reliable constraints using the above TMRSM model to obtain more robust optimal design results. Three example functions and a composite cylindrical pressure shell are given to verify the effectiveness of the TMRSM model. This pressure shell design can check the robustness of the improved RDO process.

Modeling response spectrum compatible pulse-like ground motion

The seismic response analysis of near-fault pulse-like ground motions is severely restricted due to the scarcity of pulse-like records. The requirement in regulations that the response spectra of artificial ground motions should be compatible with the target response spectrum makes the relevant studies more difficult. As a result, this study proposes a trigonometric series-based stochastic method to simulate pulse-like ground motions, with the advantage that the corresponding pseudo-spectral acceleration is compatible with the given target response spectrum. This goal is achieved by two parts. (1) The envelope function of pulse-like records obtained by the Hilbert transform is utilized as the amplitude modulation function to ensure that the simulated ground motion contains a pulse. (2) A novel iteration scheme based on random frequency parameters is proposed to guarantee the response spectrum compatibility. The velocity ground motion is first simulated since the pulse usually exists in velocity. The ground-motion acceleration subsequently obtained by differentiating the velocity is adopted to calculate the response spectrum. Two cases are implemented and verified the effectiveness of the proposed method in enriching existing pulse-like databases and generating pulse-like ground motion in areas that lack records. Moreover, the amplitude modulation function and target spectrum, as two key factors in the proposed method, determines the presence of a pulse and the pulse periods, respectively. This property makes the proposed method potentially universal applicability for stochastic pulse-like ground motion simulation in engineering.

A Monochromatic Optical Analytical Study of a Stepped Refractive Index Shell Spherical Polymeric Lens

The GRIN (gradient refractive index) Luneburg lens cannot be used in visible region applications including imaging systems and in the integrated photonic lens as static concentric symmetry lenses. Furthermore, in order to obtain free monochromatic aberration, the ray must be focused on the lens's rear surface. As a result, the detector must be movable on the rear surface. Technically, this procedure is difficult. This work presented a straightforward and practical method based on the usage of a double shell gradient stepped index of symmetry concentric polymeric lens (SCPL). The obtained results revealed that the proposed lens can be used in the visible region, as well as the shifting of the focus point from the lens's rear surface. The (SCPL) lens was optically evaluated using the trigonometric ray tracing method. The evaluation was performed using a mathematical iterative method applied in MATLAB program. The iterative method was used to determine the best shell thicknesses of shelled ball lens (double shell) for constructing the (SCPL) parameters. The constructive parameters were examined using the ZEMAX optical design program for infinite conjugate conditions and monochromatic wavelength. According to the analysis results, the Sterhl ratio is greater than (0.97), and the total transverse aberration coefficients are approaching zero.

Application of the Fast Expansion Method in Space–Related Problems

In the paper, numerical and approximate analytical solutions for the problem of the motion of a spacecraft from a starting point to a final point during a certain time are obtained. The unpowered and powered portions of the flight are considered. For a numerical solution, a finite-difference scheme of the second order of accuracy is constructed. The space-related problem considered in the study is essentially nonlinear, which necessitates the use of trigonometric interpolation methods to replace the task of calculating the Fourier coefficients with the integral formulas by solving the interpolation system. One of the simplest options for trigonometric sine interpolation on a semi-closed segment [–a, a), where the right end is not included in the general system of interpolation points, is considered. In order to maintain the conditions of orthogonality of sines, an even number of 2M calculation points is uniformly applied to the segment. The sine interpolation theorem is proved and a compact formula is given for calculating the interpolation coefficients. A general theory of fast sine expansion is given. It is shown that in this case, the Fourier coefficients decrease much faster with the increase in serial number compared to the Fourier coefficients in the classical case. This property allows reducing the number of terms taken into account in the Fourier series, as well as the amount of computer calculations, and increasing the accuracy of calculations. The analysis of the obtained solutions is carried out, and their comparison with the exact solution of the test problem is proposed. With the same calculation error, the time spent on a computer using the fast expansion method is hundreds of times less than the time spent on classical finite-difference method.

An Extended Cosine Generalized Family of Distributions for Reliability Modeling: Characteristics and Applications with Simulation Study

An extension of the cosine generalized family is presented in this paper by using the cosine trigonometric function and method of parameter induction concurrently. Prominent characteristics of the proposed family along with useful results are extracted. Moreover, two new subfamilies and several special models are also deduced. A four-parameter model called an Extended Cosine Weibull (ECW) with its mathematical properties is studied deeply. Graphical study reveals that the new model adopts right- and left-skewed, symmetrical, and reversed-J density shapes, while all possible monotone and nonmonotone shapes are exhibited by the hazard rate function. The maximum likelihood technique is exercised for parametric estimation, while estimation performance is accessed via Monte Carlo simulation study graphically and numerically. The superiority of the presented model over several outstanding and competing models is confirmed via three reliability and survival dataset applications.

Computational Solution of the Time-Fractional Schrödinger Equation by Using Trigonometric B-Spline Collocation Method

This paper proposes a numerical method to obtain an approximation solution for the time-fractional Schrödinger Equation (TFSE) based on a combination of the cubic trigonometric B-spline collocation method and the Crank-Nicolson scheme. The fractional derivative operator is described in the Caputo sense. The L1−approximation method is used for time-fractional derivative discretization. Using Von Neumann stability analysis, the proposed technique is shown to be conditionally stable. Numerical examples are solved to verify the accuracy and effectiveness of this method. The error norms L2 and L∞ are also calculated at different values of N and t to evaluate the performance of the suggested method.

Low power hardware design and its mathematical modeling for fast‐exact geolocalization system in wireless networks

SummaryA significant goal in achieving position calculation is to reduce and simplify the instructions that the Mobile Station (MS) has to accomplish in the location determination algorithm. Standard techniques widely used in many applications can be classified into Received Signal Strength Indicator (RSSI), Angle of Arrival (AoA), Time Difference of Arrival (TDoA), and Time of Arrival (ToA). However, all are almost determined by the standard trigonometric complex computation methods that are usually implemented in software where the complexity is considerable. The development of a simple and efficient technique for MS location estimation is the main objective of this paper. To design a modern hardware‐oriented algorithm, a new approach using simple Add‐and‐Shift operations in the computation is proposed and it can be implemented in a hardware target. In order to accelerate the geopositioning process, the Coordinate Rotation Digital Computer (CORDIC) algorithm has been used in this proposal. The proposed algorithm is simpler and faster than conventional ones; thus, complexity in terms of time, power consumption, and chip area has been reduced. Therefore, the algorithm is suitable for applications with limited computational and power resources. Based on comparisons with the state‐of‐the‐art algorithms, it is concluded that the proposed technique is more exact and less complex.

Effect of Leg Length Discrepancy on Lateral Center-edge Angle Measurement.

In patients with leg length discrepancy (LLD) and consequent pelvic obliquity, either the longitudinal axis of the pelvis or a line perpendicular to the ground may be used as the longitudinal reference line for measuring the lateral center-edge angle (LCEA). We aimed to (1) systematically inspect which longitudinal reference line has been used for measuring the LCEA in previous studies; (2) evaluate the frequency of change in the radiographical classification of acetabular overcoverage or undercoverage per the longitudinal reference line; and (3) validate the trigonometric method, predicting the change in the LCEA according to the LLD. Studies investigating the LCEA published between January 1976 and July 2019 in the MEDLINE database were categorized according to the longitudinal reference line used. Further, in a retrospective analysis of 238 patients surgically treated for LLD, the LCEA was first measured on standing pelvic radiographs using the longitudinal axis of the pelvis (pLCEA) and measured again using a line perpendicular to the ground (gLCEA). Femoral head coverage was categorized as undercoverage, normal, or overcoverage based on the pLCEA and gLCEA. The theoretically calculated difference between the pLCEA and gLCEA (dLCEA) as determined using a trigonometric method was compared with the dLCEA measured on radiographs. Of 229 previous studies, 188 did not specify the longitudinal reference line. The number of patients who were diagnosed with acetabular overcoverage using the pLCEA and gLCEA was one and fourteen, respectively (P<0.001). The number of patients who were diagnosed with acetabular undercoverage using the pLCEA and gLCEA was one and zero, respectively (P=1.000). There was no difference (P=0.433) between the theoretically calculated (9±5 degrees) and measured (9±5 degrees) dLCEAs. The definition of the longitudinal reference line should be clarified when measuring the LCEA. The trigonometric method can accurately predict the change in the LCEA according to LLD in concentric hips without proximal femoral and pelvic deformities. Level IV-diagnostic study.

Uneven settlement threshold of continuous beam pier based on analytic mapping relationship

Currently, there is no regulation on the settlement threshold of continuous beam bridge pier of high-speed railway (HSR), a method is proposed in the present study to calculate the settlement threshold of continuous beam pier based on the analytic mapping relationship. First, the dynamic model of the train-track-continuous beam bridge was established. Then, the mapping rail irregularities caused by pier settlement obtained from the analytic mapping relationship superimposed with the initial rail irregularities converted from the Chinese track irregularities spectrum by using the trigonometric series method. Furthermore, the proposed method was verified by comparative analysis with the measured data and the existing results. Finally, the types and thresholds of uneven settlement of continuous beam bridge pier were revealed based on the parameter discussion method. It was found that the influence of pier settlement on the train dynamic performance index had a strong correlation with the train running direction. The settlement threshold of long-span continuous beam pier was looser than that of simply supported beams with a 32 m span. The settlement threshold of continuous beam piers can be appropriately relaxed in design.

Heterogeneous effects of eccentric training and nordic hamstring exercise on the biceps femoris fascicle length based on ultrasound assessment and extrapolation methods: A systematic review of randomised controlled trials with meta-analyses.

ObjectiveTo systematically review the effects of eccentric training based on biceps femoris fascicle length using ultrasound assessment and extrapolation methods.DesignSystematic review and meta-analysis of randomised controlled trials.Data sourcesCENTRAL, CINAHL Plus with full text, PubMed and OpenGrey databases were searched on 6 July 2021.Eligibility criteria for selecting studiesRandomised controlled trials (RCTs) lasting at least four weeks and presenting data about biceps femoris (BF) fascicle length (FL) as an outcome.MethodSearching databases, screening studies, performing risk of bias assessments and determining the level of evidence (LoE) for each meta-analysis were applied during the study. PRISMA 2020 statement and Cochrane Handbook for Systematic Reviews of Interventions were used as the guidelines of this systematic review.ResultsEight randomised controlled trials included in meta-analyses. Based on the very low and low LoE, eccentric training has small (g = 0.29, 95% CI [-0.26, 0.85]), moderate (g = 0.72, 95% CI [0.17, 1.28]) and large (g = 2.20, 95% CI [0.99, 3.41]) effect sizes (ES) based on manual linear extrapolation (MLE), panoramic ultrasound scanning and trigonometric equation methods, respectively. Similarly, Nordic hamstring exercise (NHE) has small (g = 0.23 [-1.02, 1.47]), small (g = 0.38, 95% CI [-0.50, 1.27]) and large (g = 1.98, 95% CI [0.52, 3.44]) ES based on the MLE, panoramic ultrasound scanning and trigonometric equation methods, respectively.ConclusionES of eccentric training, including NHE, vary between the MLE, panoramic ultrasound scanning, and equation methods. The relevant scientific community should have a consensus on measurement standards of the BF FL measurements. Further studies can be conducted to compare the effects of eccentric training based on the ultrasound assessment and extrapolation methods.

A quartic trigonometric B-spline collocation method for a general class of nonlinear singular boundary value problems

A quartic trigonometric B-spline collocation method for a general class of nonlinear singular boundary value problems

FRACTIONAL POWER SERIES APPROACH FOR THE SOLUTION OF FRACTIONAL-ORDER INTEGRO-DIFFERENTIAL EQUATIONS

Fractional differential and integral equations are focus of the researchers owing to their tremendous applications in different field of science and technology, such as physics, chemistry, mathematical biology, dynamical system and engineering. In this work, a power series approach called Residual Power Series Method (RPSM) is applied for the solution of fractional (non-integer) order integro-differential equations (FIDEs). The Caputo sense is used for calculating fractional derivatives. Comparison of the obtained solution is made with the Trigonometric Transform Method (TTM) and Optimal Homotopy Asymptotic Method (OHAM). There is no restrictive condition on the proposed solution. The presented technique is simple in applicability and easily computable.

Numerical simulations of Zakharov’s (ZK) non-dimensional equation arising in Langmuir and ion-acoustic waves

The trigonometric quintic B-spline scheme is used in this research paper to research Zakharov’s (ZK) nonlinear dimensional equation’s numerical solution. The ZK model’s solutions explain the relationship between the high-frequency Langmuir and the low-frequency ion-acoustic waves with many applications in optical fiber, coastal engineering, and fluid mechanics of electromagnetic waves, plasma physics, and signal processing. Three recent computational schemes (the expanded [Formula: see text]-expansion method, generalized Kudryashov method, and modified Khater method) have recently been used to investigate this model’s moving wave solution. Many innovative solutions have been established in this paper to determine the original and boundary conditions that allow numerous numerical schemes to be implemented. Here, the trigonometric quintic B-spline method is used to analyze the precision of the collected analytical solutions. To illustrate the precision of the numerical and computational solutions, distinct drawings are depicted.

Solving FDE by trigonometric neural network and its applications in simulating fractional HIV model and fractional Schrodinger equation

In this paper, we put forward numerical algorithms to simulate fractional HIV infected CD4+T cell model and fractional Schrodinger equation. First, we put forward numerical algorithm to solve fractional differential systems with trigonometric neural network. Then trigonometric neural network method is used to simulate the fractional HIV infected CD4+T cell model. It is found that the infection equilibrium has the restriction condition for the number of virus particles released by each infected cell. Under this condition, the fractional system has a unique positive equilibrium. Second, we put forward numerical algorithm to solve fractional PDE with trigonometric neural network. Then this numerical method is used to simulate fractional Schrodinger equation, and some important results in fractional quantum mechanics are concluded by the corresponding numerical results.

Exponentially Convergent Galerkin Method for Numerical Modeling of Lasing in Microcavities with Piercing Holes

The paper investigates an algorithm for the numerical solution of a parametric eigenvalue problem for the Helmholtz equation on the plane specially tailored for the accurate mathematical modeling of lasing modes of microring lasers. The original problem is reduced to a nonlinear eigenvalue problem for a system of Muller boundary integral equations. For the numerical solution of the obtained problem, we use a trigonometric Galerkin method, prove its convergence, and derive error estimates in the eigenvalue and eigenfunction approximation. Previous numerical experiments have shown that the method converges exponentially. In the current paper, we prove that if the generalized eigenfunctions are analytic, then the approximate eigenvalues and eigenfunctions exponentially converge to the exact ones as the number of basis functions increases. To demonstrate the practical effectiveness of the algorithm, we find geometrical characteristics of microring lasers that provide a significant increase in the directivity of lasing emission, while maintaining low lasing thresholds.

Algorithms of multidimensional random process simulation

The article discusses two approaches to modeling signals and processes: the method of filter construction and the trigonometric method. It is shown that the later approach is more promising, since an increase in the signal/process representation dimension mathematically means adding a term to the basis function formula, which gives access to fast simulation algorithms. Examples of algorithms for multidimensional simulation of random processes using two methods are given and a software system that implements these algorithms is described. The results provided by the software system will allow you to predict characteristics of engineering projects (accuracy and speed of modeling algorithms). Due to the high relevance of and need for fundamental research of methods and algorithms for digital transformation of the component base, the digitalization of all aspects of activity, including the synthesis of new materials, the development of new methods for designing micro- and nano-systems, the article aims to expand the scope of the spectral method of simulating multidimensional processes using original algorithmic complexes.

Trigonometric Inference Providing Learning in Deep Neural Networks

Despite being heavily used in the training of deep neural networks (DNNs), multipliers are resource-intensive and insufficient in many different scenarios. Previous discoveries have revealed the superiority when activation functions, such as the sigmoid, are calculated by shift-and-add operations, although they fail to remove multiplications in training altogether. In this paper, we propose an innovative approach that can convert all multiplications in the forward and backward inferences of DNNs into shift-and-add operations. Because the model parameters and backpropagated errors of a large DNN model are typically clustered around zero, these values can be approximated by their sine values. Multiplications between the weights and error signals are transferred to multiplications of their sine values, which are replaceable with simpler operations with the help of the product to sum formula. In addition, a rectified sine activation function is utilized for further converting layer inputs into sine values. In this way, the original multiplication-intensive operations can be computed through simple add-and-shift operations. This trigonometric approximation method provides an efficient training and inference alternative for devices with insufficient hardware multipliers. Experimental results demonstrate that this method is able to obtain a performance close to that of classical training algorithms. The approach we propose sheds new light on future hardware customization research for machine learning.

A Composite Algorithm for Numerical Solutions of Two-Dimensional Coupled Burgers’ Equations

In this study, a new composite algorithm with the help of the finite difference and the modified cubic trigonometric B-spline differential quadrature method is developed. The developed method was applied to two-dimensional coupled Burgers’ equation with initial and Dirichlet boundary conditions for computational modeling. The established algorithm is better than the traditional differential quadrature algorithm proposed in literature due to more smoothness of cubic trigonometric B-spline functions. In the development of the algorithm, the first step is semidiscretization in time with the forward finite difference method. Furthermore, the obtained system is fully discretized by the modified cubic trigonometric B-spline differential quadrature method. Finally, we obtain coupled Lyapunov systems of linear equations, which are analyzed by the MATLAB solver for the system. Moreover, comparative study of these solutions with the numerical and exact solutions which are appeared in the literature is also discussed. Finally, it is found that there is good suitability between exact solutions and numerical solutions obtained by the developed composite algorithm. The technique can be extended for various multidimensional Burgers’ equations after some modifications.