Integration Interval Research Articles

Приближенное решение гиперсингулярного интегрального уравнения I рода, ограниченное на обоих концах отрезка интегрирования, с применением рядов Чебышева

A hypersingular integral equation on the interval of integration is considered. The hypersingular integral is understood in the sense of Hadamard, that is, in the finite part. The class of such equations is widely used in problems of mathematical physics, in technology, and most importantly: in recent years, they are one of the main devices for modeling problems in electrodynamics. With the use of Chebyshev polynomials of the second kind, the unknown function, the right-hand side and the kernel are replaced in the equation. The expansion coefficients of these functions are calculated using quadrature formulas of the highest algebraic degree of accuracy, i.e., Gauss quadrature formulas. Thus, the equation is discretized. The result is an infinite system of linear algebraic equations for the expansion coefficients of the unknown function. The fact that the hypersingular integral equation in the case under consideration has a unique solution in the class of sufficiently smooth functions is taken into account. The constructed computational scheme is substantiated using the general theory of functional analysis. The calculation error is estimated under certain conditions relative to the right-hand side and the kernel of the equation. The described method for solving the hypersingular integral equation is illustrated by test examples that show the high efficiency of the method.

Revisiting the nonlinear Gaussian noise model for hybrid fiber spans

We rederive from first principles and generalize the theoretical framework of the nonlinear Gaussian noise model to the case of coherent optical systems with multiple fiber types per span and ideal Nyquist spectra. We focus on the accurate numerical evaluation of the integral for the nonlinear noise variance for hybrid fiber spans. This task consists in addressing four computational aspects: (1) Adopting a novel transformation of variables (other than using hyperbolic coordinates) that changes the integrand to a more appropriate form for numerical quadrature; (2) Evaluating analytically the integral at its lower limit, where the integrand presents a singularity; (3) Dividing the interval of integration into subintervals of size π and approximating the integral over each subinterval by using various algorithms; and (4) Deriving an upper bound for the relative error when the interval of integration is truncated in orderto accelerate computation. We apply the proposed analytical model to the performance evaluation of coherent optical communications systems with hybrid fiber spans composed of quasi-single-mode and single-mode fiber segments. More specifically, the model is used to optimize the lengths of the optical fiber segments that compose each span in order to maximize the system performance. We check the validity of the optimal fiber segment lengths per span provided by the analytical model by using Monte Carlo simulation, where the Manakov equation is solved numerically using the split-step Fourier method. We show that the analytical model predicts the lengths of the optical fiber segments per span with satisfactory accuracy so that the system performance, in terms of the Q-factor, is within 0.1 dB from the maximum given by Monte Carlo simulation.

A rapid harmonic detection method for grid-connected photovoltaic when three-phase voltage is unbalanced

Abstract By using power electronic devices, photovoltaic grid-connected power generation may inject harmonics into the power system. As the photovoltaic grid-connected inverter has the same basic structure as the active power filter, so the unified control of the photovoltaic grid and active filtering can be achieved. When the current unified control system compensates harmonics of the grid side, it mainly uses ip-iq harmonic detection method, which is based on instantaneous reactive power theory. When the three-phase voltage is unbalanced, the method has a large voltage phase angle detection error and the signal of the low-pass filter tracking system is long, detection time delay and even failure occur. This paper proposes an improved fast harmonic detection method. When phase deviation or amplitude change occurs to the three-phase voltage, the positive and negative-sequence voltages are simultaneously park transformed. The negative-sequence component is filtered by the current average module to obtain the fundamental amount of the voltage, then the phase angle of the positive-sequence voltage is accurately calculated to improve the harmonic current detection accuracy. Through the study of the integral method, it is found that the least common multiple of each harmonic period can be used as the integral interval, and the integral value is also zero, so the detection delay time is reduced by replacing the low-pass filter with an integration module. The simulation results show that the proposed harmonic detection algorithm can accurately detect harmonics when the three-phase voltage is unbalanced, and about 0.057 s improve the harmonic detection speed compared with the commonly used ip-iq method.

Application of the Cauchy integral approach to singular and highly oscillatory integrals

This paper presents a method that is based on the sum of line integrals for fast computation of singular and highly oscillatory integrals , and . Where G and f are non-oscillatory sufficiently smooth functions on the interval of integration. is a product of singular factors and is an oscillatory parameter. The computation of these integrals requires f and G to be analytic in a large complex region C accommodating the interval of integration. The integrals are changed into a problem of integrals on ; which are later computed using the generalized Gauss-Laguerre rule or by the construction of Gauss rules relative to a Freud weights function with k positive. MATHEMATICA programming code, algorithms and illustrative numerical examples are provided to test the efficiency of the presented experiments.

Optimization of one-step hybrid method for direct solution of fifth order ordinary differential equations of initial value problems

This paper focuses on the derivation, analysis and implementation of a hybrid method by optimizing the order of the method by introduction of six-hybrid points for direct solution of fifth order ordinary differential equations of initial value problems (IVPs). Power series was used as the basis function for the solution of the IVP. The basis function was interpolated at some selected hybrid points whereas the fifth derivative of the approximate solution was collocated at all the interval of integration of the method to generate a system of linear equations for the determination of the unknown parameters. The derived method was tested for consistency, zero stability, convergence and absolute stability. The method was tested with two linear test problems to confirm its accuracy and usability. The comparison of the results with some existing methods shows the superiority of the accuracy of the method.

On the Regularity Order of the Pointwise Uncollided Angular Flux and Asymptotic Convergence of the Discrete Ordinates Approximation of the Scalar Flux

To determine the angular discretization error asymptotic convergence rate of the uncollided scalar flux computed with the discrete ordinates (S) method, a comprehensive theory of the regularity order with respect to the azimuthal angle of the exact pointwise SN uncollided angular flux is derived based on the integral form of the transport equation in two-dimensional Cartesian geometry. With this theory, the regularity order of the pointwise uncollided angular flux can be estimated for a given problem configuration. Our new theory inspired a novel Modified Simpson’s (MS) quadrature that converges the uncollided scalar flux faster than any of the traditional quadratures by avoiding integration across points of irregularity in the azimuthal angle. Numerical results successfully verify our new theory in four variants of a test configuration, and the angular discretization errors in the corresponding scalar flux computed with conventional angular quadrature types and with our new quadrature types are found to converge with different orders. The error convergence rates obtained with traditional quadrature types are limited by the regularity order of the exact angular flux and the quadrature’s integration intervals while our new MS quadrature types converge with order two to four times higher than traditional quadratures. A detailed study of oscillations observed in certain quadrature errors is provided by introducing the effective length of the irregular interval and the associated oscillating function.

Exploiting the Low Doppler Tolerance of Noise Radar to Perform Precise Velocity Measurements on a Short Set of Data

The extraction of velocity information from radar data by means of the Doppler effect is the driving factor for the investigations presented in this paper. A method for the quantification of the Doppler tolerance in continuous emission (CE) noise radar is introduced, addressing a current lack in literature within the frame of CE noise radars. It is shown that noise radar is highly sensitive to the Doppler effect, an issue that often results in a low Doppler tolerance especially for long coherent integration intervals. In general, the Doppler sensitivity is considered as a drawback but, in this paper, along with the absence of range-Doppler coupling in noise radar, it is turned into an advantage allowing for a very precise Doppler estimation. This new signal processing approach for Doppler extraction is detailed and its feasibility is proven on the basis of experimental data. The presented method requires much less data, i.e., target illumination time, than conventional Doppler analyses and, therefore, is beneficial in terms of radar resource management.

Development of a new numerical scheme for the solution of exponential growth and decay models

This paper presents the development of a new numerical scheme for the solution of exponential growth and decay models emanated from biological sciences. The scheme has been derived via the combination of two interpolants namely, polynomial and exponential functions. The analysis of the local truncation error of the derived scheme is investigated by means of the Taylor’s series expansion. In order to test the performance of the scheme in terms of accuracy in the context of the exact solution, four biological models were solved numerically. The absolute error has been computed successfully at each mesh point of the integration interval under consideration. The numerical results generated via the scheme agree with the exact solution and with the fifth order convergence based upon the analysis carried out. Hence, the scheme is found to be of order five, accurate and is a good approach to be included in the class of linear explicit numerical methods for the solution of initial value problems in ordinary differential equations.

Software Phase Correction Technique for Passive Radar

A traditional radar actively transmits pulses and receive the corresponding echoes. By computing the time taken to receive the echoes the system is able to detect targets and estimate their ranges. However, mainly for military applications, an active radar has a major drawback: it can be detected by the enemy since it radiates electromagnetic pulses. As such, there is currently high research and development activity in the field of passive radar systems. In this alternative scenario, the radar does transmit any signal. Instead, it uses signals already present in the environment, such as TV, radio broadcasts and satellite digital video broadcast as transmitters of opportunity. By measuring the time difference between the signal received by the transmitter of opportunity and the signal echoed by the targets, the radar can detect targets and estimate their ranges. In scenarios which use very low power illuminators of opportunity, such as satellite DVB-S or GPS, relatively long integration times are required in order to obtain reasonable values of signal to noise ratio. This implies that, besides frequency correction, the coherency of local oscillators in the receivers need to be maintained during long time intervals, which can be very difficult. The paper presents a novel phase correction technique for passive radar which uses targets of opportunity, already present in the target area, as references to maintain the coherency of the oscillators for all the integration interval. The proposed methodology is quite simple and enables the use of low-cost hardware with independent oscillators for the reference and surveillance channels which can be geographically distributed. The obtained results illustrate the effectiveness and applicability of the method.

Computing the matrix fractional power with the double exponential formula

Two quadrature-based algorithms for computing the matrix fractional power $A^\alpha$ are presented in this paper. These algorithms are based on the double exponential (DE) formula, which is well-known for its effectiveness in computing improper integrals as well as in treating nearly arbitrary endpoint singularities. The DE formula transforms a given integral into another integral that is suited for the trapezoidal rule; in this process, the integral interval is transformed into an infinite interval. Therefore, it is necessary to truncate the infinite interval to an appropriate finite interval. In this paper, a truncation method, which is based on a truncation error analysis specialized to the computation of $A^\alpha$, is proposed. Then, two algorithms are presented---one where $A^\alpha$ is computed with a fixed number of abscissa points and one with $A^\alpha$ computed adaptively. Subsequently, the convergence rate of the DE formula for Hermitian positive definite matrices is analyzed. The convergence rate analysis shows that the DE formula converges faster than Gaussian quadrature when $A$ is ill-conditioned and $\alpha$ is a non-unit fraction. Numerical results show that our algorithms achieve the required accuracy and are faster than other algorithms in several situations.

Computing the matrix fractional power with the double exponential formula

Two quadrature-based algorithms for computing the matrix fractional power $A^\alpha$ are presented in this paper. These algorithms are based on the double exponential (DE) formula, which is well-known for its effectiveness in computing improper integrals as well as in treating nearly arbitrary endpoint singularities. The DE formula transforms a given integral into another integral that is suited for the trapezoidal rule; in this process, the integral interval is transformed to the infinite interval. Therefore, it is necessary to truncate the infinite interval into an appropriate finite interval. In this paper, a truncation method, which is based on a truncation error analysis specialized to the computation of $A^\alpha$, is proposed. Then, two algorithms are presented -- one computes $A^\alpha$ with a fixed number of abscissas, and the other computes $A^\alpha$ adaptively. Subsequently, the convergence rate of the DE formula for Hermitian positive definite matrices is analyzed. The convergence rate analysis shows that the DE formula converges faster than the Gaussian quadrature when $A$ is ill-conditioned and $\alpha$ is a non-unit fraction. Numerical results show that our algorithms achieved the required accuracy and were faster than other algorithms in several situations.

Integration interval type-2 FAHP-FTOPSIS group decision-making problems for salt farmer recommendation

Integration interval type-2 FAHP-FTOPSIS group decision-making problems for salt farmer recommendation

An approximate solution of singular integral equations using the Chebyshev series on the class of functions vanishing at one end and turning into infinity at the other end of the integration interval

An approximate solution of singular integral equations using the Chebyshev series on the class of functions vanishing at one end and turning into infinity at the other end of the integration interval

ON AUTOMATIC SEGMENTATION METHOD USING IN SOLVING NONLINEAR INITIAL BOUNDARY VALUE PROBLEMS OF MECHANICS OF SOFT-SHELL STRUCTURES

An approach to improving nonlinear initial boundary value problems of thin-walled structures solution algorithm is represented in the work. The algorithm is based on using line method and parameter differentiation method and is oriented to analysis of composite structures behavior at arbitrary physical and geometrical nonlinearity. Obviously, the less restrictions the calculation algorithm imposes on problem statement, the more features of its realization must be taken into account when developing corresponding software module. The drawback of universality of the algorithm represented is that its direct implementation in some cases can lead either to wrong results, or to loss of calculation stability. To avoid these problems we suggest automatic segmentation method using as a step of the algorithm developed. The essence of the method lies in testing fulfillment of the condition of normalized integral matrices of initial and conjugate differential equation systems orthogonality at the step of problem preprocessing. In the points of integration interval where orthogonality condition is not fulfilled segmentation of the interval is carried out. For testing this approach we select nonlinear problems of dynamic inflation of an infinite cylinder of Mooney-Rivlin material and a sphere of Neo-Hookean material by suddenly applied pressure. It is shown that automatic segmentation method allows improving period and amplitude calculation accuracy, as well as increasing iteration process convergence rate. A problem of dynamic inflation of a hinged hemisphere of Neo-Hookean material by suddenly applied pressure is considered. On the basis of its solution results we suggest the approach to determining parameters of calculation algorithm which allow obtaining correct solution.

Application of computed tomography dose index as dosimetric parameter in dental tomography

ABSTRACT Objective Quantify the CTDI from a reference beam, correcting this value for several beam widths from the ratio of CTDI100 values measured in the air, and verifying the underestimation of the dose comparing CTDI100 and CTDI300, applied to dental CBCT. Methods i-CAT and Prexion 3D tomographs, 100mm pencil ionization chamber, electrometer. Firstly for beam above 40 mm, CTDIW,NT from CTDIW,Ref, multiplied by the ratio of CTDIAR measurements to N.T widths and reference, was estimated. In second, CTDI100 and CTDI300 are obtained by displacing the ionization chamber along the beam with spacing intervals equal to 100 mm to cover sufficiently large integration intervals for CBCT protocols, and a comparison is made through the modified efficiency. Results CTDIAR,100,Ref averaged 53% higher than CTDIW,Ref, due to attenuation of the beam by the simulator object, and the ratio between CTDIAR,100,Ref and CTDIW,Ref is greater than 1, being this constant relationship and validating the method for dosimetry in quality control tests. For the second method, CTDI100 greatly underestimates the dose deposited on the central axis, where CTDI300 covers all beam sizes and stands out in relation to CTDI100 to more accurately quantify the radiation levels emitted. Conclusions The IEC method is applicable to quality control, facilitating practice, and optimizing time and resources. CTDI300 is a better dose descriptor than CTDI100, and should be implemented for CBCT modalities when used.

3-D Point Cloud Generation From Airborne Single-Pass and Single-Channel Circular SAR Data

This article presents a framework to generate 3-D point clouds by very-high-resolution single-pass and single-channel circular synthetic aperture radar (CSAR). The focus is both on the precise 3-D determination of very small, detached objects and on larger buildings in complex urban scenes. Inspired by optical flow methods, our approach evaluates the tracked aspect dependent backscatter energy flow of objects above the focusing reference plane. Besides the derivation of the exact projection geometry in CSAR, we give answers to the question of how accurate 3-D information can be extracted by this approach as a function of aspect interval and the object’s signal-to-clutter ratio (SCR). A further coherent adaption uses considerably larger subapertures while focusing the data on different reference heights. The computed 3-D information from multiple aspect views is then fused to a georeferenced 3-D point cloud. This results in the first demonstration of 3-D point cloud generation from CSAR data collected with a frequency-modulated continuous-wave (FMCW) radar at the W-band. The results were validated with light detection and ranging (LiDAR) data, and the height accuracy was evaluated in relation to the aspect integration interval and the number of aspect views. For small point targets with edge lengths of 3 cm, we could demonstrate that an aspect interval of only 8.5° leads to a height accuracy below 20 cm. Building roofs reach a height accuracy of several tens of centimeters. Besides the spatial information, we can reveal the angular scattering behavior of individual objects and determine over which aspect interval they are even visible.

Burn-through prediction during in-service welding based on residual strength and high-temperature plastic failure criterion

The burn-through of in-service welding is affected by pipeline structure, welding procedure and the pressure of the internal medium, etc. It is a high-temperature failure process under the coupling action of structure field, temperature field, and stress field. Predicting the critical burn-through pressure of in-service welding is a challenging problem. In this paper, the existing burn-through criteria were comprehensively summarized and analyzed. Considering the multi-field coupling effect of in-service welding, two kinds of burn-through criteria based on the thermo-mechanical analysis model were proposed innovatively. One is the residual strength three-dimensional integral method. The residual strength in the high-temperature zone of in-service welding was calculated by this method, and the critical burn-through pressure was obtained by further calculation. The selection of effective integral temperature interval is the key to ensure the accuracy of three-dimensional integral calculation of residual strength. It is found that there is a linear relationship between welding heat input and the lower limit of the integral temperature interval. However, for different materials, the linear relation formula is different. The second criterion is the high-temperature plastic failure criterion. The most dangerous path is the one with the largest plastic failure depth along wall thickness. When the von Mises stress over 2/3 of the pipe wall along the most dangerous path is greater than the yield strength at the corresponding temperature, there is a high risk of burn-through. The accuracy of the two criteria in calculating the critical burn-through pressure was verified by in-service welding tests. The results show that the residual strength three-dimensional integration method and the high-temperature plastic failure criterion both have high accuracy and reliability in predicting the burn-through of in-service welding and will have important referential value in predicting the failure of defective pipe under complex temperature and stress conditions.

Gravitational wave signature of the fifth force for dark matter

Testing the equivalence principle (EP) is of great importance for fundamental physics. Whether the EP is held for dark matter (DM) is still unclear. Motivated by the recent discoveries of gravitational waves (GWs) by ground based detectors, we study the possibility of testing EP of DM by GW experiments. We consider a binary composed of a black hole (BH) and a neutron star (NS) with respect to the galactic DM as the attractor. Possibly, long-range fifth-force other than gravity between DM and ordinary matter may lead to the EP violation for DM, we investigate the effect of fifth-force acted by DM on the GW waveform of the BH-NS binary. The presence of the fifth-force will lead to a phase correction in the GW waveform which may be detectable, but the phase correction depends chaotically on the initial condition. We calculate the probability of detection for different initial conditions. The probability depends on the time interval of integration, the EP-violating driving force and the initial eccentricity. We find for appropriate conditions the probability can reach to a considerable value, and even approach to 1 in some extremal cases. For example, for most BH-NS binaries formed in the field locating as far as the solar system to the galactic center, the fifth force with the Eötvös parameter will result in the probability of detection >0.6 if the interval of integral time is from f = 1 Hz to the final state when the NS is disrupted by the BH. These predictions may be tested by future GW observations.

Jitter Detection and Image Restoration Based on Continue Dynamic Shooting Model for High-Resolution TDI CCD Satellite Images

Although time delay integration charge-coupled devices (TDI CCDs) have been widely used in high-resolution spaceborne optical cameras, they are sensitive to satellite jitter: the images obtained by them are affected by both distortion and blur. Therefore, according to the multistage integral imaging characteristics of TDI CCDs, this article not only proposes a continue dynamic shooting model (CDSM) to reflect the real push-broom mode of the satellite but also presents a method containing jitter detection and image restoration based on it. In the presented method, the CDSM subdivides the TDI CCD integration intervals. The subdivision number of CDSM is determined by the proposed integral transformation function (ITF). Then, it feeds back into the ITF and also contributes to the point spread function (PSF) estimation. Among the abovementioned, ITF defines the relationship between the parallax images and the jitter curve, and aims to improve the jitter detection performance. Finally, an adaptive image restoration based on context is conducted, which combines time, space, and spectrum information. Besides the simulated images, multispectral images of GaoFen-1 02 satellite were also adopted to validate the performance of the presented method. Experimental results indicate that the accuracy of the jitter detection is increased, and the geometric and radiometric qualities of restored images are also improved.

Autocorrected preconditioning regularization inversion algorithm for an atmospheric turbulence profile.

Atmospheric turbulence profiles have great significance for adaptive optics, astronomical observations, laser propagation in atmospheres, and free space optical communications. The two-aperture differential scintillation method is a recent approach for analyzing remote-sensing atmospheric turbulence profiles that utilizes active beacons to make it suitable for different measurement situations. The relationship between differential scintillation and atmospheric turbulence profiles can be modeled using the Fredholm integral equation. To address this ill-posed integration problem, the discrete forward observation equation is first analyzed to obtain better integration intervals and measurement intervals needed for inversion. Then an autocorrected preconditioning conjugate gradient normal residual (PCGNR) algorithm is proposed to acquire atmospheric turbulence profiles. The algorithm contains a developed autocorrection strategy that incorporates incremental differences, adaptive thresholds, and weighted averages to correct for artefacts and marginal errors that arise from the PCGNR method. Compared with other regularized methods, the proposed autocorrected PCGNR method is more accurate and robust in the presence of noise.