Integral Function Method Research Articles

On the study of double dispersive equation in the Murnaghan’s rod: Dynamics of diversity wave structures

This article secures the various wave structures of the fractional double dispersive equation, a significant nonlinear equation that describes the propagation of nonlinear waves within the elastic, uniform, and inhomogeneous Murnaghan’s rod. The model under discussion has a wide range of applications in science and engineering. Two recently developed analytical techniques known as the improved generalized Riccati equation mapping method and the multivariate generalized exponential rational integral function method have been applied to the proposed equation for the first time. A variety of solutions have been revealed such that dark, singular, bright-dark, bright, complex, and combined solitons. Furthermore, we include a diverse array of plots that illustrate the physical interpretation of the obtained solutions in relation to a number of significant parameters, thereby highlighting the impact of fractional derivatives. Within the context of the proposed model, these visualizations give a clear understanding of the behavior and characteristics of the solutions. This study’s results have the potential to enhance comprehension of the nonlinear dynamic characteristics exhibited by the specified system and validate the efficacy of the implemented techniques. The achieved results significantly enhance our understanding of nonlinear science and the nonlinear wave fields associated with more complex nonlinear models.

Robust stabilization for networked systems with transmission delay via integral Lyapunov functional and congruence transformation method

Uncertainties and information transmission delay exist in many real-world applications. The control design of such systems is challenging even for existing advanced control theory. Besides, if there is actuator saturation in the system and networked control strategy is considered, the problem will become even more complicated. In order to solve the problem, the stochastic Takagi-Sugeno (T-S) fuzzy delay-dependent static output feedback control is proposed in this paper, and strictly dissipative analysis is addressed for the stochastic T-S fuzzy singular information networked control systems. In the propose control scheme, the T-S fuzzy model and stochastic Bernoulli theory are employed in the controller design. The stability conditions of the closed-loop control system are summarized in the linear matrix inequalities (LMIs), then the closed-loop system is regular impulse free, stochastically admissible and strictly dissipative. Both fuzzy-basis-dependent and delay-dependent stability analysis, with the consideration of strictly dissipative performance index, are conducted to develop stability conditions in terms of LMIs based on Lyapunov stability theory. Via the LMIs optimization constraints, the nonconvex problem caused by fuzzy-basis-dependent can be solved. Finally, the simulation examples are provided to verify the effectiveness of the proposed control strategy.

ПОЛЯРИЗАЦІЙНО ТА ЧАСТОТНО СЕЛЕКТИВНІ ХАРАКТЕРИСТИКИ ХІРАЛЬНОЇ ПОВЕРХНІ, ЩО СКЛАДАЄТЬСЯ З ПЕРІОДИЧНО РОЗТАШОВАНИХ ДІЕЛЕКТРИЧНИХ КВАДРАТНИХ СПІРАЛЕЙ

Subject and Purpose. The present study is concerned with the linearly polarized electromagnetic wave transmission through a chiral metasurface composed of periodically assembled square dielectric helices. We expect that the metasurface of the kind has a wider range of functional capabilities to transform a polarized wave into a cross-polarized one when compared to a similar metasurface composed of metal helices. Methods and Methodology. To find the scattering coefficients of the considered structure, the well-established method of integral functionals is followed. A set of volume integral equations in the vectorial form is solved for the equivalent electric and magnetic polarization currents of the analyzed periodic layer. A distinctive feature of the method is that the internal electromagnetic fields of the structure are initially found, whence the fields scattered by this structure are sought. The equations are discretized in terms of integral functionals related to the polarization currents and through the use of the double Floquet–Fourier series expansion technique. Results. It has been found that the metasurface transmission coefficients depend critically on the number of bars making the square helical particle. In the case of an even bar number, the chiral metasurface exhibits the same transmission coefficients for co- polarized field components in the event that linearly x- and y-polarized waves are incident. For cross-polarized field components, the transmission coefficients differ and can reach peak values at different frequencies. Finally, transmission coefficients of these polarizers have been investigated versus dimensions of helice-making bars. Conclusion. A wide variety of transmission properties observed in the metasurfaces make them particularly attractive for use in polarization converting and separating devices. The metasurface can feature dichroic asymmetric transmission and be used as a dichroic filter with polarization transformation. It can be put to use in differential phase sections and, also, as an effective dichroic cross-polarization converter (twist polarizer)

Exponential Stabilization of Reaction–Diffusion Systems Via Intermittent Boundary Control

Exponential stabilization is studied for reaction–diffusion systems (RDSs) with a reaction term satisfying the global Lipschitz condition by means of intermittent boundary control.First, a state-dependent intermittent boundary controller is designed when system states are available. By employing the spatial integral functional method and Poincaré’s inequality, a sufficient condition ensuring exponential stability of RDSs is presented. Then, an observer is presented to estimate the system states based on boundary output, when the information of system states is not fully accessible. An observer-based intermittent boundary controller is provided aiming to stabilize the considered RDSs based on the designed observer. Furthermore, a robust observer-based intermittent boundary controller is proposed to ensure the exponential stability for RDSs with uncertainties. Examples are given to illustrate the effectiveness of results.

New soliton solutions of Simplified Modified Camassa Holm equation, Klein–Gordon–Zakharov equation using First Integral Method and Exponential Function Method

In this article, new soliton solutions of Simplified Modified Camassa–Holm (SMCH) equation and Klein–Gordon–Zakharov (KGZ) equation are acquired. First Integral Method (FIM) and Exponential Function Method (EFM) are employed to construct the soliton solutions of SMCH equation and KGZ equation. The solution by these approaches can be obtained by utilizing symbolic computational softwares like Maple and Mathematica. The results suggest that the EFM and FIM are effective and useful techniques to handle non-linear problems.

The exact solutions of conformable time-fractional modified nonlinear Schrödinger equation by first integral method and functional variable method

The exact solutions of conformable time-fractional modified nonlinear Schrödinger equation by first integral method and functional variable method

Boundary Mittag-Leffler stabilization of fractional reaction–diffusion cellular neural networks

Mittag-Leffler stabilization is studied for fractional reaction–diffusion cellular neural networks (FRDCNNs) in this paper. Different from previous literature, the FRDCNNs in this paper are high-dimensional systems, and boundary control and observed-based boundary control are both used to make FRDCNNs achieve Mittag-Leffler stability. First, a state-dependent boundary controller is designed when system states are available. By employing the spatial integral functional method and some inequalities, a criterion ensuring Mittag-Leffler stability of FRDCNNs is presented. Then, when the information of system states is not fully accessible, an observer is presented to estimate the system states based on boundary output and an observer-based boundary controller is provided aiming to stabilize the considered FRDCNNs. Furthermore, a robust observer-based boundary controller is proposed to ensure the Mittag-Leffler stability for FRDCNNs with uncertainties. Examples are given to illustrate the effectiveness of obtained theoretical results.

Intermittent boundary stabilization of stochastic reaction–diffusion Cohen–Grossberg neural networks

Cohen–Grossberg neural networks (CGNNs) play an important role in many applications and the stabilization of this system has been well studied. This study considers the exponential stabilization for stochastic reaction–diffusion Cohen–Grossberg neural networks (SRDCGNNs) by means of an aperiodically intermittent boundary control. Both SRDCGNNs without and with time-delays are discussed. By employing the spatial integral functional method and Poincare’s inequality, criteria are derived to ensure the controlled systems achieve mean square exponential stabilization. Based on these criteria, the effects of diffusion item, control gains, the minimum control proportion and time-delays on exponential stability are analyzed. Examples are given to illustrate the effectiveness of the obtained theoretical results.

Features of the Nonlinear Harmonic Distortion in AOSTFTs

In this article, we present the analysis of the nonlinear harmonic distortion (HD) characteristics for two kinds of amorphous oxide semiconductor thin film transistors (AOSTFTs) based on SiO2/indium–gallium–zinc oxide (IGZO) and HfO2/hafnium–indium–zinc oxide (HIZO), applying the integral function method (IFM). An analysis of the distortion of the transistors used as quasi-linear resistor and the variation in HD as a function of the parameter ${g}_{m}/{I}_{\text {DS}}$ are also included. As far as we know, this kind of study has not been reported before. It is shown that parameter HD2 does not present a minimum as in MOSFETs, because the transfer characteristic does not present an inflection point. The value of HD2 is higher than that of HD3 in the whole operating ranges. When the transistor is used as quasi-linear resistor for different ${V}_{G}$ bias, HD2 is greater than HD3 in more than 50 dB. Because HD2 is significantly greater than HD3, the total HD (THD) parameter can be considered practically equal to HD2. In addition, the behavior of HD around the low-power operating region is shown.

Studying additional measurement errors from control tools using an integral functional method

Our research has established that under industrial conditions the correction to the result of current measurements when an influencing parameter deviates from the rated value is rarely introduced. In a general case, the procedure for determining an additional measurement error implies that the measured values for an influencing parameter are applied to determine the degree of its deviation while a correction to the current measurement result is calculated as the product of this degree by its rated value.In a general case, a procedure for determining an additional measurement error includes two stages. At the first stage, the measured values for an influencing parameter are used to determine the degree of its deviation from the rated value. At the second stage, correction is calculated as the product of this degree by the rated value for an additional error.Such a technique to calculate a correction is time consuming and insufficiently precise, as it does not take into consideration the non-linear dependence of the additional error on a change in the influencing parameter, as well as the current value for the output signal of control tool. To determine the actual value for an influencing parameter and the additional measurement error under industrial operation of control tools, an integral functional method has been proposed. The method implies determining the difference of areas under the nominal and actual acreage static characteristics, limited to a range of measurement. The difference of areas is a function of the output signal of a control tool, a measured parameter and a change in the influencing parameter. It has been shown that the proposed method makes it possible to calculate the actual values for a technological parameter based on its measured and influencing parameters only. We have established regularities between the actual value for a measured parameter, the current value for the output signal from a control tool, and the measured value for an influencing parameter. The proposed method is important and valuable in the operation of computer-integrated control systems of technological parameters, as it makes it possible to determine the actual values for a measured parameter based on relevant algorithms without calculating corrections.

Comparative analysis of risks which are accompanied by the use of typical and boundary gases concentrations for the diagnostics of high voltage transformers

The aim of the scientific research provided in the article is to increase the operational reliability of high-voltage power transformers by reducing the possible risks when diagnosing high-voltage equipment based on the analysis of gases dissolved in oil. We described the method for determining the boundary (typical) gas concentrations by the integral function method, which is recommended by some existing standards, and the author’s method for determining the boundary concentrations of gases ensuring a minimum of possible economic damage in case of taking erroneous decisions. The analysis of boundary concentrations of gases obtained by the method of integral functions and the method of minimum risk showed that boundary values differ significantly for the same data, depending on the method of determination. To determine the reliability of decision-making we used a comparative analysis of risk values that may arise while making a diagnosis of high-voltage transformers based on the analysis of gases dissolved in oil, the boundary values of gas concentrations obtained by the integral function method and the minimum risk method, as well as the boundary values of gas concentrations regulated by known international and national standards were used. The study has revealed that the use of typical values of gas concentrations obtained by integrated distribution functions is accompanied by one of the highest risk values. The lowest risk value is provided by the boundary concentrations obtained by the minimum risk method. The method proposed for determining the boundary values of gas concentrations, taking into account the influence of the most relevant factors, allows significantly lower the values of possible risks and consequently can increase the operational reliability of high-voltage transformers, especially those that are used outside the normative service life.

Research of Errors in Measuring Instruments with Fixed Ends of Measurement Range by Integral Functional Method

The article presents the research results of additional errors in measuring instruments caused by the change of normalized influential parameters. The analysis of modern methods of additional measurement errors determination is performed, and their disadvantages are shown. A new method for research and determination of additional errors is proposed, which is based on Euler’s optimality integral functional. Applicability of such measurement errors research by the integral functional method is substantiated, the essence of which is to determine the difference of planes with nominal and current static characteristics of the measuring instrument with further definition of the integral functional and measurement errors. The research results of additional measurement errors are presented for the case when the static characteristic of the measuring instrument is linear and fixed at the initial input signal. It is shown that for measuring instruments with the linear static characteristic the change of the influence parameter does not change the characteristic linearity, but only leads to nonlinearity of the additional measurement error with increase in deviation of the influence parameter from its normalized value. The mathematical models of additional measurement errors and their graphical distribution along the measurement range are presented.

Resonance enhancement of Faraday rotation in double-periodic gyromagnetic layers analyzed by the method of integral functionals

Extraordinary enhancement of the Faraday rotation (over 1000 times) for a perfect electric conductor (PEC)-like layer with square-shaped apertures filled with yttrium iron garnet (YIG) and its smaller enhancement (17 times) for a YIG layer perforated with square-shaped air apertures are revealed at the grating-mode resonance frequencies. At the same time, the periodic perforation of a PEC-like layer with C-shaped apertures filled with YIG does not influence the Faraday effect. For our theoretical study, we develop the full-wave solution for the three-dimensional (3D) problem of the plane wave scattering from a double-periodic gyromagnetic layer obtained by the method of integral functionals. The method is based on the 3D volume integral equations for the electric and magnetic polarization currents on the periodic layer.

Harmonic distortion analysis of triple gate SOI nanowire MOSFETS down to 100 K

The linearity of triple gate nanowire transistors (NWs) implemented on a Silicon-On-Insulator (SOI) substrate is investigated in this work considering temperature (T) influence. The analysis is performed in long channel nanowire MOSFETs with different fin width (WFIN), from quasi-planar structures (WFIN=10μm) to narrow devices (9.5nm), operating as single-transistor amplifiers from room temperature down to 100K. The total, second and third order harmonic distortions (THD, HD2 and HD3, respectively) are extracted using the Integral Function Method (IFM). The analysis is divided in two parts. First, a fixed input signal is applied at the gate of the single-transistor amplifiers and, then, the output signal is fixed. Transport parameters such as effective mobility (μeff), mobility degradation coefficient (θ) and series resistance (RS) have been extracted down to 100K and correlated to the distortion to explain linearity peaks behavior with temperature and fin width. Narrow transistors have shown improved linearity mainly due to higher intrinsic voltage gain (AV) considering the entire temperature range. Low temperature operation has shown to degrade the linearity characteristics of both wide and narrow NW MOSFETs.

An approach for solving an inverse spherical two‐phase Stefan problem arising in modeling of electric contact phenomena

In this paper, a model problem that can be used for mathematical modeling and investigation of arc phenomena in electrical contacts is considered. An analytical approach for the solution of a two‐phase inverse spherical Stefan problem where along with unknown temperature functions heat flux function has to be determined is presented. The suggested solution method is obtained from a new form of integral error function and its properties that are represented in the form of series whose coefficients have to be determined. Using integral error function and collocation method, the solution of a test problem is obtained in exact form and approximately.

An analytical method for the solution of two phase Stefan problem in cylindrical geometry

Two phase Stefan problem was solved using analytical method in cylindrical domain. To solve governing equations Eigen conditions were formulated by using separation of variable technique. Eigenvalues of the eigencondition were obtained by applying corresponding boundary conditions for liquid and solid phase. Eigenvalues are graphically validated by using window size method in Mathematica. It is noted radial eigenvalues are free from imaginary values. Interface equation obtained from this method were solved and analyzed by varying the Stefan number and introducing the forced and natural convection. Conduction and convection heat transfer mechanism was studied and results obtained by varying thermal diffusivity, thermal conductivity and Stefan number were discussed. Natural convection effects were studied by introducing Rayleigh number and results showed Stefan number has significant effect than Rayleigh number during Phase transition process. Furthermore, eigen function expansion Method was compared with exact solution of Exponential Integral function method and results showed good agreement for Q = 1.

Novel Stability Criteria for Impulsive Memristive Neural Networks with Time-Varying Delays

In order to improve the ability of resisting disturbance for memristive neural networks (MNNs), a general impulsive controlled MNN with variable delays is constructed in this paper. Then, its dynamical behaviors are investigated based on the differential inclusion theories. By constructing suitable Lyapunov---Krasovskii-type functional and combining with integral and monotone function method, the global exponential stability criteria of delayed memristive neural networks (DMNNs) with impulse effects are derived. Furthermore, the impulsive controlled DMNNs can be transformed to DMNNs without impulsive effects, which promote and enrich the theoretical research results of MNNs. Finally, the effectiveness of obtained results is illustrated by two numerical examples.

Shape and cross-section optimization of plane trusses subjected to earthquake excitation using gradient and Hessian matrix calculations

ABSTRACTThis article describes a second-order shape and cross-section optimization method of plane truss subjected to earthquake excitation. The method is based on gradient and Hessian matrix calculation. First, the first and second derivatives of dynamic response with respect to design variables are calculated based on the Newmark method. Second, the inequality time-dependent constraint problem is converted into a sequence of appropriately formed unconstrained problems using the integral interior penalty function method. Then, the gradient and Hessian matrix of the integral interior penalty function are computed. Third, Marquardt's method is employed to solve the unconstrained problems. Finally, the new approach is validated through several case studies. The results show that the new optimization method is an efficient and effective approach for minimum weight design of truss structures.

The Testimony and Application of Inequality Cauchy-Schwarz

This paper discusses the form of inequality Cauchy-Schwarz in the field of calculus, giving the different methods of proof, which includes Parameter method, variable upper limit integral function method, the double integral and definite integral definition method. What’s more, through a series of examples, it also reflects that the inequality Cauchy-Schwarz can make the process of proving the inequality more simple while certifying the Integral inequality.

Optical solitons with spatio-temporal dispersion by first integral approach and functional variable method

This paper applies the first integral method and functional variable technique in order to obtain optical solitons from the governing nonlinear Schrödinger's equation with spatio-temporal dispersion. There are four types of nonlinear media that are taken into account. These are Kerr law, power law, parabolic law as well as the dual-power law nonlinearity. Several constraint conditions naturally emerge from the results obtained and these conditions are also listed.