Projection Equations Research Articles

Quiescent Optical Solitons for the Concatenation Model Having Nonlinear Chromatic Dispersion with Differential Group Delay

This study recovers quiescent optical solitons within a complex concatenation model featuring nonlinear chromatic dispersion and differential group delay. It employs the sine-Gordon equation approach and the projective Riccati equation scheme. The results include the successful recovery of soliton solutions and the identification of parameter restrictions for their existence. The novelty lies in the application of these methods to this specific problem, offering valuable insights for optical communication system design.

Deformation of High Rise Cooling Tower through Projection of Coordinates Resulted from Terrestrial Laser Scanner Observations onto a Vertical Plane

The appearance and development of new construction materials, technology and accurate geodetic instruments have led to the necessity of their inevitable use in the health monitoring, maintenance, and restoration of civil structures and special structures such as high-rise cooling towers. This paper presents an accurate practical and analytical method for determining the structural deformation and axis inclination of high rise cooling towers using terrestrial laser scanner (TLS) observations through the projection of tower surface points coordinates on a vertical plane. Four cooling towers in El-Mahla El-Kubra city, Egypt are observed and studied. Two of them with height 56 m, and the others were 36 m height. The studied four towers have different cross-section diameters along the tower height. Each tower has a cone shape with a curved wall. The equation of the tower wall is determined using TLS observations and least squares adjustment techniques. The equations of cone projection with a curved wall are derived and presented in this paper. From TLS observations, the radii and accuracy of each 2 m tower height are determined with center coordinates, and then the inclination of the tower axis is calculated and drawn. From the results of TLS observations, data processing, and analysis using the suggested techniques, it is deduced that there is a deformation in tower walls with small values. The specified technique for observations collection and TLS data analysis through projection on a vertical plane is significant and valuable for determining the structural deformation of circular high rise buildings and towers. From the results, there are obvious deformation values in some cooling towers, so maintenance work must be included. The towers also must be checked and monitored several times at brief intervals.

Exact traveling wave solutions and soliton solutions of conformable M-fractional modified nonlinear Schrödinger model

The traveling wave solutions and soliton solutions of the conformable M-fractional modified nonlinear Schrödinger equation have been devised in this work by employing the generalized projective Riccati equations technique. The proposed method is straightforward and concise to obtain exact soliton solutions of nonlinear partial differential equations in various scientific discipline. This approach has been applied successfully to a wide variety of physical and mathematical problems. In many cases, the method has been able to provide new insights into the behavior of these systems and has led to the discovery of new phenomena. Several types of soliton solutions, including hyperbolic traveling wave solution, kink-shape soliton solution, singular soliton solution and dark soliton-type traveling wave solution, have been derived from the governing equation by using this effective technique. The solutions achieved by the governing model contain significant and dynamic justifications for certain real-world physical phenomena. Some acquired soliton solutions are displayed in 3D plots, corresponding contour plots and density plots to illustrate the physical dynamics of the governing model. Furthermore, the resulting outcomes manifest that the proposed method is highly efficient for revealing wave solutions for a wide range of nonlinear partial differential equations that emerge in optics.

Identifying and updating local optimization methods in extended Kalman filter SLAM

he essence of simultaneous location and mapping (SLAM) is an estimation problem. It requires mobile robots to use the sensor information to estimate the structure of the external environment in real-time and their position in the background. Smith and Cheeseman first applied the EKF estimation method to slam in the last century. The SLAM algorithm based on an extended Kalman filter (EKF-SLAM for short) is one of the research hotspots of slam technology. The EKF is improved based on the Kalman filter to make the Kalman filter suitable for nonlinear dynamic systems to solve the estimation problem of the nonlinear system model. Therefore, the Kalman filter can be applied to slam technology. The slam model based on filtering mainly includes three equations: Firstly, is to use the equation of motion, which will increase the uncertainty of robot positioning. Then the equation for initializing the guidepost according to the observation initializes the new state quantity according to the observation information. Lastly is the projection equation of the landmark state to comment, update and correct the condition according to the observation information to reduce the uncertainty. In the implementation, there exists a problem regarding storing the obstacle location. The current standard practice is to abandon all the existing data when the location number reaches the threshold. This will cause calculation time waste and instant high error [1]. In this paper, several improvement methods try to fully utilize the received message by different algorithms built on the ROS system. These include abandoning local data logic, updating data instead of deleting data, and retaining user data when updating new data. These advanced methods designed for EKF-SLAM will provide a more precise outcome with a low error range.

Painlevé equations, integrable systems and the stabilizer set of Virasoro orbit

We study a geometrical formulation of the nonlinear second-order Riccati equation (SORE) in terms of the projective vector field equation on [Formula: see text], which in turn is related to the stability algebra of Virasoro orbit. Using Darboux integrability method, we obtain the first integral of the SORE and the results are applied to the study of its Lagrangian and Hamiltonian descriptions. Using these results, we show the existence of a Lagrangian description for SORE, and the Painlevé II equation is analyzed.

An enormous diversity of soliton solutions to the (2+1)-dimensional extended shallow water wave equation using three analytical methods

In this paper, we obtain a variety of analytical wave solutions of a ([Formula: see text])-dimensional extended shallow water wave equation. The applications of the governing equation are enormous in ocean modeling and investigation of moist-convection properties in atmospheric dynamics. Two powerful mathematical approaches, the [Formula: see text]-expansion method (EEM) and the generalized projective Riccati equations method (GPREM), were used to find closed-form traveling wave solutions. In addition, the breather wave solutions were described using the Cole-Hopf transform, derived with the help of Hirota bilinear method (HBM). Eventually, we obtain 46 closed-form solutions in explicit form. The general form of obtained solutions include trigonometric, rational, exponential and hyperbolic function solutions. To illustrate the physical significance of some novel results, we provided contour, two- and three-dimensional graphics under the suitable free choices of unknown parameters. The dynamical shapes of these solutions are one soliton, multiple solitons, kink, anti-kink, lump and periodic solutions. We believe that the applied methods of this work are well-organized, genuine and powerful mathematical tools for solving the nonlinear evolution equations (NLEEs) occurring in the study of ocean science and engineering.

Three-Dimensional NLOS VLP Based on a Luminance Distribution Model for Image Sensor

Visible light positioning (VLP) is playing a critical role in delivering better location-based services. However, traditional VLP systems rely on line-of-sight (LOS) paths and have a requirement for large numbers of light-emitting diodes (LEDs) and sensors, making them unprepared for various scenarios. This paper proposes a three-dimensional (3D) non-line-of-sight (NLOS) VLP system using a single LED and an image sensor (IS) to address the problem of obstructed LOS paths. On the one hand, an optical camera communication (OCC) subsystem is designed to receive the transmitter’s position based on NLOS links. In contrast, two highlights are visible in the picture when the image sensor captures the reflected light from the floor. Using the proposed luminance distribution model (LDM), it can be demonstrated that the two highlights can be regarded as the projections formed by two virtual LEDs. Based on the projection equations of the two virtual LEDs, an image sensing algorithm can estimate the receiver’s position. Experimental results show that the proposed system can achieve a 90th percentile accuracy of < 22 cm for 3D positioning. To the best of author’s knowledge, this is the first work to demonstrate 3D NLOS VLP using just a single LED and an IS. Additionally, the proposed LDM is the first physically based model for the first reflected light to estimate the channel gain of a NLOS link.

Wyznaczanie parametrów kinematycznych agregatu sprężarkowego w zastosowaniach z LPG

The collection and transportation of low-pressure gas released into the atmosphere from the oil fields is important in increasing gas production. This issue, which differs in its actuality, has both economic and pollution preveniton benefits. One such project, led by the UN Development Fund, was successfully implemented in the city of Siyazan, Azerbaijan. An electrically driven compressor unit of NQK-7/1-5 type with two oscillating cylinders is commonly used for such work. On visual inspection of the compressors undergoing repair, it was found that the seals in the pneumatic part and the elements of the cylinder-piston group are completely out of order, the gear teeth in the transmission part are broken, worn, the bearings are destroyed, the belts are torn, etc. Compressors sent for repair are almost always reassembled. A structural-kinematic analysis of the compressor installation was carried out to find out the causes of the existing malfunctions. For this purpose, a kinematic diagram of the conversion mechanism of the compressor – a slider-crank linkage mechanism was drawn up using the method of closed contours from projection equations relative to the X and Y axes, and based on the condition that the rotation angles of the piston and cylinder axes are equal, the instantaneous rotation angle, angular velocity and angular acceleration of the cylinder rotation, as well as instantaneous values of displacement, speed and acceleration of the piston inside the cylinder were found. It was shown that the designer's decision on the selection of the design scheme and determination of the product dimensions, which are one of the main issues in the design of the mechanism, should be made based on the results of the kinematic and dynamic analysis.

An Improved Dynamic Programming-Based Decentralized Algorithm for AC-OPF of Radially Connected Multi-Level Networks

In this paper, an improved dynamic programming (IDP)-based decentralized algorithm is proposed for the AC-optimal power flow (AC-OPF) of radially connected transmission-distribution-microgrid networks (RCTDMNs). Most of the existing decentralized algorithms need to iterate among networks, which may lead to additional computational and information exchange burden. Different from these algorithms, we decompose the original AC-OPF problem into small-scale subproblems based on dynamic programming (DP) theory, and then solve them in a decentralized manner without iteration among networks. In order to improve the computational performance of DP, its value functions are directly obtained by solving Galerkin projection equations (GPEs) instead of traversing the whole state space. Moreover, a parallel technique is proposed to further accelerate the proposed IDP algorithm when dealing with the AC-OPF problem of RCTDMNs. Case studies on various test systems demonstrate the effectiveness of the proposed method.

Study of the Energy Dissipation over the Type-A Piano Key Weir

This study assesses the effects of geometrical variations, i.e., relative width ratio, magnification ratio, and cyclic variation, on the Piano Key Weir energy dissipation performance. In order to assess this, 12 type-A PKW models were tested by conducting an experimental investigation. The analysisincludes results from 280 tests and comprehensive findings of the fluid domain. The results demonstrated that the configuration of the PKWs has a greater influence on energy dissipation. The present study's findings show that the type-A PKW dissipates energy more efficiently than nonlinear or standard weirs. The energy loss over the PKW decreases as the magnification and relative width ratios increase, although the energy dissipation increases with key cycle numbers. Furthermore, the sensitivity analysis of the weir was examined by making steps at outlet key floors and found that the energy dissipation has been increased by 6.67% at low heads while reduced by 1.59% at higher heads. In addition, the author proposed two empirical projection equations for the optimal relative residual energy configuration downstream of the PKW. The proposed equations are the function of headwater ratio, relative width ratio, and magnification ratio. The results of this investigation are in perfect correlation with earlier studies.

Traveling wave behavior of new (2+1)-dimensional combined KdV–mKdV equation

The KdV and mKdV equations have many significant applications in the field of plasma physics and ocean dynamics. In this work, the traveling wave behavior of the newly proposed (2+1)-dimensional combined KdV–mKdV equation has been examined using two direct analytical approaches. The two variable G′G,1G-expansion method and generalized projective Riccati equation method are used to determine a variety of solitons and solitary waves for the governing model. The graphical observations are made for suitable choice of parametric values. The obtained results will provide useful insight in further explorations of the wave phenomena depicted by the considered equation.

M-truncated soliton solutions of the fractional (4+1)-dimensional Fokas equation

This article aims to examine M-truncated soliton solutions of the fractional (4+1)-dimensional Fokas equation (FE), which is a generalization of the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations. The fractional (4+1)$-dimensional Fokas equation with the M-truncated derivatives is also studied first time in this study. The generalized projective Riccati equations method (GPREM) is successfully implemented. In the application of the presented method, a suitable fractional wave transformation is chosen to convert the proposed model into a nonlinear ordinary differential equation. Then, a linear equation system is acquired utilizing the GPREM, the system is solved, and the suitable solution sets are obtained. Dark and singular soliton solutions are successfully derived. Under the selection of appropriate values of the parameters, 2D, 3D, and contour plots are also displayed for some solutions.

Comment on “Optical solitons for Radhakrishnan–Kundu–Lakshmanan equation in the presence of perturbation term and having Kerr law”

We analyzed a recent paper (Ozdemir, 2022). The author used three approaches to construct solutions of Radhakrishnan–Kundu–Lakshmanan equation — the generalized projective Riccati equations method, the simple version of the new extended auxiliary equation method, and unified Riccati equation expansion method. We demonstrate that the first approach used by the author is erroneous. The second approach, which is, in fact, a modification of the Kudryashov method was used in an inefficient version with unnecessary restrictions. The third approach leads to several solutions that are equivalent to one. It is shown that the solutions presented in paper are wrong. The correct solutions of the RKL equation are presented.

Analytical solutions for nonlinear systems using Nucci's reduction approach and generalized projective Riccati equations

&lt;abstract&gt;&lt;p&gt;In this study, the Nucci's reduction approach and the method of generalized projective Riccati equations (GPREs) were utilized to derive novel analytical solutions for the (1+1)-dimensional classical Boussinesq equations, the generalized reaction Duffing model, and the nonlinear Pochhammer-Chree equation. The nonlinear systems mentioned earlier have been solved using analytical methods, which impose certain limitations on the interaction parameters and the coefficients of the guess solutions. However, in the case of the double sub-equation guess solution, analytic solutions were allowed. The soliton solutions that were obtained through this method display real positive values for the wave phase transformation, which is a novel result in the application of the generalized projective Riccati method. In previous applications of this method, the real positive properties of the solutions were not thoroughly investigated.&lt;/p&gt;&lt;/abstract&gt;

Sensitive analysis of soliton solutions of nonlinear Landau-Ginzburg-Higgs equation with generalized projective Riccati method

&lt;abstract&gt;&lt;p&gt;The study aims to explore the nonlinear Landau-Ginzburg-Higgs equation, which describes nonlinear waves with long-range and weak scattering interactions between tropical tropospheres and mid-latitude, as well as the exchange of mid-latitude Rossby and equatorial waves. We use the recently enhanced rising procedure to extract the important, applicable and further general solitary wave solutions to the formerly stated nonlinear wave model via the complex travelling wave transformation. Exact travelling wave solutions obtained include a singular wave, a periodic wave, bright, dark and kink-type wave peakon solutions using the generalized projective Riccati equation. The obtained findings are represented as trigonometric and hyperbolic functions. Graphical comparisons are provided for Landau-Ginzburg-Higgs equation model solutions, which are presented diagrammatically by adjusting the values of the embedded parameters in the Wolfram Mathematica program. The propagating behaviours of the obtained results display in 3-D, 2-D and contour visualization to investigate the impact of different involved parameters. The velocity of soliton has a stimulating effect on getting the desired aspects according to requirement. The sensitivity analysis is demonstrated for the designed dynamical structural system's wave profiles, where the soliton wave velocity and wave number parameters regulate the water wave singularity. This study shows that the method utilized is effective and may be used to find appropriate closed-form solitary solitons to a variety of nonlinear evolution equations (NLEEs).&lt;/p&gt;&lt;/abstract&gt;

Obtaining the soliton solutions of local M-fractional magneto-electro-elastic media

In this research paper, the generalized projective Riccati equations method (GPREM) is applied successfully to procure the soliton solutions of the local M-fractional longitudinal wave equation (LWE) arising in mathematical physics with dispersion caused by the transverse Poisson's effect in a magneto-electro-elastic circular rod (MEECR). Applying a wave transformation to the local M-fractional LWE, the equation can be turned into a set of algebraic equations. Solving the algebraic equation system, we procure the soliton solutions of the local M-fractional LWE. Both the obtained solution functions in the study and the graphical simulations depicted for these functions. It will assist researchers working in this field in the physical interpretation of this equation. Moreover, the reported solutions propose a rich platform to examine the local M-fractional LWE.

Enhancing predictive precision of dominant height projection equations for eucalypts by incorporating rainfall and temperature terms

Short-rotation forest stands are sensitive to extreme climate conditions during their growth period, which presents a challenge to managing forests and modelling forest growth in a constantly changing climate. We developed climate-sensitive dominant height models for the Eucalyptus grandis × Eucalyptus urophylla hybrid (GU) in South Africa. In addition, dominant height growth under three future climate scenarios was investigated. The Chapman– Richards and Gompertz models, modified by within-rotation and long-term climate data, were used to model dominant height. Model testing using independent permanent sample plot data showed that the Gompertz model modified by within-rotation bioclimatic data performed better than the other models. The climate-modified Gompertz model was used to project height growth for eucalypt stands under three future climate scenarios; ‘No change’, ‘RCP4.5’ and ‘RCP8.5’, for two periods: 2050 (years 2040 to 2060); and 2070 (years 2061 to 2080). Climate change might decelerate dominant height growth in the study area, therefore forest management plans need to be adapted accordingly.

Optical solitons of the perturbation Fokas–Lenells equation by two different integration procedures

In this study, the general projective Riccati equations and the enhanced Kudryashov’s methods are presented for the optical solitons of the perturbation Fokas–Lenells (FL) equation. For this aim, the perturbation Fokas–Lenells equation is converted to a nonlinear ordinary differential equation via wave transformations. Then, the balance number is calculated by the homogeneous balance method. Then, the used methods are applied to the given model. Finally, three-dimensional, contour, and two-dimensional figures of the results are presented. According to the obtained results, both methods are effective and powerful. The obtained results are significant in mathematical physics, soliton theory, and many other areas.

Three‐dimensional point cloud reconstruction of inverse synthetic aperture radar image sequences based on back projection and iterative closest point fusion

AbstractIn order to recover the three‐dimensional (3D) structure of the target from sequential inverse synthetic aperture radar (ISAR) images, the factorisation method is generally used. It requires a large number of high‐quality matched feature points from different ISAR images. If the number of extracted feature points is insufficient, the restored 3D structure is not obvious. Furthermore, the mismatching of feature points will greatly affect the quality of target reconstruction. However, the factorisation method only uses the information from the ISAR images, while that from imaging geometry is not sufficiently considered. ISAR imaging is a kind of projection, and the projection plane information could be taken into account for the 3D reconstruction. Hence, a new 3D reconstruction method for stable targets from sequential ISAR images is proposed in this paper. Firstly, the ISAR images are preprocessed with the CLEAN algorithm. The maximum between‐cluster variance method is applied to extract feature points from the processed images. Moreover, the image projection plane and projection equation corresponding to different ISAR images are analysed by using imaging geometry information. According to the projection equation, the feature points are back‐projected (BP) to the 3D space. Finally, the 3D point clouds obtained by the BP from multiple radar images are fused by the iterative closest point algorithm to restore the 3D structure of the target. The simulation and experiment results show the effectiveness and robustness of this method.

Bright, dark, periodic and kink solitary wave solutions of evolutionary Zoomeron equation

The modified auxiliary equation (MAE) approach and the generalized projective Riccati equation (GPRE) method are used for the first time to solve Zoomeron problem which provided different types of exact traveling wave solutions, including some new dynamical behaviors. The governing model covers unique examples of solitons with distinct properties that emerge in a variety of physical situations, including laser physics, fluid dynamics and nonlinear optics. The achieved exact traveling wave solutions include solitary wave, periodic wave, bright, dark peakon, and kink-type wave solutions. Earned results are given as hyperbolic and trigonometric functions. Moreover, the dynamical features of obtained results are demonstrated through interesting plots.