Hyperbolic Function Research Articles

Fixed-time event-triggered sliding mode time-varying formation control for multiple surface vessels with fast convergence

ABSTRACT This paper proposes a new fixed-time sliding mode time-varying formation control strategy for multiple surface vessels (MSVs) to improve control performance and ensure navigation safety. First, a fixed-time extended state observer (FxESO) is designed to provide estimations of velocities and lumped disturbances. Then, the event-triggered fixed-time distributed time-varying formation control law is designed based on prescribed performance function (PPF) and fixed-time nonsingular terminal sliding mode (FxNTSM) manifold. Finally, system stability is analyzed based on Lyapunov theory. The salient features of the proposed scheme are as follows. First, a novel hyperbolic cosecant PPF is incorporated into the control scheme to enhance control performance. Then, the proposed scheme is combined with relative threshold event-triggered strategy to reduce unnecessary updates of actuators. Finally, a uniform exact differentiator is introduced to avoid the formation control law including the acceleration of adjacent vessel. Comparative results are given to demonstrate the superiority of the proposed scheme.

Activation Functions for Neural Networks: Application and Performance-based Comparison

Past decade has seen explosive growth of Deep Learning (DL) algorithms based on Artificial Neural Networks (ANNs) and its applications in vast emerging domains to solve real world complex problems. The DL architecture uses Activation Functions (AFs), to perform the task of finding relationship between the input feature and the output. Essential building blocks of any ANN are AFs which bring the required non-linearity of the output in the Output layer of network. Layers of ANNs are combinations of linear and nonlinear AFs. Most extensively used AFs are Sigmoid, Hyperbolic Tangent (Tanh), Rectified Linear Unit (ReLU) etc to name a few. Choosing an AF for a particular AF depends on various factors such as Nature of Application, Design of ANN, Optimizers used in the network, Complexity of Data etc. This paper presents a survey on most widely used AFs along with the important consideration while selecting an AF on a specific problem domain. A broad guideline on selecting an AF based on the literature survey has been presented to help researchers in employing suitable AF in their problem domain.

A new version of trial equation method for a complex nonlinear system arising in optical fibers

In this study, the dissipation problem of nonlinear pulse in mono mode optical fibers which is governed by the Fokas system (FS) is considered. The solutions of this system have an important role in comprehending the different wave structures in physical settings. Therefore, a new version of the trial equation method (NVTEM) is employed to present the new exact wave solutions of the FS. The advantage of the NVTEM is to use different root possibilities of a polynomial which shape the solutions of the related model. Primarily this system is converted to a nonlinear ordinary differential equation (NODE) via the traveling wave transform to apply the proposed method. Various exact wave solutions to the FS are obtained such as rational function, exponential function, hyperbolic function, and Jacobi elliptic function solutions. Thus, we have revealed solutions featly which are unlike the wave solutions previously found by other analytical methods. The present results depict the formation and development of such waves and their interactions. The exhibition of the solutions is given by 3D plots together with the corresponding 2D plots. The outcomes have shown that the proposed technique is abundant in achieving different wave solutions of many nonlinear differential equations in the field of optics.

Exact solutions of the Landau–Ginzburg–Higgs equation utilizing the Jacobi elliptic functions

The Landau–Ginzburg–Higgs equation is one of the significant evolution equation in physical phenomena. In this work, the exact solutions of this equation are gained by applying an analytical method depends on twelve Jacobi elliptic functions. This equation is turned into an ordinary differential equation by the proposed method. When solving the Landau–Ginzburg–Higgs equation, an auxiliary ordinary differential equation is considered. Some theorems and corollaries utilized in the solutions of this auxiliary equation are given. Using these solutions, the elliptic and elementary solutions of the Landau–Ginzburg–Higgs equation are obtained and illustrated by tables. Many solutions are given in the form of the complex, rational, hyperbolic, and trigonometric functions. The soliton solutions and the complex valued solutions are also found by proposed method. These solutions include the largest set of solutions in the literature. Some of them are shown graphically by 2-dimensional and 3-dimensional with the help of Mathematica software. The obtained solutions are beneficial for the farther development of a concerned model. The presented method does not need initial and boundary conditions, perturbation, or linearization. Besides, this method is easy, efficient, and reliable for solutions of many partial differential equations.

Analytical study of soliton dynamics in the realm of fractional extended shallow water wave equations

Abstract In this study, we use the Khater Method (KM) as an efficient analytical tool to solve (3+1)-dimensional fractional extended shallow water wave equations (FESWWEs) with conformable derivatives. The KM transforms fractional partial differential equations to ordinary differential equations (ODEs) via strategic variable transformation. Then, series-form solutions to these ODEs are proposed, which turn them into nonlinear algebraic systems. The solution to this set of algebraic equations yields shock travelling wave solutions expressed in hyperbolic, trigonometric, exponential, and rational functions. The study's findings are corroborated by 2D, 3D, and contour graphs that show the changing patterns of the detected shock travelling waves. These findings have important significance for the discipline, offering vital insights into the intricate dynamics of FESWWEs. The effectiveness of KM is demonstrated by its capacity to produce varied solutions and contribute to a thorough knowledge of such complex phenomena.

New exact optical soliton solutions of the derivative nonlinear Schrödinger equation family

In this study, we use a systematic approach named the generalized unified method (GUM) to construct the general exact solutions of the derivative nonlinear Schrödinger (DNLS) family that also includes perturbed terms, which are the Kaup–Newell equation, the Chen–Lee–Liu equation, and the Gerdjikov–Ivanov equation. The GUM provides more general exact solutions with free parameters for nonlinear partial differential equations such that some solutions obtained by different exact solution methods, including the hyperbolic function solutions, the trigonometric function solutions, and the exponential solutions, are derived from these solutions by giving special values to these free parameters. Additionally, the used method reduces a large number of calculations compared to other exact solution methods, enabling computations to be made in a short, effortless, and elegant way. We investigate the DNLS family in this work because of its extensive applications in nonlinear optics. Particularly, the obtained optical soliton solutions of the DNLS family are useful for describing waves in optics and facilitating the interpretation of the propagation of solitons through optical fibers. Furthermore, this work not only contributes significantly to the advancement of soliton dynamics and their applications in photonic systems but also be productively used for more equations that occur in mathematical physics and engineering problems. Finally, 2D and 3D graphs of some derived solutions are plotted to illustrate behaviors of optical soliton.

Exact solutions for travelling waves using Tanh method for two dimensional stochastic Allen–Cahn equation with multiplicative noise

AbstractThe research outlined here focuses on the hyperbolic tangent (tanh) method for deriving analytical solutions for travelling wave to the two-dimensional stochastic Allen–Cahn equation with multiplicative noise. The novelty of our work is to derive these exact solutions since the previous studies focused on the solutions of numerical nature. The tanh approach, which employs a finite tanh power series, is particularly adept at modeling travelling wave profiles. A key area of interest in this study is the effect of multiplicative noise on these travelling waves dynamics, especially how high levels of noise can lead to propagation failure of waves. The results demonstrate that for weak noise, the propagation of the travelling wave is basically unaffected, while the wave fails to propagate in the strong noise regime.

Investigating wave solutions and impact of nonlinearity: Comprehensive study of the KP-BBM model with bifurcation analysis.

In this paper, we investigate the (2+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona Mahony equation using two effective methods: the unified scheme and the advanced auxiliary equation scheme, aiming to derive precise wave solutions. These solutions are expressed as combinations of trigonometric, rational, hyperbolic, and exponential functions. Visual representations, including three-dimensional (3D) and two-dimensional (2D) combined charts, are provided for some of these solutions. The influence of the nonlinear parameter p on the wave type is thoroughly examined through diverse figures, illustrating the profound impact of nonlinearity. Additionally, we briefly investigate the Hamiltonian function and the stability of the model using a planar dynamical system approach. This involves examining trajectories, isoclines, and nullclines to illustrate stable solution paths for the wave variables. Numerical results demonstrate that these methods are reliable, straightforward, and potent tools for analyzing various nonlinear evolution equations found in physics, applied mathematics, and engineering.

Symmetry-based closed-form solutions and conserved quantities of the new (2+1)-dimesnional Bogoyavlensky-Konopelchenko equation in fluid mechanics

In this investigation, we explore the mathematical intricacies of the novel (2+1)-dimensional Bogoyavlensky-Konopelchenko equation, a model with practical applications in elucidating the dynamics of internal waves within deep water. The equation’s significance spans various scientific domains, including plasma physics, nonlinear optics, and fluid dynamics. Employing a comprehensive analytical approach, specifically Lie symmetry analysis, we aim to unravel the underlying complexities of this equation and obtain new analytical solutions. To further scrutinize the equation, we apply various methods, namely Kudryashov’s method, the (G′/G)-expansion method, the simplest equation technique, and the power series method, all of which have not been applied to the equation before. Through these techniques, we successfully derive solutions in diverse functional forms, encompassing rational, trigonometric, exponential, hyperbolic, and Jacobi elliptic functions. To enhance comprehension, we present our findings visually using three-dimensional and two-dimensional plots density plots via the Mathematica tool. These graphical representations effectively communicate the intricate characteristics and nuances inherent in the solutions. Our visual representations reveal a spectrum of patterns, including periodic, singular periodic, kink-shaped structures. Additionally, our investigation extends to the determination of conserved quantities associated with the new (2+1)-dimensional Bogoyavlensky-Konopelchenko equation. This involves the application of the multiplier method and Ibragimov’s theorem, two potent techniques for identifying and understanding the conservation laws governing the model.

The geometrically m$m$‐step solvable Grothendieck conjecture for affine hyperbolic curves over finitely generated fields

The geometrically m$m$‐step solvable Grothendieck conjecture for affine hyperbolic curves over finitely generated fields

Command filter based input quantized adaptive tracking control for multi‐input and multi‐output non‐strict feedback systems with unmodeled dynamics and full state time‐varying constraints

SummaryThis paper addresses the problem of adaptive tracking control for multi‐input and multi‐output (MIMO) non‐strict feedback systems with unmodeled dynamics and full state time‐varying constraints. To tackle the interference of unmodeled dynamics, the dynamic signal generated by the auxiliary system is used. Hyperbolic tangent function is used as a nonlinear mapping tool to transform the constrained system into an unconstrained one. Hysteresis quantizer is introduced to mitigate the chattering phenomenon and quantization error in the quantization signal. The derivative of virtual signal can be approximated more efficiently by command filter. Furthermore, an error compensation mechanism is established to mitigate the error introduced by the command filter. Unknown nonlinear functions are approximated by radial basis function neural networks (RBFNNs). Stability analysis of the proposed controller is performed through the Lyapunov stability theory and the output tracking error can be constrained within a specified range. Finally, simulation results are presented to demonstrate the effectiveness of the proposed method.

Adaptive RBF Neural Network Tracking Control of Stochastic Nonlinear Systems with Actuators and State Constraints

This paper investigates the adaptive neural network (NN) tracking control problem for stochastic nonlinear systems with multiple actuator constraints and full-state constraints. The issue of system full-state constraints is tackled by a generalized barrier Lyapunov function (GBLF), and the output constraints of the system are considered to be in the form of time-varying functions, which are more in line with the needs of real physical systems. The NN approximation technique is utilized to overcome the influence of the uncertainty term on controller design due to randomness. Based on the backstepping technique, a neural adaptive fixed-time tracking control strategy is designed. Under the designed control strategy, the tracking accuracy of the controlled system can reach the expectation in a fixed time. The multi-actuator constraints are converted into a generalized mathematical model to simplify the controller design process. Using the characteristics of the hyperbolic tangent function, a new function called practical virtual control signal is designed using the virtual control signal as the input. Due to the saturation constraint property of the hyperbolic tangent function, it is theoretically ensured that no state of the system exceeds the constraints through to the new form of the virtual controller. Using the adaptive controller constructed in this paper, the controlled system is semi-global fixed-time stabilized in probability (SGFSP). Finally, the effectiveness of the proposed control strategy is further verified by simulation examples.

A HILBERT-TYPE LOCAL FRACTIONAL INTEGRAL INEQUALITY WITH THE KERNEL OF A HYPERBOLIC COSECANT FUNCTION

By using Yang’s local fractional calculus theory, the method of weight function, and real-analysis techniques in the fractal set, a general Hilbert-type local fractional integral inequality with the kernel of a hyperbolic cosecant function is established. The necessary and sufficient condition for the constant factor of the general Hilbert-type local fractional integral inequality to be the best possible is discovered. Furthermore, two equivalent inequalities with the best constant factors were obtained.

Data-Driven Adaptive Controller Based on Hyperbolic Cost Function for Non-Affine Discrete-Time Systems with Variant Control Direction

As technology evolves, more complex non-affine systems are created. These complex systems are hard to model, whereas most controllers require information on systems to be designed. This information is hard to obtain for systems with varying control directions. Therefore, this study introduces a novel data-driven estimator and controller tailored for single-input single-output non-affine discrete-time systems. This approach focuses on cases when the control direction varies over time and the mathematical model of the system is completely unknown. The estimator and controller are constructed using a Multiple-input Fuzzy Rules Emulated Network framework. The weight vectors are updated through the gradient descent optimization method, which employs a unique cost function that multiplies the error by a hyperbolic tangent. The stability analyses demonstrate that both the estimator and controller converge to uniformly ultimately bounded (UUB) functions of Lyapunov. To validate the results, we show experimental tests of force control that were executed on the z-axis of a drive-controlled 3D scanning robot. This system has a varying control direction, and we also provide comparison results with a state-of-the-art controller. The results show a mean absolute percentage tracking error smaller than one percent on the steady state and the expected variation in the system’s control direction.

Flexural-Gravity Waves in a Channel with a Compressed Ice Cover

The characteristics of linear hydroelastic waves propagating in a channel covered with compressed ice are investigated. The channel has a rectangular cross-section and is assumed to be infinite in length. The fluid in the channel is non-viscous and incompressible; its flow is potential. The ice cover is modelled by an elastic plate of constant thickness frozen to the channel walls. Principal attention is paid to the investigation of the influence of ice compression on the parameters of hydroelastic waves. The problem is solved in a coupled hydroelastic formulation. The profiles of propagating waves in the channel are sought in the form of series on the normal modes of a dry plate. The modes are defined analytically through trigonometric and hyperbolic functions. It is shown that compression in the longitudinal and transverse directions has different effects on the dispersion relations of these hydroelastic waves, their shape and phase, as well as on the critical velocities and strains distribution.

Analysis of the Surrounding Rock Full-Displacement Variation in Large-Span Mudstone Highway Tunnels

Due to the increasing development of highway reconstruction and expansion projects in China, many large-span highway tunnels are being constructed near existing highway tunnels. Tunneling underneath will inevitably cause variation in the surrounding rock displacement and may even lead to collapse. In this study, based on an analysis of extensive field monitoring data from the Gucheng tunnel, the variation law for the surrounding rock full-displacement and the influence of the tunnel-face spatial effect in a large-span mudstone tunnel are analyzed. The change in the full displacement experienced the following sequence: slow pre-displacement growth → rapid increase → slow increase → gradual stability. The displacement released by the excavation of the tunnel construction accounts for 40~60% of the total displacement, and the closer to the excavation contour, the more obvious the displacement release. The final convergence value of vertical displacement is obtained by hyperbolic function regression prediction analysis. Based on this value, Lee and Hoek equations are used for parameter analysis and field-data fitting. It is concluded that the larger the proportion of the early displacement of the surrounding rock before construction to the total displacement, the smaller the influence of the tunnel-face spatial effect on the surrounding rock. The numerical simulation results are compared with actual monitoring results, and good agreement is observed. The larger the burial depth of the tunnel, the smaller the influence range in the tunnel-face spatial effect, and the more concentrated the displacement release. The variation law and the influential range for the surrounding rock full-displacement described in this paper can provide a reference for predicting and controlling the deformation during the construction of future large-span mudstone tunnels.

Exploring nine different proofs of a famous logarithmic inequality

This article explores and analyzes nine different proofs of a fundamental logarithmic inequality, say "log(1+x)/x ≥ 1/ (1+x/2) for x>-1". Some proofs are already published; others are new or consider new approaches or new angles of research. They are based on various techniques, such as differentiation, series expansion, and the application of well-known inequalities such as the primitive, Cauchy-Schwarz, logarithmic mean, hyperbolic tangent function, Jensen, and Hermite-Hadamard inequalities. A graphical work illustrates some results. Therefore, our study clarifies the different mathematical foundations of this seemingly straightforward but important inequality and goes beyond it with some new material.

Framework for $\exists \mathbb{R}$-Completeness of Two-Dimensional Packing Problems

The aim in packing problems is to decide if a given set of pieces can be placed inside a given container. A packing problem is defined by the types of pieces and containers to be handled, and the motions that are allowed to move the pieces. The pieces must be placed so that in the resulting placement, they are pairwise interior-disjoint. We establish a framework which enables us to show that for many combinations of allowed pieces, containers and motions, the resulting problem is $\exists \mathbb{R}$-complete. This means that the problem is equivalent (under polynomial time reductions) to deciding whether a given system of polynomial equations and inequalities with integer coefficients has a real solution. We consider packing problems where only translations are allowed as the motions, and problems where arbitrary rigid motions are allowed, i.e., both translations and rotations. When rotations are allowed, we show that it is an $\exists \mathbb{R}$-complete problem to decide if a set of convex polygons, each of which has at most $7$ corners, can be packed into a square. Restricted to translations, we show that the following problems are $\exists \mathbb{R}$-complete: (i) pieces bounded by segments and hyperbolic curves to be packed in a square, and (ii) convex polygons to be packed in a container bounded by segments and hyperbolic curves.

Load-displacement appraisal and analysis for driven piles; a data-centric approach

Data-centric geotechnics is an ever-evolving field for facilitating digital transformation. The major issues for foundation design, bearing capacity, settlement and interactions of super and substructure are inherently emanated in load–displacement records. Considering the pivotal role of load–displacement behavior, it is believed to be a data center in foundation engineering. In this study, from the compiled FELADD database, including twelve foundation types load–displacement records, 71 driven piles have been gained. Aiming toward the data-centric approach, records have been processed, organized and filtered. For computing ultimate and limit load, three criteria of 10%B, Brinch-Hansen 80% and hyperbolic function are engaged. Through adopting promising criteria for normalizing load–displacement, dominant factors including embedment depth, breadth and surrounding soil type were appraised. The results indicated that higher stiffness and ultimate load are achieved for piles with higher embedment depth and larger breadth, in competent layers. Load-displacement normalization has revealed significant points. The relative displacement of 1% is recognized as the appropriate point in elastic stiffness calculation, somehow compatible with safety factor of 2. Moreover, the yielding trend is mobilized for relative displacement in the range of 5 to 10%. Overall, the load–displacement records and processing, as a data center proceeds value engineering in foundation design.

Efficient Optimization of a Support Vector Regression Model with Natural Logarithm of the Hyperbolic Cosine Loss Function for Broader Noise Distribution

While traditional support vector regression (SVR) models rely on loss functions tailored to specific noise distributions, this research explores an alternative approach: ε-ln SVR, which uses a loss function based on the natural logarithm of the hyperbolic cosine function (lncosh). This function exhibits optimality for a broader family of noise distributions known as power-raised hyperbolic secants (PHSs). We derive the dual formulation of the ε-ln SVR model, which reveals a nonsmooth, nonlinear convex optimization problem. To efficiently overcome these complexities, we propose a novel sequential minimal optimization (SMO)-like algorithm with an innovative working set selection (WSS) procedure. This procedure exploits second-order (SO)-like information by minimizing an upper bound on the second-order Taylor polynomial approximation of consecutive loss function values. Experimental results on benchmark datasets demonstrate the effectiveness of both the ε-ln SVR model with its lncosh loss and the proposed SMO-like algorithm with its computationally efficient WSS procedure. This study provides a promising tool for scenarios with different noise distributions, extending beyond the commonly assumed Gaussian to the broader PHS family.