Hyperbolic Function Research Articles

Flatness-based nonlinear internal model control with disturbances compensation via improved extended state observer for remotely operated vehicle

This paper proposes an improved Active Disturbance Rejection Control (ADRC) scheme for nonlinear control-affine systems, achieving superior robustness/ performance trade-offs. By leveraging the flatness property within a closed-loop structure, Nonlinear Internal Model Control (NLIMC) is employed as a model-based controller utilises partial plant knowledge to enhance control performance and eliminate static tracking errors. Subsequently, an improved Extended State Observer (IESO) is incorporated with NLIMC to address unmodelled dynamics, disturbances, and parameter uncertainties in addition to the full state's estimation. The IESO's added hyperbolic function dynamic ensures higher sensitivity and accuracy compared to conventional linear ESOs, especially in case of large errors. Control performances are evaluated through a Remotely Operated Vehicle (ROV) application. Numerical simulations have been carried out and the results confirm its effectiveness and superiority compared to Ocean Disturbance Observer-based Double-loop Sliding Mode Controller (ODSMC) and the linear ADRC.

Convective Heat Transfer in Uniformly Accelerated and Decelerated Turbulent Pipe Flows

This study presents a detailed investigation of the temporal evolution of the Nusselt number (Nu) in uniformly accelerated and decelerated turbulent pipe flows under a constant heat flux using direct numerical simulations. The influence of different acceleration and deceleration rates on heat transfer is systematically studied, addressing a gap in the previous research. The simulations confirm several key experimental findings, including the presence of three distinct phases in the Nusselt number temporal response—delay, recovery, and quasi-steady phases—as well as the characteristics of thermal structures in unsteady pipe flow. In accelerated flows, the delay in the turbulence response to changes in velocity results in reduced heat transfer, with average Nu values up to 48% lower than those for steady-flow conditions at the same mean Reynolds number. Conversely, decelerated flows exhibit enhanced heat transfer, with average Nu exceeding steady values by up to 42% due to the onset of secondary instabilities that amplify turbulence. To characterize the Nu response across the full range of acceleration and deceleration rates, a new model based on a hyperbolic tangent function is proposed, which provides a more accurate description of the heat transfer response than previous models. The results suggest the potential to design unsteady periodic cycles, combining slow acceleration and rapid deceleration, to enhance heat transfer compared to steady flows.

Ramanujan-Berndt-type integrals of order four and series associated with Jacobi elliptic functions

In this paper, we evaluate in closed forms two families of infinite integrals containing hyperbolic and trigonometric cosine functions in their integrands. We call them Ramanujan-Berndt-type integrals of order four since they initiated the study of similar integrals. We establish explicit evaluations of twelve classes of hyperbolic sums by special values of the Gamma function by two completely different approaches, which extend those sums considered by Xu, Yang, Zhao and the second author previously. We discover the first by refining two results of Ramanujan concerning some q-series. For the second we compare both the Fourier series expansions and the Maclaurin series expansions of a few Jacobi elliptic functions. Furthermore, by contour integrations we convert two families of Ramanujan-Berndt-type integrals of order four to the above hyperbolic sums, all of which can be evaluated in closed forms. We then discover explicit formulas for one of the two families. Moreover, we provide many examples which enable us to formulate a conjectural explicit formula for the other family of the Ramanujan-Berndt-type integrals at the end, and obtain some results on Barnes' multiple zeta function.

Rheological Analysis of Thermally Radiative Hyperbolic Tangent Fluid Flow in a Curved Channel Driven by Metachronal Ciliary Waves

Rheological Analysis of Thermally Radiative Hyperbolic Tangent Fluid Flow in a Curved Channel Driven by Metachronal Ciliary Waves

A modified Holmquist‒Johnson‒Cook (HJC) constitutive model and its application to numerical simulations of explosions and impacts in rock materials

The Holmquist‒Johnson‒Cook (HJC) model is a commonly used constitutive model for simulations of the dynamic response and damage characteristics of geotechnical materials under explosions and impacts. However, certain deficiencies in its strength, strain rate, and damage models limit its computational effectiveness. Therefore, this paper first introduced the constitutive relation of the HJC. Then, the single-strength equation of the original HJC was improved by adopting three limiting surfaces and considering the impact of the Lode-angle on the yield surface. The hyperbolic tangent function was employed to model the rate effect, and the recommended strain rate parameters for rock materials were provided. Furthermore, an exponential strain softening function-based tensile damage equation was introduced to address tensile damage under negative pressure. On this basis, a method for determining parameters for the modified HJC in rock materials was proposed. Finally, the modified model was incorporated into the LS-DYNA material library using secondary development techniques, and multiple types of impact and explosion experiments on different types of rock materials were simulated using both the original and modified HJC. Additionally, the modified HJC was compared with several other modified HJC versions from various perspectives. It has been found that the simulation results of the modified HJC are in closer agreement with the experimental results than those of the original HJC. Compared to other modified HJC models, the modified HJC proposed in this paper offers a broader range of applications and a better compromise between computational efficiency and the number of parameters.

John Cross, epidemic theory, and mathematically modeling the Norwich smallpox epidemic of 1819.

In this paper, we reintroduce Dr. John Cross' neglected and unusually complete historical data set describing a smallpox epidemic occurring in Norwich, England in 1819. We analyze this epidemic data in the context of early models of epidemic spread including the Farr-Evans-Brownlee Normal law, the Kermack-McKendrick square Hyperbolic Secant and SIR laws, along with the modern Volz-Miller random-network law. We show that Cross' hypothesis of susceptible pool limitation is sufficient to explain the data under the SIR law, but requires parameter estimates differing from the modern understanding of smallpox epidemiology or large errors in Cross' data collection. We hypothesize that these discrepancies are due to the mass-action hypothesis in SIR theory, rather than significant errors by Cross, and use Volz-Miller theory to support this. Our analysis demonstrates the difficulties arising in inference of attributes of the disease from death incidence data and how model hypotheses impact these inferences. Our study finds that, combined with Volz-Miller modeling theory, Cross' death incidence data and population observations give smallpox attributes which largely cohere to those used in modern smallpox models.

Stability analysis, modulation instability, and the analytical wave solitons to the fractional Boussinesq-Burgers system

Abstract The research is concerned with the novel analytical solitons to the (1+1)-D nonlinear Boussinesq-Burgers System (B-B S) in the sense of a new definition of fractional derivatives. The concerned system is helpful to describes the waves in different phenomenons, including proliferation of waves in shallow water, oceanic waves and many others. Authors gain the solutions involving trigonometric, hyperbolic, and rational functions by using the exp a function and the extended sinh-Gordon equation expansion (EShGEE) methods. Fractional derivative provides the better results than the present results. These results are helpful and useful in the different areas of applied sciences, including the optical fibers, telecommunications, plasma physics, fluid dynamics and many more. The solutions are shown by 2-dimensional, 3-dimensional, and contour graphs. The solutions are useful in further studies of the governing model. The stability process is performed to verify that the solutions are exact and accurate. The modulation instability is used to determine the steady-state stable results to the governing equation. The techniques utilized are both simple and effective.

Periodic, quasi-periodic, chaotic waves and solitonic structures of coupled Benjamin-Bona-Mahony-KdV system

Abstract In this paper, we mainly focus on studying the dynamical behaviour and soliton solution of the coupled Benjamin-Bona-Mahony-Korteweg–de Vries (BBM-KdV) system, which characterizes the propagation of long waves in weakly nonlinear dispersive media. The paper utilizes different tools to detect chaos, such as time series analysis, bifurcation diagrams, power spectra, phase portraits, Poincare maps, and Lyapunov exponents. This analysis helps in more accurate predictive modeling of the systems. This understanding can aid in the design of control strategies, resulting in enhancements in prediction, control, optimization, and design. Additionally, we construct the system’s solitary wave structures using the Jacobi elliptic function (JEF) method. We identify periodic wave solutions expressed in terms of rational, hyperbolic, and trigonometric functions. Certain parameter values can lead to periodic wave solutions, solitary waves (bell-shaped solitons), shock wave solutions (kink-shaped soliton solutions), and double periodic wave solutions.

A New Method for Joint Sparse DOA Estimation.

To tackle the issue of poor accuracy in single-snapshot data processing for Direction of Arrival (DOA) estimation in passive radar systems, this paper introduces a method for judiciously leveraging multi-snapshot data. This approach effectively enhances the accuracy of DOA estimation and spatial angle resolution in passive radar systems. Additionally, in response to the non-convex nature of the mixed norm, we propose a hyperbolic tangent model as a replacement, transforming the problem into a directly solvable convex optimization problem. The rationality of this substitution is thoroughly demonstrated. Lastly, through a comparative analysis with existing discrete grid DOA estimation methods, we illustrate the superiority of the proposed approach, particularly under conditions of medium signal-to-noise ratio, varying numbers of snapshots, and close target angles. This method is less affected by the number of array elements, and is more usable in practices verified in real-world scenarios.

Computational analysis of the conformable nonlinear Schrödinger equation with Kudryashov’s refractive index model and generalized non-local nonlinearity

This paper investigates the nonlinear conformable Schrödinger equation with nonlocal nonlinearities and Kudryashov’s arbitrary refractive index. The study employs the Uniform Method (UM) and the Extended Hyperbolic Function Method (EHFM) to derive various optical soliton solutions for this present conformable equation. To demonstrate the importance of the novel solutions, the paper presents two-dimensional, three-dimensional, and contour plots, illustrating kink-type, wave, bright, bell-shaped, and mixed dark–bright soliton solutions. Further, the impact of the fractional order parameter, temporal parameter, and nonlinearity coefficient on these optical solutions is analyzed, providing valuable insights into the conformable nonlinear Schrödinger model. The behavior of these optical solutions is analyzed through illustrative graphs that account for variations in the conformable order derivative, temporal parameter, and nonlinearity coefficient. The methodologies applied in this research show potential for broader application to various nonlinear Schrödinger equations in fields such as applied mathematics and nonlinear optics. The research suggests that these methods could be applied in the future to investigate other differential equations with fractional and integer orders across different fields of applied sciences. This study expands our understanding of nonlinear optics and has potential practical implications in various fields like optical signal processing, laser technology, and telecommunications.

Diverse general solitary wave solutions and conserved currents of a generalized geophysical Korteweg–de Vries model with nonlinear power law in ocean science

This article presents an analytical investigation performed on a generalized geophysical Korteweg–de Vries model with nonlinear power law in ocean science. To start with, achieving diverse solitary wave solutions to the generalized power‐law model involves using wave transformation, which reduces the model to a nonlinear ordinary differential equation. A direct integration approach is adopted to construct solutions in the beginning. This brings the emergence of interesting solutions like non‐topological solitons, trigonometric functions, exponential functions, elliptic functions, and Weierstrass functions in general structures. Besides, in a bid to secure more general exact solutions to the model, one adopts the extended Jacobi elliptic function expansion technique (for some specific cases of ). Thus, various cnoidal, snoidal, and dnoidal wave solutions to the understudied model are attained. The copolar trio explicated in a tabular form reveals that these solutions can be relapsed to various hyperbolic and trigonometric functions under certain criteria. Additionally, diverse graphical exhibitions of the dynamical attributes of the gained results are presented in a bid to have a sound understanding of the physical phenomena of the underlying model. Later, one gives the conserved vectors of the aforementioned equation by employing the standard multiplier approach.

A refined grey Verhulst model for accurate degradation prognostication of PEM fuel cells based on inverse hyperbolic sine function transformation

Accurate prognostication of degradation plays an essential role in effectively enhancing the operational lifespan of proton exchange membrane fuel cells (PEMFCs). This paper proposes a novel enhanced correctional grey Verhulst model (IHS-CRGVM-RE), designed to prognosticate the degradation process of PEMFCs using the voltage of PEMFCs stack as a health indicator. First, the inverse hyperbolic sine function transformation is employed to attain optimal smoothness in data treatment. Then, the background value within the grey Verhulst model framework is modified based on cellular automata with rectangle techniques. Finally, a residual correction mechanism is applied to delineate the influences of error outcomes concerning PEMFCs degradation. Rigorous validation is provided via a comprehensive analysis based on two distinct PEMFCs datasets. The results demonstrate that the proposed model outperforms other data-driven models in prognostication accuracy, highlighting its significant importance for prognosticating the lifespan of PEMFCs.

Several Derivative Formulas of Two Exponential Functions and Real Power of Hyperbolic Secant Function with a Generalization of a Formula for Specific Partial Bell Polynomials

In the paper, by virtue of some identities for the partial Bell polynomials and with the aid of the Faá di Bruno formula, the author presents several derivative formulas of two exponential functions and the real power of the hyperbolic secant function, and generalizes a formula for specific partial Bell polynomials.

Reinforcement learning-adaptive fault-tolerant IGC method for a class of aircraft with non-affine characteristics and multiple uncertainties

Abstract In this paper, a brand-new adaptive fault-tolerant non-affine integrated guidance and control method based on reinforcement learning is proposed for a class of skid-to-turn (STT) missile. Firstly, considering the non-affine characteristics of the missile, a new non-affine integrated guidance and control (NAIGC) design model is constructed. For the NAIGC system, an adaptive expansion integral system is introduced to address the issue of challenging control brought on by the non-affine form of the control signal. Subsequently, the hyperbolic tangent function and adaptive boundary estimation are utilised to lessen the jitter due to disturbances in the control system and the deviation caused by actuator failures while taking into account the uncertainty in the NAIGC system. Importantly, actor-critic is introduced into the control framework, where the actor network aims to deal with the multiple uncertainties of the subsystem and generate the control input based on the critic results. Eventually, not only is the stability of the NAIGC closed-loop system demonstrated using Lyapunov theory, but also the validity and superiority of the method are verified by numerical simulations.

On convergence of linear response formulae in some piecewise hyperbolic maps

Abstract When high-dimensional non-uniformly hyperbolic chaotic systems undergo dynamical perturbations, their long-time statistics are generally observed to respond differentiably with respect to the perturbation. Although important in applications, this differentiability, which is thought to be connected to the dimensionality of the system, has remained resistant to rigorous study outside of the one-dimensional setting. To model non-uniformly hyperbolic systems in multiple dimensions, we consider a family of the mathematically tractable class of piecewise smooth hyperbolic maps, the Lozi maps. For these maps, we prove that the existence of a formal derivative of the response reduces to conditional mixing of the SRB measure on the singularity set. Heuristically, this suggests that Lozi maps, and piecewise uniformly expanding maps more generally, should have linear response.

Analytic Moments of TGARCH(1,1) Models with Polynomially Adjusted Densities

Abstract This article extends He, Silvennoinen, and Teräsvirta (2008, J Finan Econ, 6, 208–230) and Francq and Zakoïan (2010, GARCH Models) by providing analytical expressions for the moments of the unconditional distribution of the TGARCH(1,1) under alternative specifications for the conditional mean and different skewed distributions for the innovations. We consider polynomially adjusted (PA) densities, such as the PA Logistic, PA hyperbolic secant, and the PA Gaussian, along with the skewed Student-t. Our results show that (i) the main driver of the skewness of the TGARCH(1,1) is the skewness of the innovations, while the excess kurtosis has a comparatively lesser impact. However, both skewness and kurtosis of the innovations significantly affect the TGARCH(1,1) kurtosis; (ii) if the conditional mean is not constant, returns can be asymmetric even if innovations are symmetric; (iii) skewed innovations can generate cross-correlations different from zero, indicating leverage effect, even when the volatility model is symmetric. Finally, we illustrate our theoretical results with an empirical application to stock indices.

Numerical Simulation of Electromagnetic Nondestructive Testing Technology for Elasto-Plastic Deformation of Ferromagnetic Materials Based on Magneto-Mechanical Coupling Effect.

A numerical tool for simulating the detection signals of electromagnetic nondestructive testing technology (ENDT) is of great significance for studying detection mechanisms and improving detection efficiency. However, the quantitative analysis methods for ENDT have not yet been sufficiently studied due to the absence of an effective constitutive model. This paper proposed a new magneto-mechanical model that can reflect the dependence of relative permeability on elasto-plastic deformation and proposed a finite element-infinite element coupling method that can replace the traditional finite element truncation boundary. The validity of the finite element-infinite element coupling method is verified by the experimental result of testing electromagnetic analysis methods using TEAM Problem 7. Then, the reliability and accuracy of the proposed model are verified by comparing the simulation results under elasto-plastic deformation with experimental results. This paper also investigates the effect of elasto-plastic deformation on the transient magnetic flux signal, a quantitative hyperbolic tangent model between Bzpp (peak-peak value of the normal component of magnetic flux signal) and elastic stress, and the exponential function relationship between Bzpp and plastic deformation is established. In addition, the difference and mechanism of a magnetic flux signal under elasto-plastic deformations are analyzed. The results reveal that the variation of the transient magnetic flux signal is mainly due to domain wall pinning, which is significantly affected by elasto-plastic deformation. The results of this paper are important for improving the accuracy of quantitative ENDT for elasto-plastic deformation.

A new statistical distribution via the Phi-4 equation with its wide-ranging applications.

This paper presents a new framework based on nonlinear partial differential equations and statistics. For the nonlinear Phi-4 equation, the probability density function of the hyperbolic secant (HS) distribution has been obtained. Our model's density has various shapes, including left-skewed, symmetric, and right-skewed. Eight distinct estimation approaches have been employed to estimate the parameters of our model. Additionally, the behavior of the HS model parameters was investigated using randomly generated data sets using these estimation techniques. Furthermore, we illustrate the applicability of the HS distribution for modeling real data by applying our results to real data. As a result, it is expected that our proposal will be of significant assistance to the community investigating new distributions based on hyperbolic functions and their applications to real-world data sets.

Modified Shooting with Discretization Validation for Non-Standard Optimal Control Problem: Case Study for Four-Stage Royalty Payment Problem

A modified shooting method augmented by discretization validation is presented in this paper to address the challenges inherent in non-standard optimal control problems. Specifically, a specific case study involving four-stage royalty payment functions is focused on, with the aim of effectively optimizing these complex, non-differentiable functions. The shooting method is adapted and enhanced to compute optimal solutions, and its accuracy is rigorously confirmed through discretization techniques. The primary objectives involve maximizing the performance index and determining optimal control strategies for scenarios where the final state variable remains unknown. In this study, the royalty payment is defined as a four-stage piecewise function. The incorporation of piecewise royalty functions introduces non-differentiability at specific time intervals, necessitating innovative approaches for finding optimal solutions. This, in turn, leads to the utilization of the continuous hyperbolic tangent (tanh) function to address the non-differentiability issue. A hybrid shooting method, combining the Newton and Golden Section Search methods, is employed in the C++ programming language to compute the unknown final state value. A new natural boundary condition, based on established theory, is introduced to further facilitate the investigation. Discretization methods such as Euler, Runge-Kutta, Trapezoidal, and Hermite-Simpson approximations are employed for validation. The validation process entails the use of the AMPL programming language with the MINOS solver. Comparative analyses reveal that the modified shooting method yields more accurate optimal results than discretization methods, thus demonstrating the method’s effectiveness in addressing non-standard optimal control problems. The significance of fundamental theories in addressing real-world challenges is underscored by this research, providing valuable insights for future researchers exploring mathematical approaches in similar contexts. The study contributes to the continued relevance of the academic field, particularly in science and mathematics education.

Exploring Kink Solitons in the Context of Klein–Gordon Equations via the Extended Direct Algebraic Method

This work employs the Extended Direct Algebraic Method (EDAM) to solve quadratic and cubic nonlinear Klein–Gordon Equations (KGEs), which are standard models in particle and quantum physics that describe the dynamics of scaler particles with spin zero in the framework of Einstein’s theory of relativity. By applying variables-based wave transformations, the targeted KGEs are converted into Nonlinear Ordinary Differential Equations (NODEs). The resultant NODEs are subsequently reduced to a set of nonlinear algebraic equations through the assumption of series-based solutions for them. New families of soliton solutions are obtained in the form of hyperbolic, trigonometric, exponential and rational functions when these systems are solved using Maple. A few soliton solutions are considered for certain values of the given parameters with the help of contour and 3D plots, which indicate that the solitons exist in the form of dark kink, hump kink, lump-like kink, bright kink and cuspon kink solitons. These soliton solutions are relevant to actual physics, for instance, in the context of particle physics and theories of quantum fields. These solutions are useful also for the enhancement of our understanding of the basic particle interactions and wave dynamics at all levels of physics, including but not limited to cosmology, compact matter physics and nonlinear optics.