Quintic Research Articles

Dark and bright hump solitons in the realm of the quintic Benney-Lin equation governing a liquid film

<p>This study explored and examined soliton solutions for the Quintic Benney-Lin equation (QBLE), which describes the dynamic of liquid films, using the Riccati modified extended simple equation method (RMESEM). The proposed approach, which is designed for nonlinear partial differential equations (NPDEs), effectively generates a large number of soliton solutions for the given QBLE, which basically captures the fundamental dynamics of the system. The rational, hyperbolic, rational-hyperbolic, trigonometric, and exponential forms of the scientifically specified soliton solutions are the main determinants of the hump solitons. We used 2D, 3D, and contour visualizations to offer accurate representations of the researched soliton phenomena associated with these solutions. These representations revealed the existence of dark and bright hump solitons in the framework of the QBLE and offer a thorough way to examine the model's behavioral characteristics in the liquid film by analyzing the QBLE model's soliton dynamics. Moreover, applying the suggested approach advances our knowledge of the unique features of the other similar NPDEs and the underlying dynamics.</p>

Numerical calculation the fracture toughness of square section beam under three-point bending test

A universal equation is proposed to determine specific fracture energy based on numerical calculations by finite element methods in the ANSYS system referring to the three-point bending of a square beam arranged edgewise on supports. Numerical dependences of the specimen compliance on the crack length were represented as the 5th and 6th degree polynomials on the relative value of Δl/h, where Δl is the crack length and h is the half length of the square diagonal. To calculate the specific fracture energy, we used the derivative of specimen compliance by the crack length. The analysis has shown that finite value of specific fracture energy at Δl→0 is obtained when the decomposition term at Δl/h is equal to 0. A satisfactory agreement between the specific fracture energy values for the 5th and 6th degree decompositions is observed only at Δl/h ≈ 1. As the values of Δl decrease, the difference becomes increasingly significant. The averaging of coefficients at corresponding Δl/h degrees of two considered decompositions allows us to find a power function that weakly depends on the Δl/h ratio and leads to a consistent dependence of the specific fracture energy on the crack length. Due to the performed calculations, a universal equation is obtained to determine the specific fracture energy according to the test data of square beams by three-point bending in a wide range of geometric dimensions.

A trajectory planning method for a casting sorting robotic arm based on a nature-inspired Genghis Khan shark optimized algorithm.

In order to meet the efficiency and smooth trajectory requirements of the casting sorting robotic arm, we propose a time-optimal trajectory planning method that combines a heuristic algorithm inspired by the behavior of the Genghis Khan shark (GKS) and segmented interpolation polynomials. First, the basic model of the robotic arm was constructed based on the arm parameters, and the workspace is analyzed. A matrix was formed by combining cubic and quintic polynomials using a segmented approach to solve for 14 unknown parameters and plan the trajectory. To enhance the smoothness and efficiency of the trajectory in the joint space, a dynamic nonlinear learning factor was introduced based on the traditional Particle Swarm Optimization (PSO) algorithm. Four different biological behaviors, inspired by GKS, were simulated. Within the premise of time optimality, a target function was set to effectively optimize within the feasible space. Simulation and verification were performed after determining the working tasks of the casting sorting robotic arm. The results demonstrated that the optimized robotic arm achieved a smooth and continuous trajectory velocity, while also optimizing the overall runtime within the given constraints. A comparison was made between the traditional PSO algorithm and an improved PSO algorithm, revealing that the improved algorithm exhibited better convergence. Moreover, the planning approach based on GKS behavior showed a decreased likelihood of getting trapped in local optima, thereby confirming the effectiveness of the proposed algorithm.

Dynamics of a damped quintic wave equation with time-dependent coefficients

<p>We present a comprehensive investigation of the long-term dynamics generated by a semilinear wave equation with time-dependent coefficients and quintic nonlinearity on a bounded domain subject to Dirichlet boundary conditions. By employing rescaling techniques for time and utilizing the Strichartz estimates applicable to bounded domains, we initially study the global well-posedness of the Shatah–Struwe (S–S) solutions. Subsequently, we establish the existence of a uniform weak global attractor consisting of points on complete bounded trajectories through an approach based on evolutionary systems. Finally, we prove that this uniformly weak attractor is indeed strong by means of a backward asymptotic a priori estimate and the so-called energy method. Moreover, the smoothness of the obtained attractor is also shown with the help of a decomposition technique.</p>

Global phase portraits of quintic reversible uniform isochronous centers

This paper studies the global phase portraits of uniform isochronous quintic centers at the origin with time reversibility such that their nonlinear part is not homogeneous. By using Poincaré compactification, we obtain all possible phase portraits of this quintic polynomial differential system.

Power based adaptive compensator of output oscillations

Power-based output feedback compensator for oscillatory systems is proposed. The average input-output power of an oscillatory signal serves as an equivalent control effort, while the unknown amplitude and frequency of oscillations are detected at each half-period. This makes the compensator adaptive and discrete, while the measured oscillatory output is the single available signal in use. The resulting discrete control scheme enables a drastic reduction of communication efforts in the control loop. The compensator is designed for 2nd order systems, while an extension to higher-order dynamics, like e.g. in case of two-inertia systems, is also provided. Illustrative experimental case study of the 5th order oscillatory system is provided.

Inhalation of gaseous hypochlorous acid and its effect on human respiratory epithelial cells in laboratory model systems.

During the disinfection of indoor spaces using gaseous hypochlorous acid (HOCl(g)), inhalation is the most common route of exposure for humans. In this study, an artificial human respiratory tract model was exposed to 12-140 ppb HOCl(g) at an aspiration flow rate of 800 mL/s for 15 h in a 1 m3 chamber. The respiratory tract model was equipped with 5th order bronchi and all gas-contact parts were made of silicone rubber with no other chlorine-consuming substances. The concentration of HOCl(g) reaching the lung pseudo-space was approximately 47.4% of the HOCl(g) concentrations in the chamber and was calculated to be very close to zero when the chamber concentration was less than 20.5 ppb. The disappearance of HOCl(g) during inhalation is likely due to the adsorption of HOCl(g) on the gas-contact silicone rubber surfaces. The cytotoxicity of HOCl(g) on respiratory epithelial cells was also examined using human air-liquid-interface airway tissue models. Human nasal epithelium and bronchiolar epithelium were exposed to 100 ppb and 500 ppb HOCl(g) for 8 h and 5 d, respectively. No significant effects of HOCl(g) on cell viability and ciliary activity were observed in any cell type, indicating that low concentrations of HOCl(g）, less than 500 ppb, had no cytotoxic effect.

О решении смешанной задачи для уравнения колебаний движущегося вязкоупругого полотна

A model initial boundary value problem of small transverse oscillations of a viscoelastic moving web with a hinged condition of fastening is considered. The vibrations of such a canvas are described by a linear differential equation of the 5th order in a spatial variable with constant coefficients. It is worth noting that the equation includes mixed derivatives of the desired function both with respect to the spatial variable and with respect to time. The paper describes a technique for constructing a solution in the form of a functional series based on a system of basis functions. To solve the initial-boundary value problem under the additional condition of conservation of energy, a condition is obtained that ensures the uniqueness of the solution. A special class of functions for which the uniqueness theorem holds is explicitly described.

Optimal control of robotic systems and biased Riemannian splines

In this paper, we study mechanical optimal control problems on a given Riemannian manifold (Q, g) in which the cost is defined by a general cometric g̃. This investigation is motivated by our studies in robotics, in which we observed that the mathematically natural choice of cometric g̃ = g* – the dual of g – does not always capture the true cost of the motion. We then, first, discuss how to encode the system’s torque-based actuators configuration into a cometric g̃. Second, we provide and prove our main theorem, which characterizes the optimal solutions of the problem associated to general triples (Q, g, g̃) in terms of a 4th order differential equation. We also identify a tensor appearing in this equation as the geometric source of “biasing” of the solutions away from ordinary Riemannian splines and geodesics for (Q, g). Finally, we provide illustrative examples and practical demonstration of the biased splines as providing the true optimizers in a concrete robotics system.

Two-point boundary value problems for 4th order ordinary differential equations

The new optimal efficient sufficient conditions are established for solvability and uniqueness of a solution of the linear and nonlinear fourth order ordinary differential equations u^(4)(t) = p(t)u(t) + q(t) for t ∈ [a,b], u^(4)(t) = p(t)u(t) + f(t, u(t)) for t ∈ [a,b], under the following two-point boundary conditions u^(i)(a) = 0, u^(i)(b) = 0 (i = 0, 1), and u^(i)(a) = 0 (i = 0, 1, 2), u(b) = 0, where p ∈ L([a,b]; R) is a nonconstant sign function and f ∈ K([a,b] × R; R).

Morphometric Status of River Mu Sub-Basin in Makurdi South, Nigeria

River Mu drainage basin morphometric status is assessed against the backdrop of its characteristics and implication on hydrologic processes operating within the watershed. Geospatial techniques of GIS and Remote Sensing was deployed in the analysis and characterization of the basin morphometric attributes. The result showed Mu drainage basin has an area of 1259.26km2, basin perimeter 240.96km, form factor 0.22, circulatory ration 0.27, and drainage texture 0.51. Mu drainage basin is of 4th order with a dendritic pattern of drainage system indicating the homogeneity in texture and lack of structural control. The stream frequency 0.10 stream segments per square kilometer, drainage density (Dd) 0.40km/km2, bifurcation ratio of 4.62 and length of overland flow of 1.25km. The drainage density, bifurcation ratio and circularity ratio value indicate that the basin is of gentle slope, elongated, high infiltration capacity, low run off and may have longer duration of flow peaks, less susceptible to flooding.

ЕКВІВАВАЛЕНТНІСТЬ ФУНДАМЕНТАЛЬНОГО СПЛАЙНУ ТА ФУНКЦІЇ ГРІНА ДЛЯ ПОБУДОВИ ТОЧНОГО СКІНЧЕННОВИМІРНОГО АНАЛОГА КРАЙОВОЇ ЗАДАЧІ ДЛЯ ЗВИЧАЙНОГО ДИФЕРЕНЦІАЛЬНОГО РІВНЯННЯ 4-ГО ПОРЯДКУ

The author considers a problem with the main and natural boundary conditions on an interval. A new method for constructing an exact discrete analog of the problem is proposed. The method deals with the projection of the differential equation on local splines, formed by the fundamental system of solutions of the Cauchy problems for the homogeneous equation. A system of linear algebraic equations with a 5-diagonal matrix is obtained for the values of the exact solutions of the original problem at the points of a uniform grid. To implement an exact analog, we recommend using high-order accuracy schemes which are formed by partial sums of series in even powers of the grid step for solving the Cauchy problems. Keywords: boundary-value problem, ordinary differential equation of the 4th order, Cauchy problem, Wronskian, local spline, superposition of solutions, exact discrete analog, system of linear algebraic equations, 5-diagonal matrix.

Purchasing Inventory Models for Deteriorating Items with Price-Growth -Exponential Demand under Inflation - in 4th order equation

Purchasing Inventory Models for Deteriorating Items with Price-Growth -Exponential Demand under Inflation - in 4th order equation

The mixed convection thermally radiated hybrid nanofluid flow through an inclined permeable shrinking plate with slip condition and inclined magnetic effect

In this paper, we consider the slip boundary condition to analyze the radiative inclined magnetohydrodynamics mixed convection hybrid nanofluid (Al2O3–Cu/H2O) flow across an inclined shrinking permeable plate. The following model’s governing PDEs are transformed into nonlinear ODEs with the help of similarity transformations. To achieve the numerical solution of ODEs, the boundary-value problem of 4th order accuracy (bvp4c) is applied. With appropriate values for the copper volume fraction, magnetic parameter, slip parameter, and radiation parameter, the numerical results are described and graphically represented in velocity and temperature profiles. Due to the shrinking plate, the dual solutions are to be observed. For higher values of copper volume fraction, slip parameter, and magnetic parameter the first solution in the velocity profile increases and the second solution in the velocity profile decreases by increasing the value of copper volume fraction and magnetic parameter. By increasing the magnetic parameter, and slip parameter first solution of the temperature profile decreases while both the solution increases by increasing the copper volume fraction and radiation parameter.

On an efficient numerical procedure for the Functionalized Cahn-Hilliard equation

The Functionalized Cahn Hilliard (FCH) equation was used to model micro-phase separation in mixtures of amphiphilic molecules in solvent. In this paper, we proposed a Tri-Harmonic Modified (THM) numerical approach for efficiently solving the FCH equation with symmetric double well potential by extending the ideas of the Bi-harmonic Modified (BHM) method. THM formulation allowed for the nonlinear terms in the FCH equation to be computed explicitly, leading to fast evaluations at every time step. We investigated the convergence properties of the new approach by using benchmark problems for phase-field models, and we directly compared the performance of the THM method with the recently developed scalar auxiliary variable (SAV) schemes for the FCH equation. The THM modified scheme was able to produce smaller errors than those obtained from the SAV formulation. In addition to this direct comparison with the SAV schemes, we tested the adaptability of our scheme by using an extrapolation technique which allows for errors to be reduced for longer simulation runs. We also investigated the adaptability of the THM method to other 6th order partial differential equations (PDEs) by considering a more complex form of the FCH equation with nonsymmetric double well potential. Finally, we also couple the THM scheme with a higher order time-stepping method, (implicit-explicit) IMEX schemes, to demonstrate the robustness and adaptability of the new scheme. Numerical experiments are presented to investigate the performance of the new approach.

Mathematical Model on Distributed Denial of Service Attack in the Computer Network

In this paper, an electronic- epidemic two-folded mathematical model is formulated with help of non-linear ordinary differential equations. Distributed Denial of Service (DDoS) attacks in the computer network are studied. The modeling of both attacking nodes and targeting nodes is performed. Botnet based malicious devices and their threats on computer networks are addressed using appropriate parameters. The basic reproduction numbers for both the attacking and the targeting population are calculated and interpreted. Local and global stability analysis is carried out for the infection-free and endemic equilibrium points. Differential equations are solved with the help of the Runge-Kutta 4th order numerical method and graphs are analyzed using MATLAB software. Simulation shows that the success or failure depends on the number of initially infected computers in the attacking group. The proposed model exhibits the phenomenon of backward bifurcation for different values of transmission parameters. This model gives the theoretical base for controlling and predicting the DDoS attack. This shows the way to minimize the attack in the network. This study will be helpful to identify the botnet devices and run the latest version of antivirus in the network to protect against DDoS attacks from attacking sources. The application of this study is to ascertain online crime and locate the attacking nodes in the field of online transactions of real-life problems that involve the internet and computer networking systems. Moreover, our model can play an important role in policy-making against the distributed attack.

Developing stem taper of Shorea robusta in the far-western Terai of Nepal

Tree taper functions expressed in terms of height and diameter at breast height (BDH) can provide accurate and timely information on current growing stock. Taper equations are very important for calculation of growing stock in forest inventories, however, in the context of Nepal, these are still unavailable at the required level. The study aimed to develop taper equations, so that those can be used to predict the diameter anywhere along the stem, and estimate tree volumes at desired sections. The input data was a destructive sampling method of 81 sample trees of Shorea robusta distributed in ten locations of the far western Terai of Nepal (Kailali and Kanchanpur districts).
 Using two independent B-Spline and 5th-degree polynomial taper models, the upper stem diameters were imputed. Both models were applied on the whole dataset, irrespective of DBH size, to get common fitted taper models. Later, the same models were tested for three different DBH classes to compare and evaluate the best-fit model. 
 Taper models, developed under the B-spline polynomial model of 3rd degree, were found to be highly dependent on the tree (DBH) sizes. Thus, better models could be developed by classifying the whole dataset based on different DBH classes. On the other hand, developed models under the 5th degree polynomial taper model did not exhibit their dependency on the tree (DBH) sizes and a better model was found for the unclassified datasets, i.e. for all DBH sizes.

Vibration-polynomial-based optimal trajectory planning for mobile concrete pumping equipment with state constraints

To improve the operating stabilities of the flexible hydraulic manipulator in concrete pumping equipment, a trajectory planning method based on optimal vibration energy is presented. First, predefined trajectories are generated by the quintic polynomial method, and kinematic inversions are established by unifying the velocity and angle constraints into the velocity level. Then, relationships between the boom vibration and the vibration energy are solved in modal space. Specifically, hydraulic actuators are regarded as spring-damping systems to calculate generalized forces into the vibration equation. The vibration equation can solve the modal coordinate and vibration model function, such that the elastic potential energy to cause vibration is completely described. Next, floating values are captured by a genetic algorithm to minimize vibration energy, and then introduced to the predefined trajectory to weaken the boom vibration. Last, the proposed trajectory planning method is verified by a concrete pumping spreader with a 13 m hydraulic manipulator.

Entropy generation in free convective micropolar couple stress regime in vertical channel

The study reports an application of homotopy perturbation method to investigate entropy generation in the steady micropolar couple stress fluidics inside a vertical porous medium channel. The channel is composed of two isothermal parallel walls maintained at different temperatures. The governing equations are framed in the Cartesian system. The boundary value problem depicting the thermofluidic configuration is treated by the homotopy perturbation method (HPM) and for its validation, the 4th-order Runge-Kutta scheme is also invoked. The quantities of interest, viz. skin friction, wall couple stress, Nusselt number computed by HPM, and 4th order R-K method are found to be in good agreement. After getting the velocity, temperature, and microrotation regime, the entropy involved with the thermodynamic process was modeled to facilitate future thermodynamic optimizations.

Современные подходы к хирургии катаракт с высокой остротой зрения на основании анализа волнового фронта

Relevance. Until recently, visual acuity was the main criterion for determining the timing of surgery, but already in the early stages of cataract development, patients may suffer from reduced image quality, which affects the quality of life (“night” myopia, light scattering, blurring). Deterioration in visual function in these patients may be explained by refractive errors. Analysis of internal higher-order aberrations resulting from slight opacification of the lens is important for determining the effect of media opacity on wavefront aberrations (WF) and for determining the indications and timing of surgical treatment, especially in patients with high visual acuity. Purpose. Evaluate the optical system of the eye with initial cataracts and high visual acuity. Evaluate the wavefront and its aberrations, contrast sensitivity and Shtrel coefficient. Material and methods. The study included 111 eyes with cataracts and 42 eyes without signs of cataracts (control group). Patients aged on average 57±5.3 years. Wavefront analysis was performed using an iTrace ray tracing aberrometer (Tracey Tech., USA) in a dark room under maximum drug-induced mydriasis. In each group, total, corneal, and internal wavefront aberrations were measured at a pupil diameter of 6.0 mm. Higher order aberrations were analyzed using Zernike polynomials (3rd to 5th order). The modulation transfer function (MTF) at different spatial frequencies (0–30 c/deg) and the Strehl coefficient in patients in the study groups were also assessed. Results. Total and internal higher-order wavefront aberrations were significant in different cataract groups compared with control patients, while higher-order corneal aberrations were not statistically different. We noted a higher level of total and internal aberrations in the group of patients with posterior subcapsular cataract. In this regard, patients with posterior subcapsular cataract require surgical treatment at an earlier stage, despite high visual acuity. Conclusion. We recommend assessing the optical quality of vision and wavefront aberrations when observing early cataracts in order to decide whether to perform surgery at an earlier date. Key words: initial cataract, quality of vision, higher order aberrations, internal wavefront aberrations