Separation Of Variables Research Articles

Sturm–Liouville problem for a one-dimensional thermoelastic operator in Cartesian, cylindrical, and spherical coordinate systems

The problem of constructing eigenfunctions of a one-dimensional thermoelastic operator in Cartesian, cylindrical, and spherical coordinate systems is considered. The corresponding Sturm–Liouville problem is formulated using Fourier’s separation of variables applied to a coupled system of thermoelasticity equations, assuming that the heat transfer rate is finite. It is shown that the eigenfunctions of the one-dimensional thermoelastic operator are expressed in terms of well-known trigonometric, cylinder, and spherical functions. However, coupled thermoelasticity problems are solved analytically only under certain boundary conditions, whose form is determined by the properties of the eigenfunctions.

THE MEDIATING ROLE OF EARNINGS MANAGEMENT IN THE RELATIONSHIP BETWEEN THE FIRM LIFE CYCLE AND THE COST OF EQUITY: EVIDENCE FROM BANKS IN IRAQ

The present study seeks to examine the interaction among three separate variables. The initial variable relates to the organization's life cycle and financial attributes. The second variable pertains to the procedures of firm management that influence performance and financial condition, as demonstrated by earnings management. Investors manage the impact of the previous two factors by weighing the risks and potential gains, which the corporation terms as "the cost of equity." We utilized these variables to analyze 15 Iraqi commercial banks that have been listed on the Iraq Stock Exchange for 10 years, equivalent to 150 observations.Our quantitative assessment of the study variables, conducted using models from the relevant literature, has yielded significant findings. We have uncovered a direct correlation between the life cycle and earnings management, indicating a beneficial association. Moreover, we have revealed that each factor exerts a unique and favourable influence on the cost of equity. The correlation between the life cycle and earnings management has led to an increase in the favourable influence of earnings management on the cost of equity. Conversely, the life cycle now has an inverse impact on the cost of equity, with the cost of equity being mainly influenced by the maturity stage. These findings have important implications for stock market traders, as they guide them to direct their investments towards banks that offer higher returns relative to the level of risk, considering the age group and the administration's approach to managing announced profits. This study's implications for stock market traders are significant, as it provides them with valuable insights for making informed investment decisions. It also serves as a guide for future studies in this field.

Two-Layer Electroosmotic Flow in a Parallel Plate Microchannel with Sinusoidal Corrugation

This study investigates the electroosmotic flow (EOF) of a two-layer Newtonian fluid system in a parallel plate microchannel with sinusoidal corrugated walls. The upper fluid is conducting, while the lower fluid is nonconducting. This analysis is performed under the Debye–Hückel approximation, utilizing perturbation expansion and the separation of variables. The potential distribution, velocity field, and the dependence of average velocity on roughness are derived. It is observed that the velocity distribution w(x, y), is significantly influenced by the phase difference θ between the corrugations on the upper and lower walls. The velocity w(x, y) decreases with an increase in the viscosity ratio μr of the bottom to top fluid, and w(x, y) is directly proportional to the dimensionless pressure gradient G and the zeta potential ratio ζ. The variation of the average velocity increment (roughness function) u2m related to wall roughness tends to decrease with the increase of the corrugation wave number λ, the electrokinetic width K, the depth ratio hr of the bottom to top fluid, the zeta potential ratio ζ and the dimensionless pressure gradient G; and increases with the increase of the viscosity ratio μr of the bottom to top fluid. Furthermore, the effect of uI2m is smaller than that of uII2m.

Hamiltonian System-Based Symplectic Framework for Analytical Vibration Analysis of Microplates

Abstract The classical elasticity is inadequate for the dynamic analysis of microplates due to the size effect. This study incorporates a higher-order strain gradient theory into the Hamiltonian system-based symplectic framework and derives new analytical solutions for the free vibration of microplates. The analytical solutions are obtained using rigorous mathematical techniques, including separation of variables, symplectic eigen expansion, and superposition, without relying on predetermined solution forms. Hence, they are not restricted to Lévy-type boundary conditions. Using these analytical solutions, we present comprehensive vibration results for microplates and perform detailed parametric studies to explore the impact of length scale parameters on the natural frequencies. Given the growing demand for microplates in advanced engineering applications, the obtained analytical solutions are expected to facilitate their design and performance optimization.

From Observers to Interveners: Examining Bystander Intentions in Facebook Live Videos

This study investigated the bystander effect by examining intentions to intervene in simulated Facebook Live scenarios. A 4 (number of other viewers: 0 vs. 4 vs. 49 vs. 224) x 2 (level of urgency: non-urgent vs. urgent video) between-subjects experimental design was conducted with 210 participants to examine how the number of bystanders and the perceived urgency of a situation influence self-reported bystander intentions. With individual empathy levels controlled, an ANCOVA test revealed no significant main effect of the number of other viewers on the intention to intervene, but a significant difference was found between the 0 and 224 viewer conditions. Urgent situations elicited higher intervention intentions compared to non-urgent ones, without an interaction effect with viewer number. With the number of other viewers and urgency level as separate independent variables, mediation analysis found patterns differing from the traditional five-step bystander intervention model. These findings enhance our understanding how bystander intentions manifest in social media contexts and highlight the influence of perceived danger and audience size. The implications for online bystander behavior and future research directions are discussed.

Approximate Solutions of the Boussinesq Equation for Horizontal Unconfined Aquifers During Pure Drainage Phase

In this work, conceptual approximations of the Boussinesq equation were introduced and analyzed, resulting into a very accurate and well-applicable model for horizontal unconfined aquifers during the pure drainage phase, without any recharge and zero-inflow conditions. The model was constructed by employing a variety of methods that included wave solution, variable separation, and series expansion, and its analysis and performance against the Boussinesq equation, at early and later times, providing fruitful insights enlightening the main mechanisms and physical characteristics of the drainage phase. The modeled non-linear forms were finally linearized, concluding with explicit analytical expressions that accurately incorporated most of the basic characteristics regarding the evolution of the water table and the outflow from the exact Boussinesq equation under different initial conditions. The endeavors of this work can be utilized for theoretical and modeling purposes related to this problem.

Analytical and numerical calculation and simulation of heat transfer in a composite wall made of polyethylene terephthalate bottle

The dynamics of heat transfer in composite walls made of Polyethylene Terephthalate (PET) bottles is studied using the energy conservation model. Ten walls of different structures were studied, including two control walls made of compressed earth brick (CEB) and cementitious material, cement-sand brick (CSB) and eight composite walls made of PET bottles filled with thermal insulation and/or mechanical stabilisation materials. The model equation is solved analytically using the separation of variables method and numerically using the finite difference method, and they are confirmed using CASTEM finite element software. As boundary conditions, a constant temperature (Text) is applied at one end of the wall, and the constant temperature (T0) is applied at any other point at the initial time. The effect of PET bottles filling on the thermal behaviour of the wall was then carried out to determine the optimum design that would give the wall better thermal inertia. The results obtained with CASTEM is fairly in agreement with the results obtained analytically and numerically, with a maximum relative error closed to that of the spatial evolution and 3.310−2 for the temporal one. It is found that heterogeneous filling of PET bottles provides better thermal inertia at the wall.

Variable Splitting and Fusing for Image Phase Retrieval.

Phase Retrieval is defined as the recovery of a signal when only the intensity of its Fourier Transform is known. It is a non-linear and non-convex optimization problem with a multitude of applications including X-ray crystallography, microscopy and blind deconvolution. In this study, we address the problem of Phase Retrieval from the perspective of variable splitting and alternating minimization for real signals and seek to develop algorithms with improved convergence properties. An exploration of the underlying geometric relations led to the conceptualization of an algorithmic step aiming to refine the estimate at each iteration via recombination of the separated variables. Following this, a theoretical analysis to study the convergence properties of the proposed method and justify the inclusion of the recombination step was developed. Our experiments showed that the proposed method converges substantially faster compared to other state-of-the-art analytical methods while demonstrating equivalent or superior performance in terms of quality of reconstruction and ability to converge under various setups.

Nonlinear mechanical response of UHSS rectangular plate under thermo-mechanical-metallurgical coupling

The mechanical properties of ultrahigh-strength steel (UHSS) produced through the gradient thermoforming process (GTP) are significantly influenced by residual stress and deformation. However, the precise distribution of these factors during GTP remains unclear. To address this issue, a detailed thermal model is developed, incorporating key heat transfer mechanisms, including coupled heat conduction and radiation, using the method of separation of variables. Furthermore, the displacement and stress fields of UHSS rectangular plates are derived using the differential quadrature method (DQM), accounting for phase transitions and external load. The combination of the separation of variables and DQM methods significantly improves computational efficiency. Furthermore, this study presents the first analysis of the combined effects of thermal load, phase transitions, and external forces on UHSS rectangular plates. Finally, through a combination of theoretical analysis, experimental validation, and case studies, the influence of material properties, geometric parameters, and external load on the temperature, stress, and displacement fields is examined. The findings reveal that heating rate, phase transitions, and external load play a critical role in the stress and displacement distribution in UHSS plates during GTP. This research provides a theoretical foundation for optimizing GTP process parameters and material design, ultimately enhancing the quality and performance of UHSS components.

ANALYSIS OF SH-WAVES IN ANISOTROPIC FIBER-REINFORCED MEDIUM OVER LINEARLY VARYING INHOMOGENEOUS SUBSTRATE UNDER NON-LOCAL ELASTICITY

An analytical technique employing the variable separable method has been used in an attempt to precisely grasp and assess the impact of the property non-local elasticity upon the wave transmission response of an anisotropic fiber-reinforced material embedded over a semi-infinite, inhomogeneous medium that changes linearly. As depth increases, the stiffness and density of a semi-infinite substrate are believed to alter linearly. Availing the Whittaker function, a relation about dispersion has been acquired to analyze the response of SH waves. The visual representation depicts a significant influence of non-local elasticity on the propagation of SH-wave modes. Special cases have been found assessing the concurrency of the model with the original form equation of Love wave. The effects of non-locality, fiber-reinforcement parameters, and inhomogeneity parameters carry implications in designing and gradation of material characteristics some important parameters on the wave characteristics of the studied model.

Love wave propagation on nano-sized layered piezoelectric plate clamped on micropolar substrate

This research paper investigates the complex dynamics of Love wave propagation on a piezoelectric plate situated atop a micropolar elastic half-space having imperfect interface. The study delves into the multifaceted effects of flexoelectricity, micro-inertia, guiding layer thickness, and interfacial imperfection on the characteristics of Love waves. The electromechanical coupling factor which is significant in analyzing SAW device performances is also studied. Variable separable method is used to solve the governing equations and desired solution is obtained analytically. The findings contribute to a deeper understanding of Love wave propagation in complex elastic structures and development of advanced sensors and communication technologies.

Heat equation for Sturm–Liouville operator with singular propagation and potential

Abstract This article considers the initial boundary value problem for the heat equation with the time-dependent Sturm–Liouville operator with singular potentials. To obtain a solution by the method of separation of variables, the problem is reduced to the problem of eigenvalues of the Sturm–Liouville operator. Further on, the solution to the initial boundary value problem is constructed in the form of a Fourier series expansion. A heterogeneous case is also considered. Finally, we establish the well-posedness of the equation in the case when the potential and initial data are distributions, also for singular time-dependent coefficients.

Free vibrations of a Continuous-discrete multi-span Beam taking into account inertial Rotation Forces

Objective. The aim of the study is to estimate free vibrations of a continuousdiscrete multi-span beam taking into account the inertial forces of rotation. The goal is to determine the spectra of natural frequencies, damping coefficients and natural modes. Method. The study is based on the methods of linear mechanics of structures; numerical and numericalanalytical calculation methods. The solution to the problem is found using the method of separation of variables. Rotational movements of particles of continuous sections are taken into account according to one of the models of the Timoshenko beam. The D'Alembert principle and hypotheses on the smallness of displacements and angles of rotation of sections are used. Result. A system of equations in matrix-vector form is obtained. The mathematical model of transverse vibrations consists of three systems of differential equations. The equation includes transverse forces, external concentrated forces, d'Alembert inertial forcesi, and linear-viscous resistance forces. The inertial forces of rotation of concentrated masses are taken into account. The boundary and other additional conditions to the equations correspond to the calculation scheme. The left end of the beam is hinged. The conjugation conditions are met at the junction of the sections. Conclusions. This random process of disturbances is very close to the processes used in deterministic problems. The amplitudes and standard deviations of displacements in the deterministic and stochastic problems almost coincide, which confirms the reliability of the proposed calculation theory. Analysis of the curves shows that the standard deviations significantly depend on the degree of correlation of the components of the vector random process of disturbances. The use of modern computing computer systems such as Matlab allows us to successfully combine the advantages of both numerical and graphical methods.

Reduction in wave shoaling over a linear transition bottom using a porous medium

Wave shoaling, which involves an increase in wave amplitude due to changes in water depth, can damage shorelines. To mitigate this damage, we propose using porous structures such as mangrove forests. In this study, we use a mathematical model to examine how mangroves, acting as a porous breakwater, can reduce wave shoaling amplitude. Shallow water equations are used as the governing equations and are modified to account for the presence of porous media. To measure the wave reduction generated by the porous media, the wave transmission coefficient is estimated using analytical and numerical approaches. The separation of variables method and the staggered finite volume method are utilized for each approach, respectively. The numerical results are then validated against the previously obtained analytical solutions. We then vary the friction and porosity parameters, influenced by the presence and extent of porous media, to evaluate their effectiveness in reducing wave shoaling.

Peakons and compactons of the (2+1)-dimensional modified dispersive water-wave system

For a higher-dimensional nonlinear dynamical system, there exist abundant coherent excitations. The variable-separated method is a powerful approach to deriving these structures, as its solutions allow for arbitrary functions. Previous works have produced numerous results, including solitons, chaos and fractals. As the molecule structure appears, constructing the multi-soliton molecule through this technology is a meaningful work, especially considering the local peakons and compactons that were seldom discussed before. In this paper, after taking the Bäcklund transformation, the variable-separated solution for the (2+1)-dimensional modified dispersive water-wave system is first derived, which is an important physical model in describing the nonlinear and dispersive long gravity waves. As a result, the multi-peakons and multi-compactons are constructed through the derived universal formula with the aid of the variable functions p and q. These solitons include two general clusters of M × N peakons and compactons, from which the multi-soliton molecules and their interactions are presented.

ЗАДАЧА ОПТИМАЛЬНОГО СТОХАСТИЧЕСКОГО УПРАВЛЕНИЯ И ОЦЕНКИ СТОИМОСТИ КОРПОРАТИВНЫХ ОБЛИГАЦИЙ, ЗАВИСЯЩИХ ОТ ВРЕМЕНИ И СЛУЧАЙНЫХ СОСТОЯНИЙ ПАРАМЕТРОВ

In the optimal stochastic control problem, one estimates, using the utility indifference method, the bond price and the premium of a default swap contract (Credit default swap) when the model parameters (market interest rate, drift coef-ficient, and volatility of risky underlying assets) are random functions of time and state. Namely, the interest rate of the risk-free asset depends on time, and the prices of risky assets are described by linear homogeneous stochastic dif-ferential equations (SDEs) with multiplicative noise. To do this, the authors consider a portfolio with a risk-free asset and a risky asset with no default risk. For each such portfolio, the authors determine the amount of risky assets that maximizes the expected utility of its final wealth. This quantity allows one to solve the parabolic partial differential equations arising from the Hamilton-Jacobi-Bellman (HJB) equation by transforming them into ordinary differential equations using the method of variable separation to obtain the instantaneous value function of each portfolio. The authors derive the bond price and the default swap-contract premium (CDS) rate, which are the amounts that provide the same level of the expected utility by investing all of one’s wealth in a portfolio that does not contain these credit instruments, or by investing these amounts in credit instruments and the remainder of one’s wealth in the portfolio.

Dynamic Modelling and Performance Analysis of Deployable Telescopic Tubular Mast

AbstractThis work presents the longitudinal and transverse coupling vibrations of a deployable Telescopic Tubular Mast (TTM), a multi-stepped structure integrated into a spacecraft system, while considering the rigid-flexible coupling phenomenon. The model is derived using the principle of virtual work and discretized via the variable separation method. The von Kármán strain is employed to incorporate geometric nonlinear effects. Semi-analytical results for the shape functions and natural frequencies of the quasi-static multi-stepped boom are obtained using the extended transfer matrix method (ETMM). These natural frequencies are validated against results from Nastran, confirming the ETMM's accuracy. In addition, the model accounts for the continuously changing natural frequencies and shape functions during the deployment phase. Finally, the dynamic phenomena of the longitudinal and transverse displacements are analyzed at various deploying states, including locking and restart behaviors. The influence of the structural damping on the vibration evolution is also contained in the numerical analysis.

Dynamics of Love-type wave propagation in composite transversely isotropic porous structures

The present study aims to analyze the propagation behavior of Love-type wave in a composite transversely isotropic porous structure. The structure comprises an inhomogeneous sandy porous layer lying between a non-homogeneous magneto-poroelastic layer and a heterogeneous fractured porous half-space. Analytic solutions of the field equations of the respective media involve the application of the variable separable method and Wentzel-Kramers-Brillouin (WKB) asymptotic approach for the conversion of partial differential equations into ordinary differential equations. Through careful imposition of boundary conditions and subsequent elimination of arbitrary constants, we derive a complex dispersion relation governing the propagation of Love-type waves. This dispersion equation yields both the phase velocity curve, corresponding to the real expression, and the damping velocity curve, derived from the imaginary expression. To represent our findings, we conduct extensive calculations and graphical simulations illustrating the influence of various material parameters such as heterogeneity, porosity, volume fraction of fractures, sandiness, magnet-oelastic coupling, angle at which wave crosses the magnetic field, and layer thickness on the dispersive nature of Love-type waves using MATHEMATICA software. Furthermore, we conduct case-specific analyses, revealing instances where the dispersion equation simplifies to the standard Love wave equation, thereby validating our mathematical framework. Our findings underscore the significant influence of the aforementioned material parameters on the phase and damping velocities of Love-type wave. This interdisciplinary investigation into different porous media opens new avenues for future research and has significant implications in various disciplines, ranging from engineering and geophysics to environmental science and beyond.

Mass and Heat Exchange in Rotating Compressor Cavities With Variable Cob Separation

Abstract Next generation aeroengines will operate at ever-increasing pressure ratios with smaller cores, where the control of blade-tip clearances across the flight cycle is an emerging design challenge. Such clearances are affected by the thermal expansion of the compressor disks that hold the blades, where acute thermal stresses govern operating life. The cavities formed by corotating disks feature a heated shroud at high radius and cooler cobs at low radius. A three-dimensional, unsteady and unstable flow structure is induced by destabilizing buoyancy forces. The radial distribution of disk temperature is driven by a conjugate heat transfer at Grashof numbers of order 1013. Such flows are further influenced by the heat and mass exchange with an axial throughflow of cooling air at low radius, where the interaction depends on the Rossby number and separation of the disk cobs. This paper is the first to study the effect of cob separation ratio on mass and heat exchange for compressor cavities. A model is developed to predict the cavity-throughflow interaction, and disk and fluid-core temperatures. The judicious use of a physics-based methodology provides reliable, reduced-order solutions to the complex conjugate problem, thereby making it appropriate for practical engine thermo-mechanical design. The model is validated by detailed experimental measurements using the Bath Compressor Cavity Rig, where variable disk cob spacings were investigated over a range of engine-representative conditions. The unsteady pressure measurements collected in the frame of reference of the rotating disks reveal new insight into the fundamentally aperiodic nature of the flow structure. This new understanding of heat transfer informs an expedient reduced-order model and enables more efficient design of future high pressure-ratio aeroengines.

New exact solutions of some (2+1)-dimensional nonlinear evolution equations and folding waves

By means of the Hirota’s bilinear method and special multi-linear variable separation ansatz, new exact solutions with low dimensional arbitrary functions of some (2+1)-dimensional nonlinear evolution equations are constructed. That is, we propose a unified method for solving the mNNV-type equations and the Burger-type equations. The key factor to the success of this method is that we have constructed some simplified Hirota’s bilinear calculation formulas in the form of variable separation of arbitrary order. Appropriate multi-valued functions are used to construct coherent structures such as the bell-type, peak-type and loop-type folding waves.