Separation Of Variables Research Articles

Mathematical Modeling and Applications of Differential Equations in Science and Engineering

Analytically, differential equations are a means of developing relationships between functions that define rates of changes and these rates in terms of the underlying function. They are crucial in modeling diverse, evolutive systems within various domains, to list but a few: physics, engineering, biology, and economics. The fundamentals of differential equations are categorized mainly into Two types of Differential Equations are Ordinary Differential Equations and Partial Differential Equations. Most of the differential equations are solved analytically and numerically techniques such as separation of variables, Fourier series and finite difference techniques. In situations, where exact solutions are tough to come across, computational tools like MATLAB, Maple and the Python libraries help towards approximations. Because they help to explain how systems behave, as well as predicting and improving them, differential equations are an essential tool for expanding the body of scientific and engineering knowledge.

The Possibility of Scientific Philosophy to Systematise the Fundamental Results of Quantum Theory

Richard Feynman wrote in [1]: "The next great era of human intelligence awakening may create a methodology for understanding the qualitative content of equations". However, he did not say how this goal could be achieved. In this paper, this goal was achieved by adopting the fundamental ideas of Descartes' scientific philosophy. That is, the idea that from the very beginning the equation of algebra and arithmetic was taken as the basis of the theory of thought. Then, when the problems of geometry, kinematics, physics are solved...And so that on this way the possibilities of the method of separation of variables and the method of abolition of variables were correctly used. The results which could be obtained on this way just turned out to be the results inherent for quantum geometry, quantum kinematics, quantum physics.... That is results on the basis of which it was really possible to put in order intellectual achievements of mankind.

Impact of spanwise rotation on flow separation and recovery behind a bulge in channel flows

Direct numerical simulations of spanwise-rotating turbulent channel flow with a parabolic bump on the bottom wall are employed to investigate the effects of rotation on flow separation. Four rotation rates, $Ro_b := 2\varOmega H/U_b = \pm 0.42$ , $\pm$ 1.0, are compared with the non-rotating scenario. The mild adverse pressure gradient induced by the lee side of the bump allows for a variable pressure-induced separation. The separation region is reduced (increased) when the bump is on the anti-cyclonic (cyclonic) side of the channel, compared with the non-rotating separation. The total drag is reduced in all rotating cases. Through several mechanisms, rotation alters the onset of separation, reattachment and wake recovery. The mean momentum deficit is found to be the key. A physical interpretation of the ratio between the system rotation and mean shear vorticity, $S:=\varOmega /\varOmega _s$ , provides the mechanisms regarding stability thresholds $S=-0.5$ and $-$ 1. The rotation effects are explained accordingly, with reference to the dynamics of several flow structures. For anti-cyclonic separation, particularly, the interaction between the Taylor–Görtler vortices and hairpin vortices of wall-bounded turbulence is proven to be responsible for the breakdown of the separating shear layer. A generalized argument is made regarding the essential role of near-wall deceleration and resultant ejection of enhanced hairpin vortices in destabilizing an anti-cyclonic flow. This mechanism is anticipated to have broad impacts on other applications in analogy to rotating shear flows, such as thermal convection and boundary layers over concave walls.

Two-phase heat conduction problems with Sturm-type boundary conditions

AbstractIn this work, the solution is justified using the method of separation of variables for initial-boundary value problems of the heat conduction equation with piecewise constant thermal conductivity coefficient under Sturm-type boundary conditions (separated boundary conditions). Various cases are considered. A system of eigenfunctions of the corresponding spectral problem is constructed and it is proved that this system of eigenfunctions forms a Riesz basis. The theorem of the existence of a unique classical solution to the problem is proved.

Efficient matrix algebra encoding for urban solar irradiation simulation: fine-grid ground-level estimation with vector data

Conducting a detailed assessment of solar irradiation at the pedestrian scale on all ground surfaces of a city can assist in identifying cooler routes for pedestrian navigation or preparing the city for potential overheating issues by pinpointing overexposed areas. This article proposes an effective method for conducting such an assessment within a GIS with metric resolution across territories exceeding 100 km². It is based on standard datasets and implements an efficient strategy that relies on separation of variables, domain decomposition, and dimensionality reduction. This strategy involves creating a synthetic representation of the facades of the surrounding buildings (spatial dimension) which accelerates the calculation of shadows based on the sun’s position (temporal dimension). To demonstrate the effectiveness of this method, we applied it to a French city, generating fourteen maps illustrating the solar irradiation of the area for different months of the year or for two given dates with specific weather conditions. The proposed strategy, along with the synthetic representation of building facades, opens up a wide range of possibilities. In addition to synthesizing machine learning labeled datasets, we can also consider calculating solar irradiation with time steps of a few minutes to update weather conditions throughout a journey.

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.

Electromagnetohydrodynamic (EMHD) flow of Jeffrey fluid through parallel plate microchannels with surface charge-dependent asymmetric slip

The investigation of electromagnetohydrodynamic (EMHD) flow of non-Newtonian fluids in microchannels represents a compelling intersection of fluid dynamics, electromagnetism and materials science, particularly demonstrating significant potential and importance in contemporary microfluidic applications. This work delves into the intricate mechanisms governing the EMHD flow of non-Newtonian Jeffrey fluids within parallel plate microchannels, with a specific focus on the influence of asymmetric slip boundary conditions that depend on surface charge. The EMHD flow is propelled by the amalgamation of the Lorentz force and a consistent pressure gradient. Analytical solutions for the velocity and volumetric flow rate of this problem are derived employing both the method of separation of variables and Cramer's rule. Initially, this study undertook comparisons with prior research, leveraging pertinent assumptions to validate the precision of our findings. Subsequently, the discussion shifts to the examination of how velocities are affected by surface charge effects under various slip boundary conditions at different Reynolds numbers. The results suggest that velocities are significantly influenced by surface charge effects under asymmetric slip boundary conditions compared to scenarios involving symmetric and single-wall type, irrespective of the Reynolds number, thereby highlighting the importance of this study. Finally, the relationships among key parameters such as surface charge density, Hartmann number, relaxation time and retardation time affecting volumetric flow rate are analyzed, while approaches and specific percentages for increasing or decreasing volumetric flow rate are provided. One of the more interesting findings is that the volumetric flow rate is expected to experience a 3.9% reduction, considering the influence of the surface charge effect. These findings are not only vital for advancing microfluidics but also have the potential to drive new innovations in related medical technologies, given the notable similarities in the physical properties of Jeffrey fluid and blood.

Hamiltonian integrable systems in a magnetic field and symplectic-Haantjes geometry

We investigate the geometry of classical Hamiltonian systems immersed in a magnetic field in three-dimensional (3D) Riemannian configuration spaces. We prove that these systems admit non-trivial symplectic-Haantjes manifolds, which are symplectic manifolds endowed with an algebra of Haantjes (1,1)-tensors. These geometric structures allow us to determine separation variables for known systems algorithmically. In addition, the underlying Stäckel geometry is used to construct new families of integrable Hamiltonian models immersed in a magnetic field.

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.

Mediterranean Diet Is a Predictor of Progression of Subclinical Atherosclerosis in a Mediterranean Population: The ILERVAS Prospective Cohort Study.

Atherosclerotic cardiovascular disease remains a major health issue, often developing silently as subclinical atherosclerotic disease (SAD). The Mediterranean diet (MDiet) is known for its cardiovascular benefits, but the combined influence of both MDiet adherence and physical activity (PA) on SAD progression has not been previously documented. Objective: We aimed to investigate how adherence to a healthy lifestyle, defined as MDiet adherence and PA level, influences SAD progression in subjects from the ILERVAS cohort follow-up. Methods: A study on 3097 participants from the ILERVAS prospective cohort was conducted. MDiet adherence was assessed using the MEDAS score, and PA categories were established using the IPAQ, both categorized into low, moderate, and high levels. Two different lifestyle scores integrating the MDiet and PA categories were built. The presence of atherosclerotic plaques was assessed by carotid and femoral ultrasound examination. Demographic, clinical, and biochemical data were also obtained. Multivariable linear, logistic, and Poisson regression models adjusted for potential confounders were used to analyze the association between the lifestyle scores and SAD progression, as well as the MDiet and PA as separate variables and number of territories with plaque. Results: A healthier lifestyle score did not show an effect on SAD progression. However, a higher MEDAS score was associated with a 3% decrease in the number of territories with plaque (IRR 0.97, 95% CI 0.96-0.99, p < 0.001), suggesting a protective effect of the adherence to the MDiet. PA did not show a significant association (IRR 1.00, 95% CI 1.00-1.00, p = 0.269). Older age, hypertension, dyslipidemia, smoking, and lower eGFR were associated with SAD progression, while the female sex was protective (IRR 0.67, 95% CI 0.63-0.72, p < 0.001). Conclusions: The findings of this study show that higher adherence to the MDiet is associated with reduced incidence of SAD, indicating its potential role in cardiovascular prevention strategies. Although a higher lifestyle score or physical activity levels did not show any significant effect, promoting the MDiet, alongside managing traditional cardiovascular risk factors, could be an effective public health intervention to prevent atherosclerosis and reduce the burden of cardiovascular disease.

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.