Separation Of Variables Research Articles

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.

Analytical Solutions for a Fully Coupled Hydraulic‐Mechanical‐Chemical Model With Nonlinear Adsorption

ABSTRACTAdsorption characteristics play a crucial role in solute transport processes, serving as a fundamental factor for evaluating the performance of clay liners. Nonlinear adsorption isotherms are commonly found with metal ions and organic compounds, which introduce challenges in obtaining analytical solutions for solute transport models. In this study, analytical solutions are proposed for a fully coupled hydraulic‐mechanical‐chemical (HMC) model that accounts for both the Freundlich and Langmuir isotherms. To mitigate the difficulties arising from the variable coefficients, the system of second‐order partial differential equations involving three variables is linearized. The method of separation of variables, theory of integration, and Fourier series are utilized to derive analytical solutions. The analytical method presented can potentially be extended to a broad spectrum of nonlinear adsorption isotherms. The results reveal a 56.5% reduction in solute breakthrough time under the Freundlich isotherm and a remarkable 2.6‐fold extension under the Langmuir isotherm when compared to the linear isotherm. The adsorption constants of the Freundlich and Langmuir isotherms exhibit a positive correlation with breakthrough time, while the exponent of the Freundlich isotherm and the maximal adsorption capacity in the Langmuir isotherm demonstrate a negative association with breakthrough time. This study enhances the precision of solute transport prediction and provides a more scientific assessment of clay liner performance.

An Analytical Insight Into Stability Analysis of Unsaturated Multi‐Layered Slopes Subjected to Rainfall Infiltration

ABSTRACTSlopes in nature usually present layered characteristics, and its stability is susceptible to rainfall events. Considering that current analytical solutions are only suited to simulate the rainfall infiltration of double‐layered infinite unsaturated slopes, an analytical procedure is hence proposed in this study to tackle the consideration of multiple layers. The variable separation method and transfer matrix method are combined to derive the analytical solution of pore water pressure (PWP) for simulating rainfall infiltration in layered infinite unsaturated slopes. After having validated the proposed model and analytical solutions by comparing with existing literature and numerical simulation, the closed‐form solution of PWP is incorporated into the limit equilibrium for assessing slope stability. A three‐layer slope is selected as an example for further discussion. PWP distribution and factor of safety are calculated, considering the effects of saturated hydraulic conductivity and thickness of the upper layer, intensity of antecedent and subsequent rainfall, and varied soil unit weight along the depth. The slope stability subjected to rainfall effects is consistent with the variation of PWP. The proposed analytical solutions provide a simple and practical avenue for computing PWP distribution and evaluating the stability of multi‐layered slopes under rainfall conditions, which can also serve as a benchmark for numerical solutions.

Symmetric structures and dynamic analysis of a (2+1)-dimensional generalized Benny-Luke equation

We study the symmetric structures and dynamic analysis of a (2 + 1)-dimensional generalized Benny-Luke (GBL) equation based on the Lie symmetry method, the GBL equation is an important non-integrable model of water waves. Specifically, we construct multiple exact solutions of the GBL equation and obtain its nonlocally related systems. Firstly, the Lie point symmetries and conservation laws of the GBL equation are computed, and then we get the reduced ordinary differential equation from one of the conservation laws. Multiple methods, for example, the dynamical systems method, the power series method, the homogeneous balancing method and generalized variable separation method, are used to solve the ordinary differential equation and abundant exact solutions of the GBL equation are got. Finally, we extend these exact solutions by discrete symmetries, and give three-dimensional graphs of partial exact solutions. In addition, we construct the nonlocally related PDE systems, which contains the potential systems from the conservation laws and an inverse system from a Lie point symmetry of the GBL equation. These findings reveal the dynamical behavior behind the GBL equation and broaden the range of nonlinear water wave model solutions.

Nanoparticle uptake by a semi-permeable, spherical cell from an external planar diffusive field. II. Numerical study of temporal and spatial development validated using FEM

In this paper, we present a mathematical study of particle diffusion inside and outside a spherical biological cell that has been exposed on one side to a propagating planar diffusive front. The media inside and outside the spherical cell are differentiated by their respective diffusion constants. A closed form, large-time, asymptotic solution is derived by the combined means of Laplace transform, separation of variables, and asymptotic series development. The solution process is assisted by means of an effective far-field boundary condition, which is instrumental in resolving the conflict of planar and spherical geometries. The focus of the paper is on a numerical comparison to determine the accuracy of the asymptotic solution relative to a fully numerical solution obtained using the finite element method. The asymptotic solution is shown to be highly effective in capturing the dynamic behaviour of the system, both internal and external to the cell, under a range of diffusive conditions.

Torsion of end-bearing pile foundations in layered anisotropic geomaterials

ABSTRACT An analysis for the torsional dynamic response of end-bearing pile foundations embedded in a layered transversely isotropic geomaterial (soil/rock) is presented. The deformation of the transversely isotropic soil or rock is described by the method of separation of variables. The elasticity theory for a viscoelastic medium with frequency independent hysteretic material damping, and the Extended Hamilton’s Principle are utilised to derive the differential equations describing pile and soil motions. The differential equations are solved analytically in an iterative algorithm. The accuracy of the analysis is verified with existing studies reported in the literature for pile foundations embedded in a homogeneous and layered soil deposit. The effect of the degree of anisotropy on the pile-soil response – dynamic pile-head stiffness, distribution of pile rotation and torque with depth, dimensionless soil displacement function for various values of pile slenderness and pile-soil stiffness ratios in a homogenous soil deposit is investigated. Design charts of static pile-head stiffness in a homogeneous soil deposit for a wide range of pile-soil stiffness and pile slenderness ratios, and degree of anisotropy are also reported. The effect of soil layering for a pile embedded in a two-layered soil deposit is also studied.

Nonlinear dynamics of a flexible rod partially sliding in a rigid sleeve under the action of gravity and configurational force

We investigate various methods of analyzing systems with moving boundaries, using as an example a flexible rod sliding in an ideal frictionless sleeve in the field of gravity. Special attention is paid to the configurational force acting on the rod at the sleeve opening and thus determining the rod’s dynamics. The non-material kinematic description used in simulations is based on the re-parametrization of the Lagrangian arc length coordinate. The variational formulation uses the energy expressions written for the entire rod, comprising the free segment and the one inside the sleeve. A novel finite element scheme is efficient for highly flexible rods, which may undergo complete ejection. A simplified two degrees of freedom model, which accelerates simulations, shows a good agreement as the bending stiffness increases. An analytical study using Hamiltonian mechanics exploits the separation of variables into fast oscillations and slow axial motion. The adiabatic invariant approach leads to approximate closed-form solutions for the slow dynamics and yields the maximum injection depth of the rod into the sleeve.

A study on possible risk factors for progressive supranuclear palsy in southern part of India.

Background: The etiological factors leading to progressive supranuclear palsy (PSP) are poorly understood. This study aims to evaluate the role of various risk factors in patients with PSP. Methods: A case-control study was conducted over a period of two years from March 2016 to March 2018. The cases were recruited independently by two senior neurologists and a consensus was then reached after discussion for their inclusion. The controls were free of parkinsonian features or dementia and matched by age (± 3 years), sex, and race with the cases. The study population was then interviewed using a standard questionnaire for various possible risk factors. Variables with a significance (P ≤ 0.05) in univariate analysis were considered for bivariate analysis, multivariate analysis, and logistic-regression analysis. Results: A total of 51 cases with an equal number of controls were included in this study. Ten separate variables that included poor educational status, well water, smoking, tapioca, bakery/fast food, tea ≥ 5 cups/day, personality, exposure to pets, exposure to cattle, and family history of stroke were found to show statistical significance after univariate analysis. Among these, tapioca consumption, fast food and bakery items consumption, type A personality, and family history of stroke were found significant after adjusting for the confounding factors. Conclusion: The possible etiological factors that have a relevance in the causation of PSP as borne out in our study include dietary habits such as tapioca, fast food, and bakery items consumption, family history of stroke, and type A personality trait.

Optimal design of laminated thermoelectric cylindrical device based on neural network and thermo-electric-mechanical coupled model

The laminated thermoelectric cylindrical shell is favorable for solar thermal energy harvesting, but the thermo-electric-mechanical coupling field is relatively complex. An optimization design model of thermoelectric structure is required to promote its practical application. In this paper, based on variable separation method, thermo-electric-mechanical coupled model of laminated thermoelectric cylindrical device under thermal shock is constructed. Transient temperature field, thermal stress and power generation efficiency are obtained. Numerical results show that both power generation efficiency and thermal stress are affected by material parameters simultaneously. In order to analyze efficiency and structural reliability comprehensively, a fitness function is introduced in this paper. Based on the data which are collected from thermoelectric coupled model, a neural network is established and trained, which can predict and evaluate the performance of thermoelectric cylindrical shell based on the fitness function. The optimal material parameters for thermoelectric shells with high efficiency and low thermal stress are obtained. Combining neural network technique and the theoretical model developed in this paper, the computation time of numerical results can be reduced from 50 to 3 min, and this neural network optimization method has higher computational efficiency. These research results will provide guidance for the optimization design of cylindrical thermoelectric device.

Optimal design and placement of a combined trench and submerged breakwater system on the coastal area of Aceh

Tsunami waves are currently one of the most devastating water hazards that threaten regions or countries near the subduction zones, including Indonesia. In this research, a novel coastal protection system combining a submerged breakwater and a trench is proposed to mitigate tsunami occurrences. The model is developed based on the Shallow Water Equations, which are then solved analytically using the Separation Variables Method, and numerically using the Staggered Finite Volume Method. The wave transmission coefficient, which indicates the wave reduction rate generated by the system, is expected to be the outcome of both approaches. The numerical scheme is validated against the analytical solutions to verify its accuracy in estimating wave reduction by the system. Further, an optimization technique is integrated into the model to obtain the optimal size and position of each structure (submerged breakwater and trench) on the real bathymetry of Aceh's coast, which has been impacted by a deadly tsunami occurrence before. The results of this study are expected to be useful in the future development of coastal protection systems against tsunami waves, especially in Indonesia.