Separation Of Variables Research Articles

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.

Analysis on acoustic modes and Brillouin gain spectra of backward stimulated Brillouin scattering in W-type acoustic waveguide optical fibers.

The acoustic Helmholtz equations of W-type acoustic waveguide fibers, including WI- and WII-type acoustic velocity of vl,2 < vl,3 < vl,1 and vl,2 < vl,1 < vl,3, separately are solved by using the method of separation of variables, and their characteristic equations are derived according to the boundary condition and the acoustic Helmholtz equations. The distribution and cut-off of acoustic modes are analyzed by introducing acoustic normalized frequencies. The dependence of the acoustic inner core radius, the acoustic velocities in the acoustic inner and acoustic outer core on acoustic modes, and Brillouin gain spectra (BGS) is investigated. The results indicate that two significant peaks of the BGS are induced by acousto-optic coupling of L01 and L02 acoustic modes with LP01 optical mode. The acousto-optic overlap factor of the L01 acoustic mode decreases first and then increases, and that of the L02 acoustic mode increases first and then decreases with the rise of acoustic inner core radius. Due to the difference of the acoustic mode field distribution, the gain peak induced by L01 mode is sensitive to the acoustic velocity of the acoustic inner core, and that of the L02 mode is sensitive to the acoustic velocity of the acoustic outer core. These conclusions agree well with the reported experimental results, and provide theoretical references for research and application of backward stimulated Brillouin scattering of acoustic waveguide optical fibers.

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.

Electromagnetohydrodynamic flow and thermal performance in a rotating rough surface microchannel

Rough surfaces in microchannels effectively enhance liquid mixing, thermal performance, and chemical reactions in electrically actuated microfluidic devices. Rotation of the microchannel with surface roughness intensifies this enhancement. We investigate the combined effects of electromagnetohydrodynamics and surface roughness on transient rotating flow in microchannels. We present a mathematical model considering the variable zeta potential, heat transfer characteristics, and entropy generation within the microchannel. We obtain analytical solutions using the separation of variables method and Fourier series expansion. The surface roughness of the microchannel, when combined with rotation, impacts the temperature enhancement. Higher rotation rates result in the formation of multiple vortices. The secondary flow pushes the primary velocity toward the boundary layer, which affects the flow pattern. Surface roughness and electroosmotic flow significantly affect secondary flow, resulting in complex flow patterns and reversals. The interaction between centrifugal and viscous forces results in maximum velocities at the boundary layers. Higher roughness and electromagnetic effects enhance temperature by intensifying fluid-solid friction and joule heating. Surface roughness causes an increase in wall shear stress and friction factor, resulting in a higher Poiseuille number. Moreover, surface roughness increases entropy production by enhancing fluid mixing and internal friction despite improved heat transfer. Higher rotation also elevates entropy generation due to additional vortices induced by secondary flow.

Boundary control problem for a parabolic equation with involution

In this paper, we consider a boundary control problem for a parabolic equation with involution in a bounded one-dimensional domain. On the part of the border of the considered domain, the value of the solution with control function is given. Restrictions on the control are given in such a way that the average value of the solution in the considered domain gets a given value. The problem given by the method of separation of variables is reduced to the Volterra integral equation of the first kind. The existence of the control function was proved by the Laplace transform method.

Novel localized waves and dynamics analysis for a generalized (3+1)-dimensional breaking soliton equation

PurposeThis paper aims to explore new variable separation solutions for a new generalized (3 + 1)-dimensional breaking soliton equation, construct novel nonlinear excitations and discuss their dynamical behaviors that may exist in many realms such as fluid dynamics, optics and telecommunication.Design/methodology/approachBy means of the multilinear variable separation approach, variable separation solutions for the new generalized (3 + 1)-dimensional breaking soliton equation are derived with arbitrary low dimensional functions with respect to {y, z, t}. The asymptotic analysis is presented to represent generally the evolutions of rogue waves.FindingsFixing several types of explicit expressions of the arbitrary function in the potential field U, various novel nonlinear wave excitations are fabricated, such as hybrid waves of kinks and line solitons with different structures and other interesting characteristics, as well as interacting waves between rogue waves, kinks, line solitons with translation and rotation.Research limitations/implicationsThe paper presents that a variable separation solution with an arbitrary function of three independent variables has great potential to describe localized waves.Practical implicationsThe roles of parameters in the chosen functions are ascertained in this study, according to which, one can understand the amplitude, shape, background and other characteristics of the localized waves.Social implicationsThe work provides novel localized waves that might be used to explain some nonlinear phenomena in fluids, plasma, optics and so on.Originality/valueThe study proposes a new generalized (3 + 1)-dimensional breaking soliton equation and derives its nonlinear variable separation solutions. It is demonstrated that a variable separation solution with an arbitrary function of three independent variables provides a treasure-house of nonlinear waves.

The exact relativistic scalar quasibound states of the dyonic Kerr–Sen black hole: quantized energy, and Hawking radiation

We consider Klein–Gordon equation in the Dyonic Kerr–Sen black hole background, which is the charged rotating axially symmetric solution of the Einstein–Maxwell–Dilaton–Axion theory of gravity. The black hole incorporates electric, magnetic, dilatonic and axionic charges and is constructed in 3+1 dimensional spacetime. We begin our investigations with the construction of the scalar field’s governing equation, i.e., the covariant Klein–Gordon equation. With the help of the ansatz of separation of variables, we successfully separate the polar part, and find the exact solution in terms of Spheroidal Harmonics, while the radial exact solution is obtained in terms of the Confluent Heun function. The quantization of the quasibound state is done by applying the polynomial condition of the Confluent Heun function that gives rise to discrete complex-valued energy levels for massive scalar fields. The real part is the scalar field relativistic quantized energy, while the imaginary part represents the quasibound states’s decay. We present all of the sixteen possible exact energy solutions for both massive and massless scalars. We also present the investigation the Hawking radiation of the Dyonic Kerr–Sen black hole’s apparent horizon, via the Sigurd–Sannan method by making use of the obtained exact scalar wave functions. The radiation distribution function, and the Hawking temperature are also obtained.

New Analytical Solutions for Free Vibration of Cracked Functionally Graded Rectangular Plates

In aerospace engineering, the existence of cracks highly alters the mechanical behaviors of structural components. As a new class of key structural components, functionally graded material (FGM) rectangular plates with an interior/side crack are focused on in this study. New analytical free vibration solutions are put forward for such cracked FGM plates. First, a cracked plate is decomposed into several subplates with the necessary continuity conditions to address the physical discontinuity issue caused by the crack. The challenge then becomes deriving analytical free vibration solutions for each subplate. To accomplish this task, a symplectic superposition method (SSM) is developed to deal with FGM rectangular plates with various boundary conditions, including the complex continuity conditions. The SSM eliminates the need for a predefined solution form, and its solution procedure, including variable separation and eigen expansion, is mathematically rigorous. Afterward, the analytical solutions obtained via the SSM are integrated to serve as the final free vibration solution of the cracked FGM plate. Comprehensive results based on the obtained analytical solutions, including natural frequencies and vibration modes, are presented and validated. The solutions are further used to investigate the effects of some important parameters, such as crack length and crack location.

Propagation phenomena for a nonlocal reaction-diffusion model with bounded phenotypic traits

In this paper we study the stability of cylinder front waves and the propagation of solutions of a nonlocal Fisher-type model describing the propagation of a population with nonlocal competition among bounded and continuous phenotypic traits. By applying spectral analysis and separation of variables we prove the spectral and local exponential stability of the cylinder waves with the noncritical speeds in some exponentially weighted spaces. By combining the detailed analysis with the spectral expansion and the special construction of sub-supersolutions, we further prove the uniform boundedness of the solutions and the global asymptotic stability of the cylinder waves for more general nonnegative bounded initial data, and prove that the spreading speeds and the asymptotic behavior of the solutions are determined by the decay rates of the initial data. Our results also extend some classical results on the stability of planar waves for Fisher-KPP equation to the nonlocal Fisher model in multi-dimensional cylinder case.

Comparative analysis of double and single porosity effects on SH-wave induced vibrations in periodic porous lattices

The manuscript presents a novel study comparing the impact of single and double porosity on horizontally polarized shear wave (SH-wave) propagation in corrugated elastic void materials. The considered mathematical model comprises two cases; the first one depicts SH-wave propagation through a void porous layer having creased boundaries and resting over a heterogeneous anisotropic fluid-saturated fractured porous half-space, whereas according to the second case, heterogeneous anisotropic fluid-saturated porous semi-infinite medium without fractures has been considered. In both cases, rigidity and density of the half-space vary quadratically with depth. The separable variable method is used to attain the complex frequency equation for each case that leads to two different dispersion relations associated with two distinct wave fronts. The first wave front depends on the void parameters, whereas the second wave front defines the propagation of SH-waves in an elastic layer without void pores. In each case, the complex dispersion relation has been separated into two equations that illustrate the dispersion and attenuation properties of SH-waves. Using the variations in the inhomogeneity, position, fluctuation, flatness, total porosity, and anisotropy parameters, case I and case II have been compared in each graphical execution. In addition, the surface response of shear stress and displacement have been implemented graphically.