Separation Of Variables Research Articles

Propagation of Non-stationary Skew-Symmetric Waves from a Spherical Cavity in a Porous-elastic Half-space

Problems of propagation and diffraction of non-stationary waves in porous-elastic mediums are of great theoretical and practical importance in such fields of science and technology as geophysics, seismic exploration of minerals, seismic resistance of structures, and many others. The work considers the problem of propagation of non-stationary skew-symmetric waves from a spherical cavity in a porous-elastic half-space saturated with liquid. To solve the problem, the integral Laplace transform in dimensionless time and the method of incomplete separation of variables were used. In the space of Laplace images in time, known and unknown functions are expanded into Gegenbauer polynomials. The problem is reduced to solving an infinite system of linear algebraic equations, the solution of which is sought in the form of an infinite exponential series. Recurrence relations for the coefficients of the series and initial conditions for them are obtained, which makes it possible to obtain a solution to the infinite system without using the reduction method. Recurrence relations make it possible to determine the coefficients of a series in the form of rational functions, which makes it possible to calculate their originals using the theory of residues. In image space, formulas are obtained for the coefficients of the series of components of the displacement vector and stress tensor. Numerical experiments were carried out, the results of which are presented in the form of graphs. The results obtained can be used in geophysics, seismology, and design organizations during the construction of structures, as well as in the design of underground reservoirs.

GROUP ANALYSES, REDUCTIONS AND EXACT SOLUTIONS OF MONGE–AMPERE EQUATION OF MAGNETIC HYDRODYNAMICS

We study the Monge–Amp`ere equation with three independent variables which occurs in electron magnetohydrodynamics. A group analysis of this strongly nonlinear partial derivative equation is carried out. An eleven-parameter transformation preserving the form of the equation is found. A formula is obtained that makes it possible to construct multiparametric families of solutions based on simpler solutions. Two-dimensional reductions leading to simpler partial differential equations with two independent variables. One-dimensional reductions are described, which make it possible to obtain self-similar and other invariant solutions that satisfy ordinary differential equations. Exact solutions with additive, multiplicative and generalized separation of variables are constructed, many of which admit representation in elementary functions. The obtained results and exact solutions can be used to evaluate the accuracy and analyze the adequacy of numerical methods for solving initial boundary value problems described by strongly nonlinear partial differential equations.

Nonlinear Schrödinger Equations with Delay: Closed-Form and Generalized Separable Solutions

Nonlinear Schrödinger equations with constant delay are considered for the first time. These equations are generalizations of the classical Schrödinger equation with cubic nonlinearity and the more complex nonlinear Schrödinger equation containing functional arbitrariness. From a physical point of view, considerations are formulated about the possible causes of the appearance of a delay in nonlinear equations of mathematical physics. To construct exact solutions, the principle of structural analogy of solutions of related equations was used. New exact solutions of nonlinear Schrödinger equations with delay are obtained, which are expressed in elementary functions or in quadratures. Some more complex solutions with generalized separation of variables are also found, which are described by mixed systems of ordinary differential equations without delay or ordinary differential equations with delay. The results of this work can be useful for the development of new mathematical models described by nonlinear Schrödinger equations with delay, and the given exact solutions can serve as the basis for the formulation of test problems designed to evaluate the accuracy of numerical methods for integrating nonlinear partial differential equations with delay.

One‐Dimensional Fully Coupled Hydro‐Mechanical‐Chemical Model for Contaminant Transport Under Step‐Loading Condition and Its Analytical Solutions

ABSTRACTStepped dumpling of waste is the main operating method used in landfill engineering. For the liners located at the bottom of the landfill, the waste can be considered as a vertical loading due to its heavy weight, causing compressive deformation of the liners and, most importantly, tending to lose the effectiveness of the geomembrane with the differential deformation. However, most experiments or numerical models use transient‐loading and ignore the influence of variable step‐loading on liner performance. In this study, a one‐dimensional fully coupled hydro‐mechanical‐chemical (HMC) model for contaminant transport under the framework of small strain, taking into account the step‐loading effect, is proposed, and its analytical solutions are derived by using the separation of variables. The behavior of contaminant transport in a single‐layer clay liner is investigated under variable step‐loading conditions. Analytical results show that the step‐loading can significantly accelerate the transport of contaminants compared to the transient‐loading. Variation of the step‐loading rate from 2 to 4 times yields an increase of 6.1 and 15.2 years of the breakthrough time. At a step‐loading rate of 50 kPa/year, the breakthrough increases by 13.37 years, and the peak settlement reduces to half that under transient‐loading. Ignoring the influence of step‐loading on contaminant transport may overestimate the performance of clay liners.

Optical wave packet compression using counterpropagating Scorer beams

Abstract We investigate the diffractive paraxial wave equation with an external potential, utilizing self-similarity and variable separation methods. The exact solution to this evolution equation, expressed through Scorer functions, gives rise to the new Scorer beams. We explore the dynamics of counterpropagating Scorer beams, as promising optical wave packets, focusing on their compression behavior. The Scorer beams are characterized by two key parameters: the attenuation factor and the initial pulse width. By appropriately adjusting these parameters, significant beam compression can be achieved. Specifically, increasing the attenuation factor enhances compression and raises pulse amplitude, while reducing the initial pulse width further amplifies these effects. Along the way, we observe interesting interference patterns of the counterpropagating Scorer beams, never seen before. This study introduces a novel approach to beam compression and opens up new possibilities for their practical applications.

Optimization schemes on manifolds for structured matrices with fixed eigenvalues

AbstractSeveral manifold optimization schemes are presented and analyzed for solving a specialized inverse structured symmetric matrix problem with prescribed spectrum. Some entries in the desired matrix are assigned in advance and cannot be altered. The rest of the entries are free, some of them preferably away from zero. The reconstructed matrix must satisfy these requirements and its eigenvalues must be the given ones. This inverse eigenvalue problem is related to the problem of determining the graph, with weights on the undirected edges, of the matrix associated with its sparse pattern. Our optimization schemes are based on considering the eigenvector matrix as the only unknown and iteratively moving on the manifold of orthogonal matrices, forcing the additional structural requirements through a change of variables and a convenient differentiable objective function in the space of square matrices. We propose Riemannian gradient-type methods combined with two different well-known retractions, and with two well-known constrained optimization strategies: penalization and augmented Lagrangian. We also present a block alternating technique that takes advantage of a proper separation of variables. Convergence properties of the penalty alternating approach are established. Finally, we present initial numerical results to demonstrate the effectiveness of our proposals.

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.

Finite dimensional Hamiltonian systems and the algebro-geometry solution of the nonlocal reverse space–time sine-Gordon equation

In this paper, we investigate the algebro-geometric solution of the novel integrable nonlocal reverse space–time sine-Gordon equation through the framework of nonlocal finite-dimensional Lie-Poisson Hamiltonian systems, which are generated by the nonlinearization method. The nonlocal reverse space–time sine-Gordon equation is introduced via the Lenard recursion equations. Under suitable constraints, this equation is nonlinearized into two nonlocal finite-dimensional Lie-Poisson Hamiltonian systems. By applying the separated variables, the nonlocal finite-dimensional Lie-Poisson Hamiltonian systems are reformulated as the canonical Hamiltonian systems with the standard symplectic structures. Furthermore, the relationship between the action–angle coordinates and the Jacobi inversion problem is established based on the Hamilton–Jacobi theory. In the end, the algebraic-geometric solution of the nonlocal reverse space–time sine-Gordon equation is obtained using the Riemann-Jacobi inversion.

Effect of tube dimensions on resonant frequency of stepped acoustic resonator

An investigation of the effects of tube dimensions on resonant frequencies of stepped acoustic resonators is conducted. A linear acoustic theory is employed to model the plane wave propagation in the stepped resonator. The linear eigenvalue problem obtained by the method of separation of variables is solved to derive a frequency equation. Then, an experimental apparatus is established, in which the tube dimensions can be changed to adjust frequency. The variation pattern of resonant frequency with different lengths and diameters of the sub-tubes is analyzed theoretically and experimentally. The results suggest that the overall length determines the approximate scope of the resonant frequencies while the exact values of resonant frequencies and the relationship between resonant frequencies of each mode are determined by the ratio of sub-tube lengths and the ratio of sub-tube diameters. The calculated results are compared with the measured data. The comparative research declares that the derived frequency equation is an effective tool to calculate the resonant frequency of stepped resonator with adjustable size used in wide-band microphone calibration.

Accurate Closed-Form Solutions for the Free Vibration and Supersonic Flutter of Laminated Circular Cylindrical Shells

According to the Donnell–Mushtari shell theory, this work presents a closed-form solution procedure for free vibration of open laminated circular cylindrical shells with arbitrary homogeneous boundary conditions (BCs). The governing differential equations of free vibration are derived from the Rayleigh quotient and solved by the iterative separation-of-variable (iSOV) method. In addition, considering axial aerodynamic pressure, simulated by the linear piston theory, the exact eigensolutions for the flutter of open laminated cylindrical shells with simply supported circumferential edges and closed laminated cylindrical shells are also achieved. The governing differential equations of cylindrical shell flutter are derived from the Hamilton variational principle and solved by the separation-of-variable (SOV) method. The influence of circumferential dimension on flutter speed is investigated for open cylindrical shells, which reveals that the number of circumferential waves in critical flutter mode increases with circumferential length, and there exists an infimum for flutter speed that is an invariant independent of circumferential length. The present results agree well with those obtained by the Galerkin method, the finite element method, and other analytical methods.

Three-dimensional fully coupled analytical solution for a water-pile-saturated soil system under vertical P-wave incident

Water-pile-soil interaction and the three-dimensional scattered wave effect play crucial roles in the seismic response of structures built on saturated seabed subjected to vertical P-wave incidents. This study proposes a fully coupled three-dimensional continuum model of water-pile-saturated media to investigate the seismic response of the system under vertical seismic excitation. By employing Helmholtz decomposition and the method of variable separation, the governing equations are decoupled based on assumed boundary and interface conditions, yielding the solution equation in the frequency domain. The effects of various pile and soil parameters on the seismic response are analyzed from multiple perspectives, along with the impact on the spatial distribution of three-dimensional scattered wave effects. Results indicate that the seismic response of piles and saturated soil is significantly influenced by the presence of water and the three-dimensional scattered wave effect.

Solution of boundary value problems for the heat equation with a piecewise constant coefficient using the Fourier method

This paper considers some initial boundary value problems for the heat equation in a bounded segment with a piecewise constant coefficient. Using the method of separation of variables, the problem under consideration is reduced to a spectral problem and eigenvalues and eigenfunctions of the resulting spectral problem are found. It is shown that the system of eigenfunctions forms a Riesz basis. Next, we prove a theorem on the existence and uniqueness of solutions to the initial-boundary value problems under consideration.

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

In 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.

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.

Biochemically plausible models of habituation for single-cell learning.

The ability to learn is typically attributed to animals with brains. However, the apparently simplest form of learning, habituation, in which a steadily decreasing response is exhibited to a repeated stimulus, is found not only in animals but also in single-cell organisms and individual mammalian cells. Habituation has been codified from studies in both invertebrate and vertebrate animals as having ten characteristic hallmarks, seven of which involve a single stimulus. Here, we show by mathematical modeling that simple molecular networks, based on plausible biochemistry with common motifs of negative feedback and incoherent feedforward, can robustly exhibit all single-stimulus hallmarks. The models reveal how the hallmarks arise from underlying properties of timescale separation and reversal behavior of memory variables, and they reconcile opposing views of frequency and intensity sensitivity expressed within the neuroscience and cognitive science traditions. Our results suggest that individual cells may exhibit habituation behavior as rich as that which has been codified in multi-cellular animals with central nervous systems and that the relative simplicity of the biomolecular level may enhance our understanding of the mechanisms of learning.

Probabilistic Keyphrase Generation From Copy and Generating Spaces.

Keyphrase generation is one of the most fundamental tasks in natural language processing (NLP). Most existing works on keyphrase generation mainly focus on using holistic distribution to optimize the negative log-likelihood loss, but they do not directly manipulate the copy and generating spaces, which may reduce the generability of the decoder. Additionally, existing keyphrase models are either unable to determine the dynamic numbers of keyphrases or produce the number of keyphrases implicitly. In this article, we propose a probabilistic keyphrase generation model from copy and generating spaces. The proposed model is built upon the vanilla variational encoder-decoder (VED) framework. On top of VED, two separate latent variables are adopted to model the distribution of data within the latent copy and generating spaces, respectively. Specifically, we adopt a von Mises-Fisher (vMF) distribution to obtain a condensed variable for modifying the generating probability distribution over the predefined vocabulary. Meanwhile, we utilize a clustering module, which is designed to promote Gaussian Mixture learning and subsequently extract a latent variable for the copy probability distribution. Moreover, we utilize a natural property of the Gaussian mixture network and use the number of filtered components to determine the number of keyphrases. The approach is trained based on latent variable probabilistic modeling, neural variational inference, and self-supervised learning. Experiments on social media and scientific article datasets outperform the state-of-the-art baselines in generating accurate predictions and controllable keyphrase numbers.

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.