Eigen Solutions Research Articles

Symplectic Method-Based Analytical Solutions of Thermal Buckling for Shear Deformable Circular Functionally Graded Plates

Thermal buckling of a shear deformable circular plate made of functionally graded materials (FGM) is studied using an exact analytical method in the Hamiltonian system. Fundamental equations for the thermal buckling of the plate under transversely nonuniform temperature rising are established based on the first-order shear deformation theory. Subsequently, by introducing the symplectic method, the differential equations are converted into canonical equations in the Hamiltonian system. Buckling loads and buckling modes correspond to symplectic eigenvalues and eigen solutions, respectively. The expressions of complete buckling modes in the form of special functions are achieved by solving canonical equations and boundary conditions analytically and exactly. The symplectic eigenvalues are obtained simultaneously and the buckling temperature increments are calculated using the inverse solution. Finally, the influences of the gradient characteristics, geometric parameters, boundary conditions, and type of thermal loading on the buckling temperature increments are discussed through parameter research.

Eigen-solutions and thermal properties of multi-parameter exponential potential

In this work, we determined an approximate eigen solutions of Modified multi-parameter exponential potential using supersymmetric quantum mechanics approach (SUSY) with improved Greene-Aldrich approximation to the centrifugal term. The energy equation and its corresponding normalised radial wave function were fully obtained. The proposed potential reduces to other useful potentials like Rosen-Morse, Hellmann, Yukawa and Coulomb potential as special cases. The thermodynamic properties like the vibrational mean energy (Uβ,V), Vibrational heat capacity (Cβ,V), vibrational entropy (Sβ,V) and vibrational free energy (Fβ,V) of the interacting potential were studied via partition function (Zβ,V) obtained from the resulting energy equation. This study was applied to three diatomic molecules: Chromium hydride (CrH), Titanium Hydride (TiH) and Thiocynate (ScN). To ascertain the high degree of our analytical mathematical accuracy, we compared the results of special cases with an existing results. These were found to be in excellent agreement with the existing results.

Unusual dynamical properties of disordered polaritons in microcavities

The strong light-matter interaction in microcavities gives rise to intriguing phenomena, such as cavity-mediated transport that can potentially overcome the Anderson localization. Yet, an accurate theoretical treatment is challenging as the matter (e.g.,molecules) are subject to large energetic disorder. In this article, we develop the Green's function solution to the Fano-Anderson model and use the exact analytical solution to quantify the effects of energetic disorder on the spectral and transport properties in microcavities. Starting from microscopic equation of motions, we derive an effective non-Hermitian Hamiltonian and predict a set of scaling laws: (i) The complex eigen-energies of the effective Hamiltonian exhibit an exceptional point, which leads to underdamped coherent dynamics in the weak disorder regime, where the decay rate increases with disorder, and overdamped incoherent dynamics in the strong disorder regime, where the slow decay rate decreases with disorder. (ii) The total density of states of disordered ensembles can be exactly partitioned into the cavity, bright-state and dark-state local density of states, which are determined by the complex eigen solutions and can be measured via spectroscopy. (iii) The cavity-mediated relaxation and transport dynamics are intimately related such that the energy-resolved relaxation and transport rates are proportional to the cavity local density of states. The ratio of the disorder averaged relaxation and transport rates equals the molecule number, which can be interpreted as a result of a quantum random walk. (iv) A turnover in the rates as a function of disorder or molecule density can be explained in terms of the overlap of the disorder distribution function and the cavity local density of states. These findings reveal the significant impact of the dark states on the transport properties of disordered ensembles in cavities.

Fuzzy c-Varieties Clustering for Vertically Distributed Datasets

Fuzzy c-varieties (FCV) is a linear fuzzy clustering method, whose clustering criterion is constructed by replacing the fuzzy c-means (FCM) prototypes with linear varieties, and has been utilized for local principal component analysis. In this paper, the FCV iterative algorithm is modified for handling vertically distributed datasets under consideration of privacy preservation. A cryptographic calculation scheme is realized by utilizing the least square criterion without eigen solutions. The characteristics of the proposed algorithm is demonstrated through numerical experiments.

Деякі особливості втрати стійкості ламінарною течією в трубі

Despite the seeming simplicity of the steady flow in a pipe of constant radius, the question of the cause and process of the transition remains debatable. Especially since the necessary condition for the stability loss of parabolic profile is not satisfied, and the linear theory of hydrodynamic stability for an axisymmetric Poiseuille flow does not give growing axisymmetric eigen solutions for any Reynolds numbers, since the terms characterizing the interaction of disturbances with the initial velocity profile drop out in the linearized equations of momentum conservation. The report presents the results of the study of stages of convective stability loss for the flow at the initial section of the pipe depending on the variable parameters based on the numerical solution of the three-dimensional system of unsteady Navier-Stokes equations and the equation energy transfer. The variable parameters in this study were: Reynolds number, magnitude and gradient sign of the dynamic viscosity coefficient arising in nonisothermal flows. An analogy of the arising secondary axisymmetric large-scale toroidal vortex structures in the near-wall region to Tollmien-Schlichting waves in the region of the transition of the laminar boundary layer on the plate is shown. The subsequent loss of axisymmetry and stability of these torus-like vortex structures is analyzed, which leads to the formation of fairly regular longitudinal vortex structures downstream, the nonlinear interaction of which leads to chaotization of the flow. The lengths of these sections are determined depending on the Reynolds number, the magnitude and sign of the gradient of the dynamic viscosity coefficient.

Negative Ion Resonance States: The Fock-Space Coupled-Cluster Way.

The negative ion resonance states, which are electron-molecule metastable compound states, play the most important role in free-electron controlled molecular reactions and low-energy free-electron-induced DNA damage. Their electronic structure is often only poorly described but crucial to an understanding of their reaction dynamics. One of the most important challenges to current electronic structure theory is the computation of negative ion resonance states. As a major step forward, coupled-cluster theories, which are well-known for their ability to produce the best approximate bound state electronic eigen solutions, are upgraded to offer the most accurate and effective approximations for negative ion resonance states. The existing Fock-space coupled-cluster (FSCC) and the equation-of-motion coupled-cluster (EOM-CC) approaches that compute bound states are redesigned for the direct and simultaneous determination of both the kinetic energy of the free electron at which the electron-molecule compound states are resonantly formed and the corresponding autodetachment decay rate of the electron from the metastable compound state. This Feature Article reviews the computation of negative ion resonances using the FSCC approach and, in passing, provides the highlights of the equivalent EOM-CC approach.

Approximate Analytical Solutions of the Klein–Gordon Equation with Generalized Morse Potential

In this study, the Klein–Gordon equation for the newly proposed generalized Morse potential (GMP) was solved using the newly proposed Nikiforov–Uvarov-functional analysis (NUFA) method for solving central potentials. The analytical expressions of Eigen solutions for GMP were obtained in closed form. The effects of the GMP parameters on the energy eigenvalues have been discussed. The wave function and probability density graphs with position have been illustrated, as regards the ground state, first, and second excited states for different deformation parameters. The effects of temperature and upper bound vibrational quantum number have been shown to vary for different thermodynamic functions considered for the selected deformation parameters. Hence, it was observed that the deformation parameter has a strong influence on the energy and thermodynamic function of the GMP model.

A symplectic analytical singular element for V-notched analyses in elastic and viscoelastic plane problems

The linear elastic and viscoelastic problems with V-shaped notches under in-plane loading are investigated in this paper. We consider two different sets of boundary conditions on the radial edges: Free-Free case and Clamped-Clamped case. A novel singular finite element (SASE) with arbitrary high-order accuracy for the notched problems is proposed. Firstly, via Laplace transform, the original viscoelastic problem is transformed into a corresponding elastic one. Then we construct the SASE by using elastic symplectic eigen solutions with higher order expanding terms. The SASE can depict the characteristics of displacement fields and singular stress fields in the vicinity of the notch vertex having arbitrary opening angle. By taking advantage of the symplectic eigen solutions in the SASE, Mode I and/or Mode II notch stress intensity factors can be determined directly without any post-processing. Fine finite element meshes are not required. Numerical examples are provided to illustrate the validity and accuracy of the present method.

Eigen solutions of the Schrodinger equation with variable mass under the influence of the linear combination of modified Woods–Saxon and Eckart potentials in toroidal coordinate

The solutions of the Schrodinger equation was investigated in space-times with nonspherical topology in the presence of the linear combination of modified Woods–Saxon and Eckart potentials. The variable mass and Supersymmetric Quantum Mechanics method were used to solve the Schrodinger equation in toroidal coordinate. This solution was used to obtain the nonrelativistic energy eigenvalue and the wave function of the system. Using the proper super potential to deal with the hyperbolic form of the effective potential, the Schrodinger equation in the radial direction part was solved analytically for modified Woods–Saxon plus Eckart potentials. The energy eigenvalue and the un-normalised ground state wave function equations were obtained. The results showed that the energy value depends on the quantum number, the parameters of potentials, and variable separation constants.

Eigen-Solutions to Schrodinger Equation with Trigonometric Inversely Quadratic Plus Coulombic Hyperbolic Potential

In this work, we applied parametric Nikiforov-Uvarov method to analytically obtained eigen solutions to Schrodinger wave equation with Trigonometric Inversely Quadratic plus Coulombic Hyperbolic Potential. We obtain energy-Eigen equation and total normalised wave function expressed in terms of Jacobi polynomial. The numerical solutions produce positive and negative bound state energies which signifies that the potential is suitable for describing both particle and anti-particle. The numerical bound state energies decreases with an increase in quantum state with fixed orbital angular quantum number 0, 1, 2 and 3. The numerical bound state energies decreases with an increase in the screening parameter and 0.5. The energy spectral diagrams show unique quantisation of the different energy levels. This potential reduces to Coulomb potential as a special case. The numerical solutions were carried out with algorithm implemented using MATLAB 8.0 software using the resulting energy-Eigen equation.

Orthogonal Directional Transforms using Discrete Directional Laplacian Eigen Solutions for Beyond HEVC Intra Coding.

We introduce new transforms for efficient compression of image blocks with directional preferences. Each transform, which is an orthogonal basis for a specific direction, is constructed from an eigen-decomposition of a discrete directional Laplacian system matrix. The method is a natural extension of the DCT, expressing the Laplacian in Cartesian coordinates rotated to some predetermined angles. Symmetry properties of the transforms over square domains lead to efficient computation and compact storage of the directional transforms. A version of the directional transforms was implemented within the beyond HEVC software and demonstrated significant improvement for intra block coding.

Mass Spectrum of Dirac Equation with Local Parabolic Potential

In this paper, we solve the eigen solutions to the Dirac equation with local parabolic potential which is approximately equal to the short distance potential generated by spinor itself. The energy spectrum is quite different from that with Coulomb potential. The mass spectrum of some bary-ons is similar to this one. The angular momentum-mass relation is quite similar to the Regge trajectories. The parabolic potential has a structure of asymptotic freedom near the center and confinement at a large distance. So, the results imply that, the local parabolic potential may be more suitable for describing the nuclear potential. The procedure of solving can also be used for some other cases of Dirac equation with complicated potential.

Bound state energies of a diatomic quantum dot molecule in a magnetic field

We investigate energy levels of a diatomic quantum dot by using an analytical method. We use Schrodinger equation for spherical quantum dot molecule in the presence of a static magnetic field and a confining potential which is Morse potential. We obtain the ground state and the excited states eigen solutions which are compared with the experimental data.

Interfacial crack analysis between dissimilar viscoelastic media using symplectic analytical singular element

In this paper, a two-dimensional interfacial crack problem between two dissimilar linear viscoelastic media is numerically investigated. The problem can be formulated as a viscoelastic problem. Firstly, the viscoelastic problem in the time domain is transformed into a corresponding elastic one in the Laplace domain using the Laplace transformation. Then a symplectic analytical singular element (SASE) in the framework of finite element method is applied to the obtained elastic problem. The SASE is constructed using bimaterial symplectic eigen solutions with higher order expanding terms. The eigen solutions are derived into explicit forms, and fracture parameters of the problem can be determined directly from expanding coefficients of the eigen solutions. Particularly we investigate how to obtain strain energy release rate, which is an important fracture parameter to viscoelastic fracture problem. Numerical examples are provided to demonstrate the accuracy and effectiveness of the present method.

Dynamic buckling analysis of functionally graded material cylindrical shells under thermal shock

This study focuses on dynamic buckling of functionally graded material (FGM) cylindrical shells under thermal shock. The transient non-uniform temperature fields in the FGM shells subjected to dynamic thermal loading are determined using an analytic method. Based on the Hamiltonian principle, the dynamic thermal buckling problem of the FGM cylindrical shells is transformed into the symplectic space for solving. At the same time, the buckling thermal loads and buckling modes corresponding to generalized eigenvalues and eigen solutions of canonical equations can be calculated via the bifurcation conditions. The dynamic thermal buckling characteristics of the FGM cylindrical shells as well as the solving processes are given by the symplectic method. A complete dynamic buckling modes space is presented for the FGM cylindrical shells. The effects of the material gradient, parameters of structural geometry and thermal loadings on the dynamic buckling temperature are discussed.

Dynamical Properties of Intensity Dependent Two-Mode Raman Coupled Model in a Kerr Medium

The present paper describes the dynamics of a single two-level atom interacting with intensity dependent quantized bimodal field via Raman type process in an ideal cavity filled with a non-linear Kerr medium. The cavity modes interact with the atom as well as the Kerr medium. A unitary transformation method that is used to solve the time-dependent problem also gives the Eigen solutions of the interaction Hamiltonian. We study the atomic population dynamics and investigate the effects of intensity dependent coupling, Kerr medium and detuning parameters on the domain of the non-classical features of the atom field state. The intensity dependent coupling is introduced by the Hermitian operator valued functions of the number operator $$ f\left({\widehat{n}}_i\right)\kern0.36em \left(i=1,2\right) $$ . We find that the intensity dependent coupling considered by the function $$ f\left({n}_i\right)=\frac{1}{\sqrt{n_i}} $$ introduces and ameliorates the non-classical features in the presence of Kerr medium.

Numerical convergence of the Lyapunov spectrum computed using low Mach number solvers

In the dynamical systems approach to describing turbulent or otherwise chaotic flows, an important quantity is the Lyapunov exponents and vectors that characterize the strange attractor of the flow. In particular, knowledge of the Lyapunov exponents and vectors will help identify perturbations that the system is most sensitive to, and quantify the dimension of the attractor. However, reliably computing these quantities requires robust numerical algorithms. While several perturbation-based techniques are available in literature, their application to commonly used turbulent flow solvers as well the numerical convergence properties have not been studied in detail. The goal of this work is to examine the convergence properties of the Lyapunov exponents when computed using a low Mach number solver, for a series of progressively complex flow problems. In particular, a manufactured solutions approach is devised using the Orr-Sommerfeld (OS) perturbation theory to extract Lyapunov exponents and vectors from OS eigen solutions. Overall the spatial convergence rates of Lyapunov exponents follow the truncation order for the discretization scheme. However, temporal convergence rates were found to be only a weak function of timestep used. Additionally, for some configurations, the convergence properties are found to depend on the Lyapunov exponent itself. These results indicate that convergence properties do not follow universal rates, and require careful analysis for specific configurations considered.

Viscoelastic crack analysis using symplectic analytical singular element combining with precise time-domain algorithm

In this study, the crack problem in linear viscoelastic material is investigated numerically. The time dependent two-dimensional (2D) viscoelastic crack problem is treated by the precise time-domain expanding algorithm (PTDEA), such that the original problem is transformed into a series of quasi-elastic crack problems. The relationships among these quasi-elastic problems are expressed in terms of the time-domain expanding coefficients of displacement and stress in an improved recursive manner. Then a symplectic analytical singular element (SASE) which has been demonstrated to be effective and efficient for 2D elastic fracture problem is applied to solve the quasi-elastic crack problems obtained above. The SASE is constructed by using the symplectic eigen solutions with higher order expanding terms. An improved convergence criterion employing both displacement and stress for PTDEA is proposed. Taking advantage of the SASE, the stress intensity factors, crack opening and sliding displacements (COD and CSD) and strain energy release rate of the studied problem can be solved directly without any post-processing. Numerical examples show that the results of the present method can be solved accurately and effectively.

Determination of stress state of anisotropic plates with rigid inclusions based on singular integral equations

In the paper singular integral equations for anisotropic plates with inclusions (the Dirichlet problem) are developed in the simple form on the basis of established dependencies between the Lekhnitskii complex potentials, stress and displacements. The numerical method for solving integral equations is developed on the basis of the quadrature method for the inclusion system. The eigen solutions of the problem were taken into consideration in this method. Simplicity, accuracy of the approach and stability of the obtained systems of algebraic equations are illustrated during: finding stress with controlled accuracy for systems of the large number of rigid inclusions, determining the high stress concentration near inclusions with corner points (the asymptotic method is additionally used).

Eigen solutions of the Schrӧdinger equation and the thermodynamic stability of the black hole temperature

Abstract The approximate analytical solutions of the Schrӧdinger equation for Eckart potential is obtained via supersymmetry shape invariance approach. The energy equation and the corresponding wave function are obtained in a closed and compact form. The wave function was used to calculate the Renyi entropy. The results of the Renyi entropy was used to study the mass energy parameter, temperature and heat capacity of the black hole. From the results obtained, the temperature of the black hole becomes stable as the two Eckart potential parameters increases respectively.