Eigen Solutions Research Articles

Eigen solutions of the D-dimensional Schrӧdinger equation with inverse trigonometry scarf potential and Coulomb potential

The approximate analytical solutions of the D-dimensional space of the Schrӧdinger equation is studied with a newly proposed potential model. The proposed potential is a combination of Coulomb potential and inverse trigonometry scarf-type potential. The energy equation and the corresponding wave function are obtained using parametric Nikiforov–Uvarov method. The energy equations for Coulomb potential and inverse trigonometry scarf-type potential are respectively obtained by changing the numerical values of the potential strengths. It is found that the results obtained are equivalent to that previously obtained for Hellmann potential which is a combination of Coulomb potential and Yukawa potential. It is also found that the results for inverse trigonometry scarf potential are equivalent to the results previously obtained for Yukawa potential. Also, the Onicescus information energy of a system under the influence of the newly proposed potential is investigated in detail.

A Symplectic Analytical Singular Element for Steady-State Thermal Conduction With Singularities in Anisotropic Material

Modeling of steady-state thermal conduction for crack and v-notch in anisotropic material remains challenging. Conventional numerical methods could bring significant error and the analytical solution should be used to improve the accuracy. In this study, crack and v-notch in anisotropic material are studied. The analytical symplectic eigen solutions are obtained for the first time and used to construct a new symplectic analytical singular element (SASE). The shape functions of the SASE are defined by the obtained eigen solutions (including higher order terms), hence the temperature as well as heat flux fields around the crack/notch tip can be described accurately. The formulation of the stiffness matrix of the SASE is then derived based on a variational principle with two kinds of variables. The nodal variable is transformed into temperature such that the proposed SASE can be connected with conventional finite elements (FE) directly without transition element. Structures of complex geometries and complicated boundary conditions can be analyzed numerically. The generalized flux intensity factors (GFIFs) can be calculated directly without any postprocessing. A few numerical examples are worked out and it is proven that the proposed method is effective for the discussed problem, and the structure can be analyzed accurately and efficiently.

Eigen solutions and entropic system for Hellmann potential in the presence of the Schrödinger equation

By using the supersymmetric approach, we studied the approximate analytic solutions of the three-dimensional Schrodinger equation with the Hellmann potential by applying a suitable approximation scheme to the centrifugal term. The solutions of other useful potentials, such as Coulomb potential and Yukawa potential, are obtained by transformation of variables from the Hellmann potential. Finally, we calculated the Tsallis entropy and Renyi entropy both in position and momentum spaces under the Hellmann potential using integral method. The effects of these entropies on the angular momentum quantum number are investigated in detail.

Some Characteristic Functions for the Eigen Solution of Nonlinear Spinor

In this paper, we solve the eigen solutions to some nonlinear spinor equations, and compute several functions reflecting their characteristics. The numerical results show that, the nonlinear spinor equation has only finite meaningful eigen solutions, which have positive discrete mass spectra and anomalous magnetic moment. The nonlinear potential and interactions yield different contributions to the total energy, and these components of the energy lead to different energy-speed relation. The magnitude of these components can be detected by elaborate experiments. The weird properties of the nonlinear spinors might be closely related with the elementary particles and their interactions, so some deeper investigations on them are significant.

Edge localized modes of cold neutrons in periodic condensed media

It is found that for certain energies of discreet cold neutrons, quasi-stationary eigen solutions of the corresponding Schrodinger equation, which are localized in the layer of a periodic medium, exist. The localization time of these solutions is strongly dependent on the layer thickness, being finite for a finite layer thickness and increasing indefinitely upon a infinite growth of the layer thickness as the third power of the layer thickness. The problem has been solved in the two-wave approximation of the dynamic diffraction theory for the neutron propagation direction coinciding with the periodicity axes (normal incidence of the neutron beam on the layer). The expressions for neutron eigenwave functions in a periodic medium, the reflection and transmission coefficients, and the neutron wavefunction in the layer as a function of the neutron energy incident on the layer have been determined. It turns out that for the certain discrete neutron energies, the amplitudes of the neutron wavefunction in the layer reach sharp maxima. The corresponding energies are just outside of the neutron stop band (energies forbidden for neutron propagation in the layer) and determine the energies of neutron edge modes (NEMs) localized in the layer, which are direct analogs of the optical edge modes for photonic crystals. The dispersion equation for the localized neutron edge modes has been obtained and analytically solved for the case of thick layers. A rough estimate for the localization length L is L ~(db N)–1, where b is the neutron scattering length, d is the crystal period, and N is the density of nuclei in the crystal. The estimates of the localized thermal neutron lifetime show that acheaving of a lifetime close to the free neutron lifetime seems nonrealistic due to absorption of thermal neutrons and requires a perfect large size crystal. Nevertheless, acheaving the localized neutron lifetime exceeding by ~104 times the neutron time of flight through the layer appears as experimentally attainable. The perspectives of the NEM observation are briefly discussed. It is proposed to use NEM for ultrahigh thermal neutron monochromatization by means of NEM excitation in perfect single crystals.

A symplectic analytical singular element for the inclined crack terminating at the material interface of composite structures

The inclined bimaterial interface crack problem is studied from theoretical and computational aspects. The original problem is transformed into an eigen value problem by using the symplectic approach, and the eigen solution are obtained. A Symplectic Analytical Singular Element (SASE), of which the interior fields are described by the obtained eigen solutions, is developed. The stress intensity factors (SIFs) can be calculated directly without any post-processing. Numerical example on a square bimaterial plate containing an inclined crack is provided to validate the proposed method. The comparison of the predicted results with the existing result indicates that the proposed method is accurate for the modelling of the inclined bimaterial interface crack.

Eigen solutions, Shannon entropy and fisher information under the Eckart Manning Rosen potential model

We solved the Schrodinger equation with a certain approximation to the centrifugal term for an arbitrary angular momentum state with the Eckart Manning Rosen potential. The bound-state energy eigenvalues and the corresponding wave functions have been approximately obtained using the parametric Nikiforov Uvarov method. The solutions of the Schrodinger equation for the Eckart potential, Manning Rosen potential, and Hulthen potential have been obtained using a certain transformation. The concepts of the Shannon entropy and the Fisher information of a system under the Eckart Manning Rosen potential are investigated in detail. The behavior of the screening parameter and the quantum number n for Fisher information and the Shannon entropy are also investigated.

Analytical solution of the second Stokes problem with variable amplitude on behavior of gas over an oscillation surface

The analytical solution of the second Stokes problem is found. The case of the variable amplitude of fluctuation of a surface is considered. The linear kinetic equation with boundary conditions is derived. Eigen solutions of the equation are found. Properties of dispersion function are investigated. The general solution of the kinetic equation with boundary conditions in terms of the eigen solutions decomposition is composed.

Coupled mode analysis of a periodic one-dimensional multimodal fiber bundle

This contribution is a mathematical analysis of the coupled mode equations of a one dimensional infinite periodic lattice of multimodal adjacent fibers that are fused together (a “fiber bundle”). As such, it provides a systematic and detailed derivation of the coupled mode equations and their eigen (modal) solutions within a matrix-based framework and using Z-transform spectral-based formulation. The resulting solution is general in the sense that it is not restricted to a particular dielectric profile of the fibers. Moreover, under a weak coupling assumption, the modal solution clearly identifies the physical building blocks of the solution.

A size independent enriched finite element for the modeling of bimaterial interface cracks

The failure of structures made of multiple materials is often caused by bimaterial interface cracks. Due to the complex nature of the stress field around the interface crack tip, the faithful modeling of such cracks requires numerical methods with high accuracy and stability. For this purpose, an enriched element is proposed in this study. The element’s interior fields are described only by using the Williams eigen solutions. The element’s stiffness matrices for both plane and antiplane deformations are proven to be independent of its size, and the fracture parameters can be obtained accurately for a wide range of mesh densities.

Closed form solution of Eigen frequency of monopile supported offshore wind turbines in deeper waters incorporating stiffness of substructure and SSI

Offshore wind turbines (OWTs) are dynamically loaded structures and therefore the estimation of the natural frequency is an important design calculation to avoid resonance and resonance related effects (such as fatigue). Monopiles are currently the most used foundation type and are also being considered in deeper waters (>30m) where a stiff transition piece will join the monopile and the tapered tall tower. While rather computationally expensive, high fidelity finite element analysis can be carried to find the Eigen solutions of the whole system considering soil–structure interaction; a quick hand calculation method is often convenient during the design optimisation stage or conceptual design stage. This paper proposes a simplified methodology to obtain the first natural frequency of the whole system using only limited data on the WTG (Wind Turbine Generator), tower dimensions, monopile dimensions and the ground. The most uncertain component is the ground and is characterised by two parameters: type of ground profile (i.e. soil stiffness variation with depth) and the soil stiffness at one monopile depth below mudline. In this framework, the fixed base natural frequency of the wind turbine is first calculated and is then multiplied by two non-dimensional factors to account for the foundation flexibility (i.e. the effect of soil–structure interaction). The theoretical background behind the model is the Euler–Bernoulli and Timoshenko beam theories where the foundation is idealised by three coupled springs (lateral, rocking and cross-coupling). 10 wind turbines founded in different ground conditions from 10 different wind farms in Europe (e.g. Walney, Gunfleet sand, Burbo Bank, Belwind, Barrow, Kentish flat, Blyth, Lely, Thanet Sand, Irene Vorrink) have been analysed and the results compared with the measured natural frequencies. The results show good accuracy (errors below 3.5%). A step by step sample calculation is also shown for practical use of the proposed methodology.

Dynamics for the collective model of two atoms interacting with two-mode quantized radiation fields in a Raman type process

We obtain exact solutions for the collective model of two identical two-level atoms interacting with two quantized cavity electromagnetic fields in a Raman type process. The unitary transformation method that is used to solve the time dependent problem, also gives the eigen solutions of the interacting Hamiltonian at the same time. We study the dynamics of atomic population and photon statistics in the two cavity modes. We obtain evidence of co-operative effects in antibunching and anticorrelations in the modes.

Eigen analysis of functionally graded beams with variable cross-section resting on elastic supports and elastic foundation

The free vibration of functionally graded material (FGM) beams on an elastic foundation and spring supports is investigated. Young\'s modulus, mass density and width of the beam are assumed to vary in thickness and axial directions respectively following the exponential law. The spring supports are also taken into account at both ends of the beam. An analytical formulation is suggested to obtain eigen solutions of the FGM beams. Numerical analyses, based on finite element method by using a beam finite element developed in this study, are performed in order to show the legitimacy of the analytical solutions. Some results for the natural frequencies of the FGM beams are given considering the effect of various structural parameters. It is also shown that the spring supports show the greatest effect on the natural frequencies of FGM beams.

SPH for incompressible free-surface flows. Part II: Performance of a modified SPH method

Building on Part I, we propose a modified method, mSPH, which retains the weak compressibility and kernel interpolation of the basic SPH method, but suppresses the two sources of spurious high-frequency dynamics in the presence of weak compressibility: the (stable) acoustic eigen solutions, and the unstable depth-oscillatory modes (associated with non-uniform density). We achieve this by using a form of the initial and boundary conditions consistent with the governing equations; and employing a robust dissipative scheme, in the form of periodic smoothing. We quantify the effect and efficacy of the latter in terms of the numerical parameters: the artificial sound speed, the kernel bandwidth, the Courant condition, and the smoothing frequency. Further, we obtain a global error metric that quantifies the spectral amplitudes of the high-frequency dynamics and identifies the initiation and growth of temporally unstable modes. This metric is used as an independent measure for the validity of the weak compressibility assumption, without the need for calibration with external data. We demonstrate the performance of mSPH, and the usefulness of the error metric in four illustrative applications: the hydrostatic problem, the collapse of a liquid column, the standard dam-break benchmark, and sloshing in a swaying tank. It is shown that mSPH is robust and obtains convergent and accurate kinematics and dynamics compared to theory and experiments.

An orthogonal subspace minimization method for finding multiple solutions to the defocusing nonlinear Schrödinger equation with symmetry

AbstractAn orthogonal subspace minimization method is developed for finding multiple (eigen) solutions to the defocusing nonlinear Schrödinger equation with symmetry. As such solutions are unstable, gradient search algorithms are very sensitive to numerical errors, will easily break symmetry, and will lead to unwanted solutions. Instead of enforcing a symmetry by the Haar projection, the authors use the knowledge of previously found solutions to build a support for the minimization search. With this support, numerical errors can be partitioned into two components, sensitive versus insensitive to the negative gradient search. Only the sensitive part is removed by an orthogonal projection. Analysis and numerical examples are presented to illustrate the method. Numerical solutions with some interesting phenomena are captured and visualized by their solution profile and contour plots. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

The Squeeze Film Damping Effect on Dynamic Responses of Micro-Electromechanical Resonators

The air squeeze film damping effect on the dynamic responses of clamped micro- electromechanical resonators is investigated in this study. A dynamic model for a clamped micro- electromechanical resonator with the damping consideration is derived using Lagrange’s equation. The corresponding resonator eigen solutions are formulated and solved by employing the assumed-mode method. The effect of different parameters; i.e. the resonator size, ambient temperature and pressure on the squeeze film damping characteristics were simulated and investigated. The results indicate that the squeeze film damping effect may significantly affect the dynamic responses of micro-scale electromechanical resonator.

２次元一様流中の半没平板まわりの流れに関する解析解について

The perturbed complex potentials representing flows around a vertical plate semi-submerged in a uniform stream are derived in analytical forms by the reduction method. They are composed from the regular solution and the weak singular eigen solutions. The linear combinations of them represent some flows such as regular flow, zero-vertical flux flow, flow satisfying Kutta condition and wave-free flow. The wave resistances of the flows are also obtaied in analytical forms. The analytical solution obtained by Bessho-Mizuno(1962) has a possibility that it dose not satisfy the boundary condition on the plate.

Eigen solutions of internal waves over subcritical topography

Diapycnal mixing plays an important role in the ocean circulation. Internal waves are a kind of bridge relating the diapycnal mixing to external sources of mechanical energy. Difficulty in obtaining eigen solutions of internal waves over curved topography is a limitation for further theoretical study on the generation problem and scattering process. In this study, a kind of transform method is put forward to derive the eigen solutions of internal waves over subcritical topography in two-dimensional and linear framework. The transform converts the curved topography in physical space to flat bottom in transform space while the governing equation of internal waves is still hyperbolic if proper transform function is selected. Thus, one can obtain eigen solutions of internal waves in the transform space. Several examples of transform functions, which convert the linear slope, the convex slope, and the concave slope to flat bottom, and the corresponding eigen solutions are illustrated. A method, using a polynomial to approximate the transform function and least squares method to estimate the undetermined coefficients in the polynomial, is introduced to calculate the approximate expression of the transform function for the given subcritical topography.

Modeling of Transistor's Tracking Behavior in Compact Models

We present a novel method to model the tracking behavior of semiconductor transistors undergoing across-chip variations in a compact Monte Carlo model for SPICE simulations and show an enablement of simultaneous tracking relations among transistors on a chip at any poly density, any gate pitch, and any physical location for the first time. At smaller separations, our modeled tracking relation versus physical location reduces to Pelgrom's characterization on device's distance-dependent mismatch. Our method is very compact, since we do not use a matrix or a set of eigen solutions to represent correlations among transistors.

Solutions of a Particle with Fractional δ-Potential in a Fractional Dimensional Space

A Fourier transformation in a fractional dimensional space of order λ (0<λ≤1) is defined to solve the Schrödinger equation with Riesz fractional derivatives of order α. This new method is applied for a particle in a fractional δ-potential well defined by V(x)=−γ δ λ(x), where γ>0 and δ λ(x) is the fractional Dirac delta function. A complete solutions for the energy values and the wave functions are obtained in terms of the Fox H-functions. It is demonstrated that the eigen solutions are exist if 0<λ<α. The results for λ=1 and α=2 are in exact agreement with those presented in the standard quantum mechanics.