Eigen Functions Research Articles

On the quadratic closeness of eigenfunctions of «unperturbed» and «perturbed» differentiation operators on an interval

The article considers the eigenvalue problem of the differentiation operator, when the spectral param- eter is also present in the boundary condition with an integral perturbation, where the integrand has the property of limited variation and has a value of unity at the ends of the segment [-1, 1]. The derivatives with respect to the time variable included in the boundary condition naturally arise when solving (by the Fourier method) initial boundary value problems for evolution equations. The conjugate operator is constructed. It is shown that the spectral questions of the conjugate operator have a similar structure. The characteristic determinant of the original direct spectral problem with an integral perturbation of the boundary condition and for the eigenvalue problem of a loaded first-order differential equation on a segment with a periodic boundary condition, which is an entire analytical function of the spectral parameter, is constructed. Based on the formula of the characteristic determinant, conclusions are drawn about the asymptotic behavior of the spectrum of the original «perturbed» differentiation operator and the loaded first-order differential equation on the segment. A special feature of the operator under consideration is the non-self-adjointness of the operator in L2(–1,1). The quadratic proximity of the systems of eigen functions of the «unperturbed» and «perturbed» differentiation operators and the Riesz basis property of these systems are proved. In this case, the system of eigenfunctions of the «perturbed» operator is not orthonormal.

On the Structure and Properties of Eigen Functions When Solving One Class of Integro-Differential Equations

On the Structure and Properties of Eigen Functions When Solving One Class of Integro-Differential Equations

Analyzing the Nuclear Structure of 13O-13B and 13N-13C Mirror Nuclei

In the context of the shell model, electromagnetic transitions were used to analyze the nuclear structure of a mirror nucleus with the same mass number, A = 13. The shell model investigation was performed by calculating the root mean square (rms) for the proton, neutron, matter and charge radii, occupancies, the excitation energies, as well as electromagnetic moments, using the elements of a body density matrix of the PSDMOD two-body effective interactions carried out in the psdpn-shell model space. The present results adopted the harmonic oscillator's single particle eigen functions and Hartree-Fock approximation. In addition, the effect of core polarization was added using the effective charge and effective g factors to calculate electromagnetic moments. To acquire a fair analysis of the information, the core polarization has to be present. The outcomes were compared with experimental data

Thermal stresses associated with a thermosensitive multilayered disc analysed due to point heating

In this paper, analytical solutions are presented for temperature and thermal behavior of a thermosensitive multilayered annular disc due to point heat source. Convective heating is applied to both the innermost and outermost layers. The nonlinearity of the thermal diffusivity equation is dealt using Kirchhoff’s transformation technique. A finite integral transform in the form of Bessel’s function is used to deal with the radial variable r. Fourier transform and angular eigen functions are also used to solve the thermal diffusivity equation. Deflection, resultant forces, shearing forces, resultant moments and thermal stresses are obtained. A mathematical representation is formulated for a 3-layered disc, with the inner, middle and outer layers composed of copper, zinc and aluminum respectively. The results are depicted graphically.

Optical properties of spherical quantum dot using screened Kratzer potential

Here we investigate the optical properties of a spherical quantum dot using the screened Kratzer potential. The density matrix formalism is employed to study refractive index changes, along with linear and nonlinear optical absorption coefficients. To this effect, the radial Schrödinger equation is solved using the Nikiforov–Uvarov functional analysis method, resulting in analytical expressions for the energy eigenvalues and the associated eigen functions. Effects of dot size, potential parameters, and screening length on these properties are analyzed in detail for a GaAs quantum dot. The obtained results reveal that the optical response of the system is highly sensitive to the dot size and potential parameters, and thus it can be tuned accordingly. The impact of external magnetic and Aharonov–Bohm flux fields on the optical properties of the GaAs quantum dot is also discussed. The findings of this study may provide useful insights in design and optimization of quantum dot-based optoelectronic devices.

Traveling wave solution and the stability of critical points of an enzyme-inhibitor system under diffusion effects: with special reference to dimer molecule

Enzymes are absolutely essential biological catalysts in human body that catalyze all cellular processes in physiological network. However, there are certain low molecular weight chemical compounds known as inhibitors, that reduce or completely inhibit the enzyme catalytic activity. Mathematical modeling plays a key role in the control and stability of metabolic enzyme inhibition. Enzyme stability is an important issue for protein engineers, because of its great importance and impact on optimal utility of material in biological tissues. In this outlook, we have first determined the existence of traveling wave solution for the enzyme-inhibitor system and then emphasized the stability of critical points that arise in the reactions. The study of traveling wave solution of an enzyme-inhibitor system with reaction diffusion equations involve quite complex mathematical analysis. The results obtained in this model indicate that the traveling wave solution may give a well explained method for improving enzyme kinetic stability. The present study will be helpful in understanding the stability of critical points of an enzyme-inhibitor system to give an idea about the inhibition of less stable enzymes. Moreover, the role of diffusion on the enzyme activity has been exhaustively discussed using mathematical tools related to eigen values and eigen function analysis.

Investigation of linear and nonlinear optical properties in quantum dots: Influence of magnetic and AB flux fields

In the present work, analytical expressions for the energy eigenvalues and eigen functions are obtained by solving the radial Schrödinger equation for inverse quadratic Hellmann plus Yukawa (IQHY) potential using Nikiforov-Uvarov functional analysis method. As an application part of this work, expressions for the linear and third-order nonlinear absorption coefficients and refractive index changes are computed to study the optical properties of the spherical GaAs quantum dots. We extend the investigation of the IQHY potential by incorporating the effects of magnetic and Aharonov–Bohm (AB) flux fields. Our results reveal that the optical properties of the GaAs quantum dots are strongly influenced by variations in the dot radius, potential height and intensity of incident photons. Specifically, we observe that the introduction of magnetic and AB flux fields significantly alters the energy eigenvalues and eigenfunctions, leading to distinct optical characteristics. The results of present study are found in consistent with other theoretical studies available in the literature.

A THEORY OF MULTIDIMENSIONAL FREDHOLM INTEGRAL EQUATIONS HAVING SEPARABLE KERNELS: SOLVABLE IN A REGION SURROUNDING BY THE HYPERPLANES

In this article, we present a theory of multidimensional Fredholm integral equations, having separable kernels, are solvable in a region surrounded by hyperplanes. In derivation of their solutions, we employ the generalized Hilbert-Schmidt theory involving eigenvalues and corresponding normalized eigen functions obtained by separable kernels in a region surrounded by the hyperplanes. Finally, we apply two variables Gegenbauer polynomials and derive a result on the inequality of the solution for the double symmetric Fredholm integral equation.

Exploring electronic properties of zinc oxide using pure and hybrid density functional theory and Compton profiles

To check the viability and applicability of available experimental Compton profiles (CPs) of ZnO, we have computed Mulliken population (MP) and CPs using pure and hybrid density functional theory (DFT) within the basis of linear combination of atomic orbitals (LCAO) scheme. Pure DFT have been employed within local density approximation, generalized gradient approximation (GGA) and second order GGA (SOGGA) whereas for hybrid exchange and correlation energies the standard WC1LYP and B1WC hybrid schemes are adopted. A quantitative comparison of theoretical and the available experimental CPs unambiguously identify WC1LYP as a better choice for reproducing experimental momentum densities, and thereafter hybrid approach for generating accurate Eigen functions of ZnO. Further, MP analysis reports the dominancy of ionic character in the compounds. We have also scaled the CPs of ZnO and CdO using WC1LYP for equal-valence-electron-densities and concluded about more ionic character in ZnO than that in CdO.

A physics-based machine learning technique rapidly reconstructs the wall-shear stress and pressure fields in coronary arteries.

With the global rise of cardiovascular disease including atherosclerosis, there is a high demand for accurate diagnostic tools that can be used during a short consultation. In view of pathology, abnormal blood flow patterns have been demonstrated to be strong predictors of atherosclerotic lesion incidence, location, progression, and rupture. Prediction of patient-specific blood flow patterns can hence enable fast clinical diagnosis. However, the current state of art for the technique is by employing 3D-imaging-based Computational Fluid Dynamics (CFD). The high computational cost renders these methods impractical. In this work, we present a novel method to expedite the reconstruction of 3D pressure and shear stress fields using a combination of a reduced-order CFD modelling technique together with non-linear regression tools from the Machine Learning (ML) paradigm. Specifically, we develop a proof-of-concept automated pipeline that uses randomised perturbations of an atherosclerotic pig coronary artery to produce a large dataset of unique mesh geometries with variable blood flow. A total of 1,407 geometries were generated from seven reference arteries and were used to simulate blood flow using the CFD solver Abaqus. This CFD dataset was then post-processed using the mesh-domain common-base Proper Orthogonal Decomposition (cPOD) method to obtain Eigen functions and principal coefficients, the latter of which is a product of the individual mesh flow solutions with the POD Eigenvectors. Being a data-reduction method, the POD enables the data to be represented using only the ten most significant modes, which captures cumulatively greater than 95% of variance of flow features due to mesh variations. Next, the node coordinate data of the meshes were embedded in a two-dimensional coordinate system using the t-distributed Stochastic Neighbor Embedding (-SNE) algorithm. The reduced dataset for -SNE coordinates and corresponding vector of POD coefficients were then used to train a Random Forest Regressor (RFR) model. The same methodology was applied to both the volumetric pressure solution and the wall shear stress. The predicted pattern of blood pressure, and shear stress in unseen arterial geometries were compared with the ground truth CFD solutions on "unseen" meshes. The new method was able to reliably reproduce the 3D coronary artery haemodynamics in less than 10 s.

A time variant uncertainty propagation method for high-dimensional dynamic structural system via K-L expansion and Bayesian deep neural network.

In this paper, a time variant uncertainty propagation (TUP) method for dynamic structural system with high-dimensional input variables is proposed. Firstly, an arbitrary stochastic process simulation (ASPS) method based on Karhunen-Loève (K-L) expansion and numerical integration is developed, expressing the stochastic process as the combination of its marginal distributions and eigen functions at several discrete time points. Secondly, the iterative sorting method is implemented to the statistic samples of marginal distributions for matching the constraints of covariance function. Since marginal distributions are directly used to express the stochastic process, the proposed ASPS is suitable for stationary or non-stationary stochastic processes with arbitrary marginal distributions. Thirdly, the high-dimensional TUP problem is converted into several high-dimensional static uncertainty propagation (UP) problems after implementing ASPS. Then, the Bayesian deep neural network based UP method is used to compute the marginal distributions as well as the eigen functions of dynamic system response, the high-dimensional TUP problem can thus be solved. Finally, several numerical examples are used to validate the effectiveness of the proposed method. This article is part of the theme issue 'Physics-informed machine learning and its structural integrity applications (Part 1)'.

An Extended Model for Short Wavelength Peristaltic Flow

Peristaltic flow theory is often based on the long wavelength assumption, which can be approximated by small wave number cases. This paper presents a unique approach to analyze the peristaltic flow for both short and long wavelength domains. The governing equation considering wave number is solved by combining eigen function expansion and generalized inversion methods, which is first proposed to solve peristaltic flow problems. In particular, the modified Navier-Stokes equations, considers the effects of the Trouton’s elongational viscosity. The result indicates that this computational method efficiently reduces the dimensions and condition number of the matrix. The sketch of stream function shows that a trapped bolus also occurs in short wave peristaltic flow. The validity of this computational approach is demonstrated by comparing these results with existing theories.

Analytical solution for Fredlund–Hasan unsaturated consolidation using mode superposition method

AbstractThis note presents an exact analytical solution of Fredlund–Hasan consolidation for unsaturated soils under an arbitrary loading using the mode superposition method. Air pressure and water pressure are expressed in a series of products of Eigen functions with respect to depth and normal coordinates with respect to time, respectively. A pressure function substitutes for the related homogeneous problem. Eigen functions can be first obtained based on two pairs of drainage condition. Then, the nonhomogeneous governing equations of unsaturated soils under constant loading, exponential loading, and sine loading are considered. After matrix algebraic manipulation, the normal coordinates can be obtained based on the initial conditions that compose the final solutions of the Fredlund–Hasan unsaturated consolidation. This solution can avoid inconvenient Laplace transform and inverse Laplace transform. Through comparisons with a classic example, the validation of the proposed analytical solution is verified against other analytical solutions and the finite difference method solutions.

The classical and quantum implications of the Raychaudhuri equation in f(T)-gravity

The present work deals with the classical and quantum aspects of the Raychaudhuri equation (RE) in the framework of f(T)-gravity theory. In the background of homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker space-time, the RE has been formulated and used to examine the focusing theorem and convergence condition for different choices of f(T). Finally in quantum cosmology, the wave function of the Universe has been shown to be the energy eigen function of the time-independent Schrödinger equation of a particle. Also probability measure on the minisuperspace has been examined at zero volume for singularity analysis in the quantum regime. Lastly, the Bohmian trajectory for the present quantum system has been explicitly determined for some particular choices.

원전용 밸브 패킹응력 해석을 위한 모델링 연구

The valve used in nuclear power plant must be qualified by the known experimental equation, which is the requirement of ASME QME-1 standard. Up to now, the stress on packing is assumed to be constant in design and experimental analysis. This paper deals with the development of analysis model for the packing stress. In the analysis model, the stress is calculated with the love function, whose eigen function is deduced by applying the separation method of axial and radial variables to biharmonic equation with boundary conditions. And the unkown parameters of love function are calculated by applying the remaining boundary conditions. The distribution of radial and axial stress are not constant and the big variation of stress is found at the inner and outer face. The radial stress after boundary layer of radial stress is 45% of gland packing stress. Considering the discussion that the double packing stress makes the radial stress in the packing as same as the fluid pressure, it is proper that the industrial design guide has been used to select the gland packing stress as twice as much as fluid pressure. And the force on stem can be calculated by applying shear stress of packing on stem area. Later, the load factor, which is used in the experimental analysis for consideration of stress distribution, must be related to the real stress distribution by this research result.

Study the Ground State Perturbed Eigenfunction with Modified Order Parameters of Hydrogen Atom

Studying the eigen value function of a perturbed hydrogen atom in its ground state with various orders is the goal of this investigation (1st, 2nd, 3rd, 4th, 5th, 6th and 7th). The new equation is created using the theoretic way to investigate the nature of eigen function. At various perturbed order parameters, it is discovered that the nature rapidly decreases with distance from the Bohr’s radius. Additionally, the nature with perturbed order parameters was discovered to be linear at various separations beyond Bohr’s radius. This demonstrates that while a large amount of energy is needed to perturbed a system close to the Bohr’s radius, very little energy is needed to perturbed a system farther away from the Bohr’s radius.

Mathematical Eigenfunctions Analysis for 2nd Kind of Fredholm Integral equations

The current paper is a trial to investigate the Eigen functions appears through mathematical treatment of 2nd kind of Fredholm integral equations by making use a new developed an Inverse Iterative Numerical Scheme (IINS). To test the applicability and accuracy of the proposed IINS, two numerical examples were solved and compared with previous results. The discussion of the results gave a good coincides up to some decimal places of results with the available ones.

Mathematical Model of the On-Board Antenna of Reentry Spacecraft Taking into Account Surface Waves

Introduction. The onboard antennas of the returned hypersonic aircraft are weakly directional, which is achieved by radiation from the open end of the waveguide. When passing through dense layers of the atmosphere, they are exposed to aerodynamic heating, for protection from which a heat-resistant radio-transparent thermal protection is used. The case of uniform thermal protection of the antenna can be interpreted as the absence of heating or heating uniform in the thickness of the thermal protection.Aim. The problem of obtaining an analytical description of the radiation characteristics of a circular waveguide closed by a flat homogeneous dielectric plate is solved. Since in such a formulation it is necessary to consider the resonant domain, a strict solution of Maxwell's equations is required.Materials and methods. Of the known analytical methods of solution, it is possible to use the method of integral transformations and the method of eigen functions. Both of these methods are used in the work. In this case, the assumption is used that the electrical parameters of the dielectric plate (thermal protection) and the geometric dimensions do not depend on time.Results. The relations describing the directional pattern of a circular waveguide with dielectric thermal protection and taking into account the electrical parameters of thermal protection and its thickness are obtained. Expressions are also obtained for the fields of lateral, surface and outflow waves, from which it is possible to calculate the power output by these fields. Relations for the separation of singular points of integrand expressions into poles corresponding to surface, outflow and lateral waves are obtained. Surface, outflow and lateral waves can have a certain effect on the radiation pattern. To determine this conclusion, analytical relations are obtained for determining the poles of integrand expressions that fully describe surface, outflow and side waves. All the analytical results obtained correspond to Numerical calculations were carried out on some of the obtained ratios.Conclusion. The results showed that the power of the side waves is zero. It also follows from the calculations carried out that there is no radiation field of surface and outgoing waves, i.e. there is no contribution of them to the radiation pattern.

Optimization of Circular Silencers Internally Hybridized with Multiple Reverse Chambers Using the GA Method

A noise abatement method using an eigen function in conjunction with an optimizer (Genetic Algorithm) under a space-constrain situation is proposed in this study concerning a high order wave propagation effect. An eigen function is used to describe an acoustical field and shorten the acoustical model in a four-pole transfer matrix form. The system’s four-pole matrix estimates the Transmission Loss (TL) for the silencers by multiplying individual four-pole matrices. Additionally, a case study of an air compressor’s broadband noise elimination is introduced. Three types of circular reverse silencers (silencers A-C) internally hybridized with multiple straight chambers are adopted to efficiently suppress the noise wave. Considering broadband characteristics and space-constrained situations, an average-deviation-integrating form used as an objective function is presented and linked to an optimizer-GA (Genetic Algorithm). Acoustical models for silencers A to C are checked before the numerical optimization is performed. A pure tone optimization is executed to assure the consistency of the GA scheme. The result shows that the overall noise drop is enhanced and the number of straight expansion chambers installed at the inlet/outlet of the circular reverse silencer increases. Consequently, concerning a space-constrained situation in the real world and the characteristics of the broadband noise spectrum, an efficient method is proposed in this paper for designing an optimally shaped hybrid muffler using the objective function in conjunction with an optimization scheme.

On the solutions and applications of the generalised Cornell potential

This work presents the solutions to the non-relativistic Schrödinger equation for the generalised Cornell potential using the Nikiforov–Uvarov functional analysis method and the Greene–Aldrich approximation. Energy eigenvalues and eigen functions are obtained in closed form. Eigenvalue spectra of some physical potentials are also deduced from the generalised results. Further, within the framework of the Kratzer and the Coulomb perturbed potentials, two variants of the generalised potential, energy eigenvalue spectra of diatomic molecules viz CH, ScH, N, I and H and the mass spectra of and mesons are determined, respectively. We also compute the expectation values of position inverse and its square, kinetic energy, and square of momentum for the above diatomic molecules by employing the Hellman–Feynman theorem. A close agreement is observed between the present results and some other past works.