Frobenius Method Research Articles

Analytical Solution of Non-Linear Oscillator and Thermal Properties of Interacting System of Real Polyatomic Molecules

The study examined the solution of nonlinear oscillator and thermal properties of interacting system of real polyatomic molecules. A nonlinear oscillator potential was substituted into the radial part of the Schrodinger wave equation and solved using the Frobenius method to obtain the energy eigenvalues of the system. The partition function and the energy eigenvalues are used to determine three thermal properties for four polyatomic molecules which showed that added energy is required in the absence of the nonlinear term as the orbital and magnetic quantum numbers are increased and degeneracy completely removed. Other findings also reveal that as the virial coefficients of the four polyatomic molecules vary, the additive property of entropy is observed as well as the enhancement of the mean energy approximately proportional to the volume of the molecules. The four polyatomic molecules under consideration are environmentally impactful and are indispensable to man.

Frobenius method for Mahler equations

Using Hahn series, one can attach to any linear Mahler equation a basis of solutions at 0 reminiscent of the solutions of linear differential equations at a regular singularity. We show that such a basis of solutions can be produced by using a variant of Frobenius method.

Analysis of axisymmetric hollow cylinder under surface loading using variational principle

In this work, a variational principle–based approach has been adopted to analyze one of the classical linear elasticity problem of the axisymmetric cylinder under surface loading. The use of variational principle results in a set of governing partial differential equations with associated boundary conditions. The equations have been solved using the separation of variable approach and the Frobenius method. A general solution has been derived and used to solve two test cases. The proposed solution is capable of meeting all the boundary conditions. The solution has been validated by comparing it with a finite element–based numerical solution and considering a special limiting condition of a solid cylinder, for which results are available in the literature. Further various studies have been carried out to understand the robustness and limitation of the presented solution.

Stochastic modelling of Karlotoxin influence on prey

Abstract Karlodinium veneficum is type of dinoflagellate plankton present in coastal regions. Harmful algae blooms resulting from uncontrolled growth of K. veneficum often leads to fish kills. They release a toxin named Karlotoxin that is known to affect their prey’s bio-locomotion by stunning and slowing them down. We investigate whether the toxin plays a role in aggregating the prey around the a predator, thereby leading to a local increase in prey density. To achieve this, we closely examine the toxin’s influence on the prey’s probability density distribution with different assumptions on their relative speed in 1D, with either the predator being stationary or swimming at a constant speed. When the predator is stationary, we fully solve the prey’s density distribution for all times, and verify the result by a Monte-Carlo simulation. For a swimming predator, we find the steady-state density distribution of prey analytically. When the predator’s speed $s$ is strictly greater (or less) than the prey ($s-1>0$ or $s-1<0$), the results are verified by Monte-Carlo simulations; when their relative speed $s-1$ has roots, we use the Frobenius method to perform a local analysis for the prey’s density at steady state near the roots, and use the result to derive a scheme for finding the analytical solution. This solution is then verified numerically using a finite difference method. When the roots $x_{1}$ and $x_{3}$ satisfy $s^{\prime}(x_{1})<0$ and $s^{\prime}(x_{3})>0$, we show that the probability density for the prey has a form $|x-x_{1}|^{-s^{\prime}(x_{1})^{-1} -1}$ near the root $x_{1}$, leading to either an integrable singularity or a local maximum. Near the root $x_{3}$, the prey’s density can be represented as a Taylor series and is smooth. In most of the cases mentioned above, toxin leads to the aggregation of prey, however the maximum density does not always occur where the toxin has the highest concentration.

Semi-analytical finite ray-tracing through the quadratic symmetric GRIN lens.

The propagation of light within a gradient index (GRIN) media can be analyzed with the use of differential equations for a given non-homogenous refractive index profile. Numerical methods are often necessary to perform ray-tracing in GRIN media; however, analytical solutions exist for several types of GRIN lenses. In this paper, paraxial and non-paraxial differential equations are derived to calculate the ray path in a GRIN lens. It is shown that the paraxial equation has an analytical solution for a GRIN media with a quadratic profile within the paraxial region. The analytical solution can be obtained by using Legendre polynomials or by the Frobenius method involving a power series. Using the Legendre or Frobenius solution, one can calculate the refractive indices along the ray path. A new recursive relationship is proposed to map the trajectory of light at finite heights. To illustrate the finite ray-tracing method utilizing a non-paraxial differential equation, two lenses (with spherical and elliptical iso-indicial contours) are considered. The lenses' back focal distances, for rays entering the lenses at varying finite heights, are calculated. For each lens, its spherical aberration is estimated. The effective focal length and the shape of the principal surface are also obtained. The accuracy of the results is then compared to the numerical ray-tracing using an optical design software, Zemax OpticStudio. The predicted spherical aberration for the spherical lens differs from numerical ray-tracing by less than λ14 at the marginal zone, while the error for the effective focal length is less than λ100.

Low Energy Analysis of Phase Shifts and Cross-Section for n – 3He System

"Considering the Hulth´en potential as the nuclear part of interaction regular solution is derived by exploiting the Frobenius method. The Jost function is constructed by utilizing its integral representation in terms of regular solution. The scattering phase shifts and cross-sections are calculated for the n−3He system from the on-shell Jost function. The results obtained are in close conformity with the previous works that exist in the literature."

Frobenius methods for analytic second order linear partial differential equations

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

Mapping complete eigensolution space for the analysis of stress transfer in and around the broken fiber: A variational principle based approach

One of the critical reasons for the failure of the Fiber Reinforced Composite (FRC) is fiber breakage. The stress transfer in and around this breakage has remained the topic of research for the last couple of decades. The two-phase composite model is generally considered standard to gain insight into the mechanics. Most of the literature simplifies the problem by considering the shear lag approximations or some other a-prior and/or ad hoc assumptions on various field variables. Moreover, the majority of the study in the literature considers both matrix and fiber to be isotropic. On the contrary, most of the fibers in FRC are transversely isotropic. In the present manuscript, an attempt has been made to provide a solution for this problem with an anisotropic fiber satisfying all the boundary conditions and continuity requirements in an exact sense. The solution strategy adopts the derivation of the solution for individual phases under axial traction and then imposes the boundary and the continuity conditions. The formulation of the problem for the individual phase is based on the variational principle and the solution is based on the method of separation of the variables and the Frobenius method. To validate the solution the numerical results for the various stresses have been compared against the results from the Finite Element Analysis (FEA). The comparison shows a good agreement between them. Further, the effect of the volume fraction and the fiber-matrix stiffness ratio on the Ineffective length, maximum shear stress at the interface, and the far-field stresses in the fiber has been studied.

Analytical solution for nuclear plus atomic Hulthén potential — A perturbative approach

In this work, we construct the analytical solution of the Schrödinger equation for a combined nuclear plus atomic Hulthén potential with different range parameters, using the Frobenius method. The atomic Hulthén potential acts as the screened Coulomb potential to represent the very short-range electromagnetic interaction. It is intended to emphasize how the impact of the combined potential in these situations is routinely examined within the context of nuclear physics. The Jost function is calculated from its integral representation in terms of the regular solution. From the phase of the Jost function, the scattering phase shifts for different partial waves, and further differential scattering cross-section and total cross-sections are calculated for alpha-proton and proton–proton systems. On comparing with the existing experimental data, we conclude that the results obtained are in close conformity with the previous works that exist in the literature.

Transient Translational–Rotational Motion of an Ocean Current Converter Mooring System with Initial Conditions

Systems of ocean current power generation are generally moored deep in the seabed. The mooring system is like ropes. The ropes are very long and can provide tension but not compression, and their dynamic displacement is large and unstable, which is different from traditional structures. To generate high-efficiency ocean current power generation, it is necessary to design a stable mooring system. Maintaining the stability and small dynamic displacement of the ocean current invertor is significantly helpful for the high efficiency of the invertor. In addition, the stability of the mooring system and a small dynamic tension, high safety factor, and long life of the mooring ropes are essential. In this study, we investigate the transient behavior of a mooring system composed of an inverter platform, pontoon, and ropes under initial conditions. An analytical method is proposed. The transient translational and rotational displacements are composed of 36 independent normalized fundamental solutions. The composition depends on the initial conditions. Each fundamental solution is derived by using the Frobenius method. This study proposes the replacement of the traditional single-rope mode with the double-rope parallel mode, which can maintain a high fracture strength and low effective spring constant in the rope. It is verified that this design can decrease instantaneous tension and increase the safety factor of the rope. Additionally, high hydrodynamic damping coefficients can significantly increase the stability of the mooring system.

Kernel well-posedness and computation by power series in backstepping output feedback for radially-dependent reaction–diffusion PDEs on multidimensional balls

Recently, the problem of boundary stabilization and estimation for unstable linear constant-coefficient reaction–diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. Some early success has been achieved under simplifying conditions, such as radially-varying reaction coefficients under revolution symmetry, on a disk or a sphere. These particular cases notwithstanding, the problem remains open. The main issue is that the equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations. Traditionally, well-posedness of these equations has been proved by transforming them into integral equations and then applying the method of successive approximations. In this case, with the resulting integral equation becoming singular, successive approximations do not easily apply. This paper takes a different route and directly addresses the kernel equations via a power series approach (in the spirit of the method of Frobenius for ordinary differential equations), finding in the process the required conditions for the radially-varying reaction (namely, analyticity and evenness) and showing the existence and convergence of the series solution. This approach provides a direct numerical method that can be readily applied, despite singularities, to both control and observer boundary design problems.

Comment on: “Effects of the Kaluza–Klein theory on a Klein–Gordon particle with position-dependent mass”

In this paper, we argue that the results shown in [Mod. Phys. Lett. A 34, 1950319 (2019)] are not correct. The truncation of the Frobenius method only yields some particular bound states from which one cannot derive any sound physical conclusion. The existence of allowed model parameters is an artifact of such procedure. Besides, the authors failed to obtain most of the solutions given by the truncation of the Frobenius series as well as their connection with the actual eigenvalues of the problem.

Aspects of thermal one-point functions and response functions in AdS black holes

We revisit the problem of analytically computing the one-point functions for scalar fields in planar AdS black holes of arbitrary dimension, which are sourced by the Weyl squared tensor. We analyze the problem in terms of power series expansions around the boundary using the method of Frobenius. We clarify the pole structure of the final answer in terms of operator mixing, as argued previously by Grinberg and Maldacena. We generalize the techniques to also obtain analytic results for slowly modulated spatially varying sources to first nontrivial order in the wave vector for arbitrary dimension. We also study the first-order corrections to the one-point function of the global AdS black hole at large mass, where we perturb in terms that correspond to the curvature of the horizon.

Using Local Smith Normal Form for Numerical Implementation of the Generalized Frobenius Method

Using Local Smith Normal Form for Numerical Implementation of the Generalized Frobenius Method

Klein–Gordon oscillator with scalar and vector potentials in topologically charged Ellis–Bronnikov-type wormhole

In this work, we study the Klein–Gordon oscillator with equal scalar and vector potentials in a topologically charged Ellis–Bronnikov wormhole space-time background. The behaviour of a relativistic oscillator field is studied with a position-dependent mass via transformation [Formula: see text] and vector potential through a minimal substitution in the wave equation. Simplifying the Klein–Gordon oscillator equation for three different types of potential, such as linear confining, Coulomb-type, and Cornell-type potential and we arrive at a second-order differential equation known as the biconfluent Heun (BCH) equation and the corresponding confluent Heun function. Finally, we solve the wave equation by the Frobenius method as a power series expansion around the origin and obtain the energy levels and the wave function.

Closed-Form Solution for the Natural Frequencies of Low-Speed Cracked Euler–Bernoulli Rotating Beams

In this study, two closed-form solutions for determining the first two natural frequencies of the flapwise bending vibration of a cracked Euler–Bernoulli beam at low rotational speed have been developed. To solve the governing differential equations of motion, the Frobenius method of solution in power series has been used. The crack has been modeled using two undamaged parts of the beam connected by a rotational spring. From the previous results, two novel polynomial expressions have been developed to obtain the first two natural frequencies as a function of angular velocity, slenderness ratio, cube radius and crack characteristics (depth and location). These expressions have been formulated using multiple regression techniques. To the knowledge of the authors, there is no similar expressions in the literature, which calculate, in a simple way, the first two natural frequencies based on beam features and crack parameters, without the need to know or solve the differential equations of motion governing the beam. In summary, the derived natural frequency expressions provide an extremely simple, practical, and accurate instrument for studying the dynamic behavior of rotating cracked Euler–Bernoulli beams at low angular speed, especially useful, in the future, to establish small-scale wind turbines’ maintenance planes.

An alternative expression of the time domain free surface Green function for deep water

An alternative expression of the time domain free surface Green’s function is proposed. Its source and image source contribution satisfies the homogeneous Neumann condition on the mean free surface, therefore it can be useful for the boundary element method based on the double-body flow linearization. Furthermore, it is proved that it is also the solution of Clement’s 4th order Ordinary Differential Equation (Clément, 1998). The Frobenius method with time step movement proposed by Chuang et al. (2007) is adopted for accurate and efficient evaluation.

Dynamic response of an axisymmetric transversely isotropic medium with its modulus varying with depth subjected to LWD load

ABSTRACT In this paper, the dynamic response of an axisymmetric transversely isotropic medium with its modulus varying with depth subjected to the lightweight deflectometer (LWD) load is calculated by an analytical solution method. In this model, the analytical solutions are expressed as power series, and both cases of vertical modulus, increasing and decreasing with depth, are considered. This study can be divided into three key steps. Firstly, the ordinary differential equations are obtained via the Hankel–Laplace transform, and the general solutions of dynamic response are obtained according to the Frobenius method. Secondly, combined with the boundary conditions, the dynamic responses of the subgrade in the time domain are calculated. The accuracy and reliability of this dynamic model are proved by parameter convergence analysis, finite element method and comparison validation. Lastly, the effect of thickness, non-uniform distribution of modulus and transverse isotropy subjected to LWD load are analysed and discussed. The calculation methods and the corresponding results are meaningful for the design and non-destructive monitoring of the subgrade.

Steady blood flow through a porous microchannel with the influence of an inclined magnetic field

This research deals with steady blood flow through a porous microchannel with the influence of an inclined magnetic field. In the research, we formulate time-independent differential equations which represent the blood momentum, energy equation, and mass concentration in the flood governing the flow. The governing equations were scaled to be dimensionless using some prescribed dimensionless parameters in the work. The reduced dimensionless governing equations were further solved using the Frobenius method (FM), where the blood velocity, mass concentration and blood temperature profiles were obtained. Numerical simulation was carried out using Wolfram Mathematica, version 12, to investigate the effect of the variation of the values of pertinent parameters on the flow profiles. In the investigation, it was revealed that the variation of Grashof number, Schmidt number, and porosity parameters increased the blood velocity before decreasing to zero when the boundary layer thickness was at its peak, while the variation of magnetic field and Prandtl number reduced blood velocity. However, the magnetic field angle of inclination initially increased the blood velocity before decreasing. In a similar vein, as the Schmidt number and chemical reaction parameters decrease the mass concentration level in the fluid, the blood temperature increases for the variation of the Schmidt number. In conclusion, the problem was formulated, solved, and simulation was done successfully, where various pertinent parameters were checked. These results are very useful for clinicians and scientists who are interested in investigating the role of some pertinent factors in understanding blood circulation problems theoretically.

Quasinormal modes and stability of accelerating Reissner-Norsdtröm AdS black holes

We investigate the numerical stability of accelerating AdS black holes against linear scalar perturbations. In particular, we study the evolution of a probe non-minimally coupled scalar field on Schwarzschild and Reissner-Nordström AdS black holes with small accelerations by computing the quasinormal modes of the perturbation spectrum. We decompose the scalar field Klein-Gordon equation and study the eigenvalue problem for its angular and radial-temporal parts using different numerical methods. The angular part is written in terms of the Heun solution and expanded through the Frobenius method which turns out to give eigenvalues qualitatively similar to the ones obtained through the spherical harmonics representation. The radial-temporal evolution renders a stable field profile which is decomposed in terms of damped and purely imaginary oscillations of the quasinormal modes. We calculate the respective frequencies for different spacetime parameters showing the existence of a fine-structure in the modes, for both real and imaginary parts, which is not present in the non-accelerating AdS black holes. Our results indicate that the Schwarzschild and Reissner-Nordström AdS black holes with small accelerations are stable against linear scalar perturbations.