Frobenius Method Research Articles

On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard program

We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, and we establish the classical gluing method of Mazzeo and Pacard for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of an ODE, and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. No traditional phase-plane analysis is available here. Instead, first, we provide a rigorous construction of radial fast-decaying solutions by a blow-up argument and a bifurcation method. Second, we use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be. Third, for the linear theory, we examine a fractional Schr\"{o}dinger equation with a Hardy type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a two-dimensional kernel as in the second order case.

Vibration analysis of transversely isotropic hollow cylinder considering three different theories using the matrix Frobenius method

Purpose The purpose of this paper is to deal with the three-dimensional analysis of free vibrations in a stress-free and rigidly fixed homogeneous transversely isotropic hollow cylinder in the context of three-phase-lag (TPL) model of hyperbolic thermoelasticity. Design/methodology/approach The matrix Frobenius method of extended power series is employed to obtain the solution of coupled ordinary differential equations along the radial coordinate. Findings The natural frequency, dissipation factor and inverse quality factor in the stress-free and rigidly fixed hollow cylinder get significantly affected due to thermal vibrations and thermo-mechanical coupling. Originality/value The modified Bessel functions and matrix Frobenius method have been directly used to study the vibration model of a homogeneous, transversely isotropic hollow cylinder in the context of TPL model based on three-dimensional thermoelasticity.

Frobenius method for the solution of Klein-Gordon-Fock equation with equal scalar and vector oscillator plus potential

In this paper, the solution of Klein Gordon Equation is sought. Frobenius method was used to solve the Klein Gordon (KG) equation with equal scalar and vector harmonic oscillator plus inverse quadratic potential for s - waves. A corresponding un-normalized wavefunction was obtained for the Frobenius equation in the form of a power series.

A New Integral Transform for Solving Higher Order Linear Ordinary Laguerre and Hermite Differential Equations

In this work a new integral transform is introduced and applied to solve higher order linear ordinary Laguerre and Hermite differential equations. We compare present transform with other method such as Frobenius Method.

Analysis of Free Vibrations of Axisymmetric Functionally Graded Generalized Viscothermoelastic Cylinder Using Series Solution

The analysis of free vibrations of axisymmetric functionally graded isotropic viscothermoelastic hollow cylinder has been investigated in the radial direction. The material of viscothermoelastic cylinder has been assumed to be graded in the radial direction due to simple exponent law. The governing partial differential equations are transformed into ordinary differential equations with th e help of time harmonic vibrations. The extended power series solution of matrix Frobenius method has been applied to derive the analytical solutions for stresses, displacement and temperature change. The frequency equations of various types of modes of vibrations have been derived in a compact form. To investigate the numerical features of vibrations, the analytical results of frequency equations have been further solved with the help of fixed-point numerical iteration technique using MATLAB software tools. The polymethyl methacrylate material has been used for numerical computations. The numerical results have been presented for frequency shift, natural frequencies, thermoelastic damping, stresses, displacement and variation of temperature. With the increase in the values of grading index, variation of vibrations in field functions go on decreasing. The numerical results show alternate variations in homogenous materials in contrast to inhomogeneous materials. The behaviour of deformation, temperature change, frequency shift and thermoelastic damping have been monitored (increase or decrease) with the help of grading index (in-homogeneity parameter).

Simplified models for determining the response of an isotropic, continuously nonhomogeneous half-plane to a moving distributed line load

The problem of a vertical distributed line load moving with constant speed on the surface of an isotropic half-plane with shear modulus varying continuously with depth has been very recently solved analytically by the authors in an exact manner. The same problem is solved here also analytically under various reasonably simplified assumptions that effectively reduce the coupled system of the two governing partial differential equations of motion into an uncoupled one. The assumptions are zero horizontal displacement, or zero horizontal normal stress, or zero horizontal normal stress plus zero derivative of the horizontal displacement with respect to the vertical coordinate. The method of complex Fourier series involving the horizontal coordinate and the time is used to reduce the uncoupled two partial differential equations to ordinary ones with variable coefficients, which are solved by the method of Frobenius in closed form. Comparison of the approximate solutions corresponding to the aforementioned simplified assumptions against the ‘exact’ solution by means of parametric studies serves to assess the degree of their accuracy for both cases of the shear modulus increasing and decreasing with depth.

Forced Convection Heat Transfer of a Giesekus Fluid in Circular Micro-Channels Subjected to a Constant Wall Temperature

Abstract In this paper, an exact analytical solution for forced convective heat transfer of nonlinear viscoelastic fluid in isothermal circular micro-channel is presented. The nonlinear Giesekus constitutive equation is used to model the Giesekus fluid heat transfer in micro-channel with constant wall temperature, which is the main innovative aspect of the current study. This constitutive equation is a powerful tool and able to model the fractional viscometric functions, extensional viscosity, and elastic property. The solution of temperature profile and Nusselt number is obtained based on the Frobenius method. The effects of Weissenberg number, mobility factor, slip coefficient, and Navier index on temperature distribution, velocity profile, and Nusselt number are investigated in detail. The results show that the increases in both slip coefficient and Navier index cause the increases in slip velocity and maximum dimensionless temperature at the wall and the micro-channel center, respectively. Moreover, the Nusselt number has an upward trend with increases in slip coefficient and Navier index parameters. The results are indicated that the flow and temperature fields have a complex relation with mobility factor which controls the level of the nonlinearity of the Giesekus model. Additionally, three correlations for Nusselt number of Giesekus flow in micro-channel are presented.

Free flexural vibration of tapered beams

The free flexural vibration behaviour of a range of tapered beams is investigated by making use of the exact solutions of the governing differential equations and then imposing the necessary boundary conditions. This research is prompted by a recently published paper in Computers and Structures which used the Frobenius method of series solution but needed 130 to 300 terms in the series when solving the free vibration problem of tapered beams using the transfer matrix method. This paper investigates the same problem using a direct approach, but with a twofold purpose. The first is to show that an exact solution for the problem is possible by using Bessel functions rather than relying on a series solution which is somehow unnecessary and from a computational standpoint inefficient. The second reason for writing this paper is to broaden the scope of the recently published paper by not focusing on only one type of cross-section, but on a range of cross-sections and yet retaining the exactness of the results. The application of Bessel functions is demonstrated when developing the theory and computing the numerical results. The results presented can be used as an aid to validate the finite element and other approximate methods.

Forced Vibration Analysis in Axisymmetric Functionally Graded Viscothermoelastic Hollow Cylinder Under Dynamic Pressure

The forced vibrations of axisymmetric inhomogenous isotropic viscothermoelastic hollow cylinder under periodic dynamic pressure have been studied. The material was taken inhomogenous due to easy exponent law in radial direction. By applying time harmonics variation technique, the partial differential equations were converted into ordinary differential equations. These ordinary differential equations were solved by applying series solution of matrix Frobenius method analytically to represent displacement, temperature and stresses. Numerically simulated outcomes were presented graphically to express the effect of functionally graded material disk for different values of grading parameter. With the increase in value of grading index, variation of field functions go on decreasing. The variation of field functions show high variation in homogenous materials in contrast to low variation in inhomogeneous materials.

Three-dimensional vibration analysis in transversely isotropic cylinder with matrix Frobenius method

This article deals with the study of three-dimensional vibrations in stress free as well as rigidly fixed, thermally insulated (or isothermal), homogeneous transversely isotropic solid cylinder under the purview of three-phase lag model of generalized thermoelasticity. The displacement potential functions have been introduced in the equations of motion and heat conduction in order to decouple the purely shear and longitudinal motions. The matrix Frobenius method of extended power series is employed to obtain the solution of coupled ordinary differential equations along the radial coordinate. Circumferential wave propagation in cylindrical curved plate is discussed. To illustrate the analytic results, the numerical solution of various relations and equations have been carried out to compute the lowest frequency, inverse quality factor, and dissipation factor of vibrations and the computer-simulated results are presented graphically for different thermoelastic models.

On the numerical solution of matrix Bessel equations

AbstractThis article deals with the numerical evaluation of certain systems of second order differential equations that are analogous in structure to the scalar Bessel equations, i.e. . The governing equation includes three distinct coefficient matrices that we assume to be real, symmetric, and positive definite. However, allowance is also made for G to be semi‐definite. We first review the conditions under which rigorous closed‐form solutions are possible, and this occurs when the various matrices share a common set of eigenvectors, even if not the eigenvalues. We find in that case that the response consists in a linear combination of conventional Bessel functions. More generally, we propose three alternative methods to accomplish the evaluation: An ascending series solution based on the Frobenius method. An effective asymptotic solution that provides results in very close agreement with the first method. It is stable and accurate for a wide range of values of the argument ‐ more so than the first method. A direct numerical integration solution using the fifth order Runge‐Kutta‐Nyström algorithm. It works well, except for some idiosyncrasies that depend on the presence of eigenvalues with widely different magnitudes, as will be explained. We determined that all of these techniques produce highly accurate response functions when measured in terms of the residual to the differential equation, and in all cases we found that residual to be negligible.

Dynamic response of an isotropic elastic half-plane with shear modulus varying with depth to a load moving on its surface

Abstract The dynamic response of a half-plane soil medium to a load moving on its surface is obtained analytically-numerically. The soil is assumed to be linear elastic, isotropic and continuously non-homogeneous with its shear modulus varying with depth. Both cases of the shear modulus increasing and decreasing with depth are considered. The load is assumed to be distributed and moves with constant speed. The load and the soil displacements are expanded in complex Fourier series with respect to the horizontal direction x and the time t and thus the partial differential equations governing the soil motion are reduced to ordinary differential equations with variable coefficients, which are solved analytically by the method of Frobenius. The correctness of the solution in series form is checked by using it to solve some special case problems for which there exist closed form solutions in the literature. Parametric studies are conducted to assess the effect of non-homogeneity on the response for both cases of the shear modulus increasing and decreasing with depth.

The Fractional Laguerre Equation: Series Solutions and Fractional Laguerre Functions

In this paper, we propose a fractional generalization of the well-known Laguerre differential equation. We replace the integer derivative by the conformable fractional derivative of order $0 < \alpha<1.$ We then apply the Frobenius method with the fractional power series expansion to obtain two linearly independent solutions of the problem. For certain eigenvalues, the infinite series solutions truncate to obtain the singular and non-singular fractional Laguerre functions. We obtain the fractional Laguerre functions in closed forms, and establish their orthogonality result. The applicability of the new fractional Laguerre functions is illustrated.

K-Hypergeometric Series Solutions to One Type of Non-Homogeneous k-Hypergeometric Equations

In this paper, we expound on the hypergeometric series solutions for the second-order non-homogeneous k-hypergeometric differential equation with the polynomial term. The general solutions of this equation are obtained in the form of k-hypergeometric series based on the Frobenius method. Lastly, we employ the result of the theorem to find the solutions of several non-homogeneous k-hypergeometric differential equations.

АНАЛІЗ ЗАГАЛЬНОГО РОЗВ’ЯЗКУ ОДНОГО ЛІНІЙНОГО ОДНОРІДНОГО ДИФЕРЕНЦІАЛЬНОГО РІВНЯННЯ З ПРАВИЛЬНОЮ ОСОБЛИВОЮ ТОЧКОЮ

У роботі виявлено, що обране для дослідження лінійне однорідне диференціальне рівняння другого порядку, яке має правильну особливу точку, за допомогою перетворення Куммера-Ліувілля зводиться до виродженого гіпергеометричного рівняння, яке подане в формі Уіттекера. Встановлені співвідношення між коефіцієнтами заданого рівняння, виродженого гіпергеометричного рівняння у загальному вигляді та у формі Уіттекера. На основі цих співвідношень показано, що досліджуване рівняння має лише один незалежний фундаментальний розв’язок у вигляді виродженої гіпергеометричної функції. Другий фундаментальний розв’язок може бути знайдений або за допомогою метода Фробеніуса у вигляді узагальненого степеневого ряду, або виражений через функцію Трікомі. В останньому випадку загальний розв’язок досліджуваного рівняння є лінійною комбінацією функцій Уіттекера.

Massive nonminimally coupled scalar field in Reissner-Nordström spacetime: Long-lived quasinormal modes and instability

Here we show that the phenomenon of arbitrarily long-lived quasinormal modes (called quasiresonances) of a massive scalar field in the vicinity of a black hole is not an artifact of the test field approximation, but takes place also when the (derivative) coupling of a scalar field with the Einstein tensor is taken into consideration. We observe that at large coupling and high multipole numbers, the growing modes appear in the spectrum, which are responsible for the eikonal instability of the field. For small coupling, when the configuration is stable, there appear the purely imaginary quasinormal modes which are nonperturbative in the coupling constant. At the sufficiently small coupling the nonminimal scalar field is stable and the asymptotic late-time tails are not affected by the coupling term. The accurate calculations of quasinormal frequencies for a massive scalar field with the derivative coupling in the Reissner-Nordstr\"om black-hole background are performed with the help of Frobenius method, time-domain integration and WKB expansion.

An exact analysis on heat convection of nonlinear viscoelastic flows in isothermal microtubes under slip boundary condition

In this paper, forced convective heat transfer of Phan–Thien–Tanner fluid in isothermal microtubes is studied analytically. The nonlinear Navier slip law is used to model the slip boundary condition at microtubes wall. The second-order eigenvalue form of related ODE is solved using the Frobenius method and the temperature distribution obtained in the form of power series. The Nusselt number is also calculated using the modal analysis technique. The effects of Weissenberg number, slip coefficient, and Navier index on velocity profile, temperature distribution and Nusselt number are investigated in detail. The results show that the increases in both slip coefficient and Navier index cause to increases in slip velocity and maximum dimensionless temperature at the wall and the microtube center, respectively. Moreover, the Nusselt number is enhanced by increasing the slip coefficient and Navier index parameters.

On Neutral and Unstable Modes and on the Role of Reynolds Stress in the Stability Problem of Density Stratified Shear Flows in Channels with Variable Bottom

The stability problem of inviscid incompressible stratified shear flows in sea straits of arbitrary cross sections is formulated without the Boussinesq approximation. Series solutions of the stability equation are found by the method of Frobenius and these solutions are used to study the existence of neutral modes when the Richardson number is larger than one quarter. The energy aspect of the problem is studied by discussing the role of the Reynolds stress and its variation in the vertical direction for both unstable modes and neutral modes adjacent to the unstable modes. The interfacial conditions at layers of discontinuity of the basic velocity, density or topography are derived and these are used in the study of instability of two examples of basic flows.

A transfer matrix method for in-plane bending vibrations of tapered beams with axial force and multiple edge cracks

This paper proposes a transfer matrix method for the bending vibration of two types of tapered beams subjected to axial force, and it is applied to analyze tapered beams with an edge or multiple edge open cracks. One beam type is assumed to be reduced linearly in the cross-section height along the beam length. The other type is a tapered beam in which the cross-section height and width with the same taper ratio is linearly reduced simultaneously. Each crack is modeled as two sub-elements connected by a rotational spring, and the method can evaluate the effect of cracking on the desired number of eigenfrequencies using a minimum number of subdivisions. Among the power series available for the solutions, the roots of the differential equation are computed using the Frobenius method. The computed results confirm the accuracy of the method and are compared with previously reported results. The effectiveness of the proposed methods is demonstrated by examining specific examples, and the effects of cracking and axial loading are carefully examined by a comparison of the single and double tapered beam results.

Application of the Frobenius Method to the Schr？dinger Equation for a Spherically Symmetric Hyperbolic Potential

In this paper, an efficient technique for computing the bound state energies and wave functions of the Schrodinger Equation (SE) associated with a new class of spherically symmetric hyperbolic potentials is developed. This technique is based on a recent approximation scheme for the orbital centrifugal term and on the use of the Frobenius method (FM). The bound state eigenvalues are given as zeros of calculable functions. The corresponding eigenfunctions can be obtained by substituting the calculated energies into the recurrence relations for the expanding coefficients of the Frobenius series representing the solution. The excellent performance of this technique is illustrated through numerical results for some special cases like Poschl-Teller potential (PTP), Manning-Rosen potential (MRP) and Poschl-Teller polynomial potential (PTPP), with an application to the Gaussian potential well (GPW). Comparison with other methods is presented. Our results agree noticeably with the previously reported ones.