Second-order Homogeneous Differential Equations Research Articles

Uniform Asymptotic Approximation Method with Pöschl–Teller Potential

In this paper, we study analytical approximate solutions for second-order homogeneous differential equations with the existence of only two turning points (but without poles) by using the uniform asymptotic approximation (UAA) method. To be more concrete, we consider the Pöschl–Teller (PT) potential, for which analytical solutions are known. Depending on the values of the parameters involved in the PT potential, we find that the upper bounds of the errors of the approximate solutions in general are ≲0.15∼10% for the first-order approximation of the UAA method. The approximations can be easily extended to high orders, for which the errors are expected to be much smaller. Such obtained analytical solutions can be used to study cosmological perturbations in the framework of quantum cosmology as well as quasi-normal modes of black holes.

A coupled hygro-elastic 3D model for steady-state analysis of functionally graded plates and shells

Abstract This 3D coupled hygro-elastic model proposes the three-dimensional (3D) equilibrium equations associated with the 3D Fick diffusion equation for spherical shells. The primary unknowns of the problem are the displacements and the moisture content. This coupled 3D exact shell model allows to understand the effects of the moisture field in relation with the elastic field on stresses and deformations in different plates and shells. This model is specifically developed for configurations including functionally graded material (FGM) layers. Four different geometries are analyzed using an orthogonal mixed curvilinear reference system. The main advantage of this reference system for spherical shells is the degeneration of the equations to those for simpler geometries. The solving method is the exponential matrix method in the thickness direction. The closed-form solution is possible because of simply supported sides and harmonic forms for displacements and moisture content. The moisture content amplitudes are directly applied at the top and bottom outer faces through steady-state hypotheses. The final system is based on a set of coupled homogeneous second-order differential equations. The moisture field effects are evaluated for the static analysis in terms of displacement, strain, and stress components. After preliminary validations, used to better understand how to properly define the calculation of the curvature-related terms and FGM properties, four new benchmarks are proposed for several thickness ratios, geometrical data, FGM configurations, and moisture values imposed at the external surfaces. From the results, it is clear the accordance between the uncoupled hygro-elastic model and this new coupled hygro-elastic model when the 3D Fick diffusion law is employed. Both effects connected with the thickness layer and the embedded material are included in the 3D hygro-elastic analyses proposed. The 3D coupled hygro-elastic model is simpler than the uncoupled one because the 3D Fick diffusion law does not have to be separately solved.

Influence of logarithmic radial inhomogeneity of the elastic modulus on the stress-strain state of a hollow cylinder

The paper is devoted to the formulation and solution of the boundary value problem for a hollow circular cylinder whose properties continuously depend on the radial coordinate. It is shown that the mathematical model of such a problem is a homogeneous second-order differential equation with variable coefficients and two boundary conditions. In the case where the Poisson's ratio is constant and the Young's modulus logarithmically depends on the radial coordinate, the resulting equation can be reduced to the Whittaker equation, the solution of which is a linear combination of confluent hypergeometric functions. Based on this, an analytical solution of the boundary value problem is constructed in this particular case. Functions describing displacements, strains, and stresses occurring in the cylinder are found.

Direct connection between Navier and spherical harmonic kernels in elasticity

<abstract> <p>Linear isotropic elasticity is an interesting branch of continuum mechanics, described by the fundamental laws of Hooke and Newton, which are combined in order to construct the governing generalized Navier equation of the displacement within any material. Implying time-independence and in the absence of external body forces, the latter is reduced to the corresponding form of a homogeneous second-order partial differential equation, whose solution is given via the Papkovich differential representation, which expresses the displacement field in terms of harmonic functions. On the other hand, spherical geometry provides the most widely used framework in real-life applications, concerning interior and exterior problems in elasticity. The present work aims to provide a little progress, by producing ready-to-use basic functions for linear isotropic elasticity in spherical coordinates. Hence, we calculate the Papkovich eigensolutions, generated by the spherical harmonic eigenfunctions, obtaining connections between Navier and spherical harmonic kernels. A set of useful results are provided at the end of the paper in the form of examples, regarding the evaluation of displacement field inside and outside a sphere.</p> </abstract>

Three-Dimensional Acoustic Analysis of a Rectangular Duct with Gradient Cross-Sections in High-Speed Trains: A Theoretical Derivation

Rectangular ducts used in the air-conditioning system of a high-speed train should be carefully designed to achieve optimal acoustic and flow performance. However, the theoretical analysis of the rectangular ducts with gradient cross-sections (RDGC) at frequencies higher than the one-dimensional cut-off frequency is rarely published. This paper has developed the three-dimensional analytical solutions to the wave equations of the expanding and shrinking RDGCs. Firstly, a homogeneous second-order variable coefficient differential equation is derived from the wave equations. Two coefficients of the solution to the differential equation are set to zero to ensure convergence. Secondly, the transfer matrices of the duct systems composed of multiple RDGCs are derived from the three-dimensional solutions. The transmission losses of the duct systems are then calculated from the transfer matrices and validated with the measurement. Finally, the acoustic performance and flow efficiency of the RDGCs with different geometries are discussed. The results show that the REC with double baffles distributed transversely has good performance in both acoustic attenuation and flow efficiency. This study shall provide a helpful guide for designing rectangular ducts used in high-speed trains.

A discrete scheme of the fluid motion equation based on the pore-scale SPH method

To study the phenomenon of convection diffusion and solute migration in porous media, the smoothed particle hydrodynamics method is used to solve the N–S equation describing pore water flow and the linear homogeneous second-order differential equation describing the solute diffusion process. The accuracy of the simulation method is verified by solving the one-dimensional definite solution of the convection diffusion equation at a low Peclet (Pe) number. Under the condition of a high Pe number, the simulation experiment of solute breakthrough porous media at the pore scale is simulated. Through the analysis of the velocity distribution of fluid particles in pore channels of porous media and the mechanical dispersion process of solutes in porous media, it is found that the flow velocity presents a parabolic distribution in the pore channel, and the particles with high velocity are mainly concentrated in the middle of the pore channel between the solid particles. The solute dispersion is obvious in porous media, and some fluid particles containing solutes penetrate porous media earlier along fast-flowing pore channels. Finally, the solute breakthrough curve and the motion path length of the particle are obtained through the simulation experiment. The calculation methods of the dispersion degree and tortuosity of porous media are further given, and the dispersion degree of porous media in this paper is 0.000 95 m.

Dynamic analysis of functionally graded sandwich beams using a semi-analytic method named scaled boundary finite element method

In this research, free and forced vibrations of the functionally graded material (FGM) sandwich beams are investigated by using the scaled boundary finite element method (SBFEM). The innovation of this method is that the discretization only exists in the axial direction of the beam, which reduces the spatial dimension of problem by one, the time-consuming mesh generation processes are greatly reduced. The homogeneous second-order differential equations of the beams are derived by the method of virtual work principle with the radial coordinates as independent variables, which realizes the characteristics of the FGM changing along the radial direction. By using the state space method and the Padé series expansion method, the governing equations are further solved analytically, and the mass and stiffness matrices of each layer of the FGM sandwich beams are obtained. The combination between layers of the FGM sandwich beams is realized according to continuity conditions of displacement. Therefore, the free and forced vibrations using Newmark time integration method of the FGM sandwich beams are further realized. A special attention is given to two different types of the FGM sandwich beams, one is the beam with FGM face sheets and the other is the beam with FGM core. Three different types of dynamic loadings are considered in the analysis of forced vibration. The convergence of the proposed method for solving the dynamic characteristics of the beams is verified by increasing the order of the higher order spectral elements. In order to prove the numerical stability of the current method for dynamic analysis of the FGM sandwich beams, the frequency of the FGM sandwich beams under various parameters and the transient vertical displacement of the beam loading points under three different loadings are solved. The numerical results of the current method are compared with the results obtained by using different theories in the other literature, which shows a good agreement. In addition, on the basis of the research, the doped FGM core in the uniform core is discussed by using the current method, which provides some guidance for the optimal design of the dynamic characteristics of the beams.

Hyers\u2013Ulam stability of second-order differential equations using Mahgoub transform

The aim of this research is investigating the Hyers–Ulam stability of second-order differential equations. We introduce a new method of investigation for the stability of differential equations by using the Mahgoub transform. This is the first attempt of the investigation of Hyers–Ulam stability by using Mahgoub transform. We deal with both homogeneous and nonhomogeneous second-order differential equations.

Indefinite integrals involving Jacobi polynomials from integrating factors

ABSTRACT A method was presented recently for deriving integrals of special functions using two kinds of integrating factor for the homogeneous second-order linear differential equations which many special functions obey. The classical orthogonal polynomials are well suited for this method, and results are given here for Jacobi polynomials. Results have been previously presented for Gegenbauer, Hermite and Laguerre polynomials. All the integrals presented here appear to be new and have been checked using Mathematica.

Indefinite integrals for some orthogonal polynomials obtained using integrating factors

ABSTRACT A method has been presented recently for deriving integrals of special functions using two kinds of integrating factor for the homogeneous second-order linear differential equations which many special functions obey. The classical orthogonal polynomials are well-suited for this method, and results are given here for the Gegenbauer, Hermite and Laguerre polynomials. All the integrals presented here appear to be new and have been checked using Mathematica. Results for other orthogonal polynomials will be presented separately.

Analytic Models of Roll Call Voting Dynamics

Roll call modeling is an essential component of analyzing a political system. Current models focus on individual decision-making, and most of them do not take advantage of voting dynamics. Some political systems, such as Ukraine’s Verkhovna Rada, are inherently dynamic and should be modeled as such. Therefore, a roll call model is developed from a linear second-order homogeneous differential equation. This model equation is fit to Verkhovna Rada votes from the seventh and eighth convocations. The model determines whether or not bills will reach the passing threshold with 77% and 85% accuracy for the seventh and eighth convocations, respectively. It is shown that the dynamic legislative model is slightly less accurate than a neural network, but it is significantly more interpretable. This interpretability is vitally important, as it is what makes models meaningful beyond their predictive power. It is found that bills sponsored by the president show quantitatively different behavior than ordinary bills and the ordinary bills are largely decided in the first two votes. Furthermore, our models have intuitive theoretical implications, some of which are back by prior work. The models suggest that MPs are less willing to change their vote on bills as iterations increase and they are more sensitive to change the public opinion if the bill is sponsored by the president. While the majority of bills are modeled well, about 25% of votes have greater than 10% error. Investigation of these votes indicates that some votes may be impossible to predict without a more complex model which incorporates contextual information. Finally, the information from a bill’s first two votes is also leveraged through a vote switching network. This directed network gives insight into who sends the most powerful signals and who follows them. An ensemble of centrality members is then used to identify the legislator’s most influential members.

Eigensolutions, scattering phase shift and thermodynamic properties of Hulthẻn-Yukawa potential

The amendibility of a spin-0 and spin-1 particle with a combined potential in the presence of the Duffin-Kemmer-Petiau wave equation is highly recommendable. Thus, the approximate bound state of the Duffin-Kemmer-Petiau equation and Schrӧdinger equation were obtained with a combination of Hulthẻn and Yukawa potentials in the framework of asymptotic iteration method and parametric Nikiforov-Uvarov method respectively for any arbitrary angular momentum quantum number J using a suitable approximate scheme to the centrifugal term. This was done when the second-order homogeneous differential equation was transformed to a form of recurrence relation from which a quantization condition obtained was used to calculate the eigenvalue energy equation and the corresponding wave function. In other to apply more application to this work, the scattering phase shift of the Duffin-Kemmer-Petiau equation was calculated and the thermodynamic properties of the potential under consideration were also calculated in view of the Schrӧdinger equation. It is noted that the results obtained by varying the two strengths of the potential differs due to the effect of the screening parameter.

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.

Taylor approximation to treat nonlocality in the scattering process

Study of scattering process in the nonlocal interaction framework leads to an integro-differential equation. The purpose of the present work is to develop an efficient approach to solve this integro-differential equation with high degree of precision. The method developed here employs Taylor approximation for the radial wave function which converts the integro-differential equation in to a readily solvable second-order homogeneous differential equation. This scheme is found to be computationally efficient by a factor of 10 when compared to the iterative scheme developed in J.~Phys.~G~Nucl.~Part.~Phys.~{\bf 45},~015106~(2018). The calculated observables for neutron scattering off $^{24}$Mg, $^{40}$Ca, $^{100}$Mo and $^{208}$Pb with energies up to 10 MeV are found to be within at most 8$\%$ of those obtained with the iterative scheme. Further, we propose an improvement over the Taylor scheme that brings the observables so close to the results obtained by iterative scheme that they are visually indistinguishable. This is achieved without any appreciable change in the run time.

Nonintegrability of Parametrically Forced Nonlinear Oscillators

We discuss nonintegrability of parametrically forced nonlinear oscillators which are represented by second-order homogeneous differential equations with trigonometric coefficients and contain the Duffing and van der Pol oscillators as special cases. Specifically, we give sufficient conditions for their rational nonintegrability in the meaning of Bogoyavlenskij, using the Kovacic algorithm as well as an extension of the Morales–Ramis theory due to Ayoul and Zung. In application of the extended Morales–Ramis theory, for the associated variational equations, the identity components of their differential Galois groups are shown to be not commutative even if the differential Galois groups are triangularizable, i. e., they can be solved by quadratures. The obtained results are very general and reveal their rational nonintegrability for the wide class of parametrically forced nonlinear oscillators. We also give two examples for the van der Pol and Duffing oscillators to demonstrate our results.

An Exact Solution of the Second-Order Differential Equation with the Fractional/Generalised Boundary Conditions

We analysed the initial/boundary value problem for the second-order homogeneous differential equation with constant coefficients in this paper. The second-order differential equation with respect to the fractional/generalised boundary conditions is studied. We presented particular solutions to the considered problem. Finally, a few illustrative examples are shown.

Restoration of the signal of permissions caused by aperiodic process, when defining constant useful signal

Reviewed the general solution of the homogeneous second-order differential equation that describes the aperiodic processes. Coming out of the system of the four transcendental equations, which is composed from the primary to the fourth order inclusive of this solution, determined the amplitudes and damping decrements of these processes. Knowledge of these parameters allows us to describe the interference signal as a function of time.

CFRP reinforcement adhesive joint performance on the transversely damaged steel elements under axial tension

The analytic solution for analysis and design of bonded CFRP-reinforcement of tension elements with discontinuity type damage are presented. Analytical expressions are obtained from integrating a homogeneous second-order differential equation describing the equilibrium in the adhesive layer. A simplified equilibrium model is considered with only shear stresses in the adhesive layer taken into account. Comparison of numerical results obtained on the basis of analytical expressions and on the basis of FEM in the MSC Nastran software is performed. The graphical and tabular dependences for the joint adhesive layer parameters are presented. It is shown that the bearing capacity of the adhesive joint is significantly influenced not only by the strength of the glue, but also, and to a great extent, by its shear modulus. The corresponding effective length of the adhesive layer is calculated which increase does not result in the bearing capacity of the joint increase.

Indefinite integrals of Lommel functions from an inhomogeneous Euler–Lagrange method

ABSTRACTA method given recently for deriving indefinite integrals of special functions which satisfy homogeneous second-order linear differential equations has been extended to include functions which obey inhomogeneous equations. The extended method has been applied to derive indefinite integrals for the Lommel functions, which obey an inhomogeneous Bessel equation. The method allows integrals to be derived for the inhomogeneous equation in a manner which closely parallels the homogeneous case, and a number of new Lommel integrals are derived which have well-known Bessel analogues. Results will be presented separately for other special functions which obey inhomogeneous second-order linear equations.

On a remarkable sequence of Bessel matrices

A sequence of matrices whose elements are modified Bessel functions of the first kind is considered. Such a sequence arises when studying certain ordinary linear homogeneous second-order differential equations belonging to the family of double confluent Heun equations. The conjecture that these matrices are nonsingular is discussed together with its application to the problem of the existence of solutions analytic at the singular point of the equation referred to above.