Order Non-homogeneous Differential Equations Research Articles

Common Features of Free Particle Wave Functions in Curved Space-times

Quantum equations for massless particles of any spin and for massive spin one-half particles are considered in curved space-times. It is demonstrated that in stationary axially symmetric space-times the angular wave functions up to a normalization function are the same as in a Minkowski space-time. The radial wave functions satisfy second order nonhomogeneous differential equations with three nonhomogeneous terms which depend in a unique form on the ratio of time to space curvatures. For a Dirac spin one-half particle, in addition to these three terms a fourth term which depends on the particle rest mass is added.

Elastic Analysis of Steel-Concrete Composite Beams with Partial Interaction

The paper presents an exact analytical method for the elastic analysis of steel-concrete composite beams with partial interaction. Accepting the basic assumptions of the Newmark analytical model and adopting the axial force in the concrete slab as the main unknown, the second order nonhomogeneous differential equation of the steel-concrete composite element with partial interaction is derived. Further, the complete solutions for simply supported and fixed-ended composite beams subjected to concentrated and uniform loads respectively, are developed. The solution of the homogeneous equation is determined by imposing proper Dirichlet or Neumann boundary conditions depending on the static scheme of the element. The particular solutions are then derived for the considered loading conditions. It is shown that the internal axial force in concrete slab associated to composite beams with partial interaction can be expressed as a fraction of the axial force in concrete slab under full interaction through a non-dimensional function f(aL) which takes into account the connection’s stiffness, the mechanical properties and also the length of the element. Moreover, the solutions are included in a flexibility-based approach to derive the force-displacement relations of the beam element with partial interaction. For the resulted 2-noded beam-column element with 6DOF, the stiffness matrix is derived, showing that the partial composite action may be included at the element level by means of a series of correction factors applied to the standard full-interaction stiffness matrix coefficients. A numerical example is provided to demonstrate the accuracy and performance of the proposed method. Within the elastic range, the predicted load-midspan deflection curve is in very good agreement with both experimental and other numerical results retrieved from international literature. A parametric study was conducted to investigate the influence of the shear connection degree on the beam’s midspan deflection and the results were compared with those computed by using code provisions.

The growth of solutions of non-homogeneous linear differential equations

In this paper, we have considered second order non-homogeneous linear differential equations whose coefficients are entire functions. We have established conditions ensuring the non-existence of finite order solution of such type of differential equations. Moreover, we have also extended our results to higher order non-homogeneous differential equations.

Quantum Theory of Massless Particles in Stationary Axially Symmetric Spacetimes.

Quantum equations for massless particles of any spin are considered in stationary uncharged axially symmetric spacetimes. It is demonstrated that up to a normalization function, the angular wave function does not depend on the metric and practically is the same as in the Minkowskian case. The radial wave functions satisfy second order nonhomogeneous differential equations with three nonhomogeneous terms, which depend in a unique way on time and space curvatures. In agreement with the principle of equivalence, these terms vanish locally, and the radial equations reduce to the same homogeneous equations as in Minkowski spacetime.

Closeness-to-convexity of solutions of a second order nonhomogeneous differential equation

Розгляуто неоднорідне диференціальне рівняння Шаха z 2 w '' + ( β 0 z 2 + β 1 z ) w ' + ( γ 0 z 2 + γ 1 z + γ 2 ) w = A ( z ) , де A ( z ) = ∑ n = 0 ∞ a n z n , а радіус збіжності останнього степеневого ряду R [ A ] ≥ 1 . Визначено умови на коефіцієнти a n степеневого розвинення функції A ( z ) та на параметри β 0 ,   β 1 , γ 0 , γ 1 , γ 2 , , за яких неоднорідне рівняння Шаха має близькі до опуклих в одиничному крузі розв'язки. Окремо розглянуто випадки γ 2 = 0 та γ 2 > 0 , кожен із яких теж розпадається на підвипадки.

Linear directional differential equations in the unit ball: solutions of bounded L-index

AbstractWe study sufficient conditions of boundedness ofL-index in a directionb∈ ℂn∖ {0} for analytic solutions in the unit ball of a linear higher order non-homogeneous differential equation with directional derivatives. These conditions are restrictions by the analytic coefficients in the unit ball of the equation. Also we investigate asymptotic behavior of analytic functions of boundedL-index in the direction and estimate its growth. The results are generalizations of known propositions for entire functions of several variables.

Using Undetermined Coefficients to Solve Certain Classes of Variable-Coefficient Equations

This paper considers the question of when undetermined coefficients may be applied to compute the particular solution of certain second order nonhomogeneous differential equations with variable coefficients. A few examples illustrate the results of this investigation.

Numerical solution of third order linear differential equations using generalized one-shot operational matrices in orthogonal hybrid function domain

The present work proposes a new set of orthogonal hybrid functions (HF) which evolved from synthesis of orthogonal sample-and-hold functions (SHF) and triangular functions (TF). The HF set is used to approximate a time function in a piecewise linear manner. For such approximation, the mean integral square error (MISE) is much less than block pulse function based approximation which always provides staircase solutions and it is found that HF based approximation is a strong contender of approximations based upon orthogonal polynomials like Legendre polynomials. The respective MISE’s and elapsed times for MISE computations for these three kinds of approximations are also studied in detail.The one-shot operational matrices for integration of different orders in HF domain are also derived. These matrices are employed for more accurate multiple integration. An example is treated to determine efficiency of these matrices. Finally, these matrices are employed for solving third order non-homogeneous differential equations followed by a numerical example. The results, obtained via MATLAB, are compared with the exact solution as well as the results obtained via 4th order Runge–Kutta method.

Polynomial solutions of N th order non-homogeneous differential equations

Polynomial solutions of N th order non-homogeneous differential equations

Shear Lag in Box Girders

An initial approximate method is presented to evaluate the shear lag phenomenon in reinforced concrete single-box girder bridges. Assuming an ideal cross-section and considering a symmetrical deformation of the top and bottom slabs, but preserving continuity conditions at the web-slab intersection points, a fourth order non-homogeneous differential equation with constant coefficients was obtained. Making further simplifications, quick results are obtained for uniformly distributed loads. In spite of all the simplifications these results were in good agreement with the German specifications for shear lag provided that midspan reduction factors are desired. the factors are the most important anyway, because they are valid practically almost for the whole span length.

Lie Theory and Special Functions Satisfying Second Order Nonhomogeneous Differential Equations

A Lie algebraic technique is given for the systematic study of families of special functions which satisfy second order nonhomogeneous differential equations such that the solutions of the homogeneous equations are of hypergeometric type. Among the functions obtained by this technique are the functions of Struve, Lommel and various nonhomogeneous hypergeometric, confluent hypergeometric and parabolic cylinder functions.

Asymptotic expansions in certain second order nonhomogeneous differential equations

In the nonoscillation theory of ordinary differential equations the asymptotic behaviour of the solutions has often been described by exhibiting asymptotic expansions for these solutions. A fundamental illustration of this technique may be found in Hille [6] wherein the linearly independent solutions of a second order homogeneous differential equation were described by single termed asymptotic expansions. For the dominant solution, this result was successively extended by Waltman in [8] for a second order equation and in [9] for an n th order equation. A further generalization of these results appears in [3] where a complete n th order nonhomogeneous nonlinear differential equation was considered; again, asymptotic representations were given to describe the behaviour of the solutions of the differential equation. Moore and Nehari [7], Wong [10], [11], and Hale and Onuchic [2], also use asymptotic representations in discussing the behaviour of the solutions of certain differential equations. All of the above results are essentially perturbation problems with the unperturbed linear differential equation having the form y ( n ) = h ( t ) for some n and h ( t ).

Further results for the solutions of certain third and fourth order differential equations

In this paper we investigate certain generalized forms of third and fourth order non-homogeneous differential equations. Ezeilo (l) derived interesting results for the problem.We show similar results for:,and.