scholarly journals A Note on the Orthogonality Properties of the Pseudo-Chebyshev Functions

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1273
Author(s):  
Diego Caratelli ◽  
Paolo Emilio Ricci
Keyword(s):  
Differential Equation ◽  
Half Integer ◽  
Analytical Properties ◽  
Governing Differential Equation ◽  
Recurrence Formulae ◽  
Chebyshev Functions ◽  
Rational Degree

A novel class of pseudo-Chebyshev functions has been recently introduced, and the relevant analytical properties in terms of governing differential equation, recurrence formulae, and orthogonality have been analyzed in detail for half-integer degrees. In this paper, the previous studies are extended to the general case of rational degree. In particular, it is shown that the orthogonality properties of the pseudo-Chebyshev functions do not hold any longer.

On linear wave motions in magnetic-velocity shears

1977 ◽  
Vol 285 (1328) ◽  
pp. 607-636 ◽  
Keyword(s):  
Differential Equation ◽  
Critical Level ◽  
Velocity Shear ◽  
Linear Wave ◽  
Wave Amplification ◽  
Isothermal Fluid ◽  
Governing Differential Equation ◽  
Boussinesq Fluid ◽  
Propagation Properties ◽  
Background State

The propagation properties of linear wave motions in magnetic and/or velocity shears which vary in one coordinate z (say) are usually governed by a second order linear ordinary differential equation in the independent variable z. It is proved that associated with any such differential equation there always exists a quantity A which is independent of z. By employing A a measure of the intensity of the wave, this result is used to investigate the general propagation properties of hydromagnetic-gravity waves (e.g. critical level absorption, valve effects and wave amplification) in magnetic and/or velocity shears, using a full wave treatment. When variations in the basic state are included, the governing differential equation usually has more singularities than it has in the W.K.B.J. approximation, which neglects all variations in the background state. The study of a wide variety of models shows that critical level behaviour occurs only at the singularities predicted by the W.K.B.J. approximation. Although the solutions of the differential equation are necessarily singular at the irregularities whose presence is solely due to the inclusion of variations in the basic state, the intensity of the wave (as measured by A) is continuous there. Also the valve effect is found to persist whatever the relation between the wavelength of the wave and the scale of variations of the background state. In addition, it is shown that a hydromagnetic-gravity wave incident upon a finite magnetic and/or velocity shear can be amplified (or over-reflected) in the absence of any critical levels within the shear layer. In a Boussinesq fluid rotating uniformly about the vertical, wave amplification can occur if the horizontal vertically sheared flow and magnetic field are perpendicular. In a compressible isothermal fluid, on the other hand, wave amplification not only occurs in both magnetic-velocity and velocity shears but also in a magnetic shear acting alone.

scholarly journals New Bi-modular Material Approach to Buckling Problem of Reinforced Concrete Columns

2016 ◽  
Vol 6 (1) ◽  
pp. 19 ◽  
Author(s):  
Ahmad Salah Edeen Nassef ◽  
Mohammed A. Dahim
Keyword(s):  
Differential Equation ◽  
Reinforced Concrete ◽  
Differential Equations ◽  
Concrete Columns ◽  
Neutral Axis ◽  
Reinforced Concrete Columns ◽  
New Approach ◽  
Codes Of Practice ◽  
Governing Differential Equation ◽  
Transverse Deflection

<p class="1Body">This paper was investigating the buckling problem of reinforced concrete columns considering the reinforced concrete as bi – modular material. Governing differential equations was driven. The relation between the non-dimensional transverse deflection and non-dimensional distance between centroid axis and the neutral axis "eccentricity" was drawn to enable the solution of the governing differential equation. The new approach was verified with different experimental results and different codes of practice.<strong></strong></p>

Chebyshev functions of half-integer order

2007 ◽  
Vol 18 (10) ◽  
pp. 743-749
Author(s):  
J. García Ravelo ◽  
R. Cuevas ◽  
A. Queijeiro ◽  
J. J. Peña ◽  
J. Morales
Keyword(s):  
Integer Order ◽  
Half Integer ◽  
Chebyshev Functions

scholarly journals An Analytical Study of Weakly Nonlinear Dynamics of a Walters’ Liquid B around a Flexible Sheet Undergoing Super Linear Stretching

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
P. G. Siddheshwar ◽  
A. Chan ◽  
U. S. Mahabaleswar
Keyword(s):  
Nonlinear Dynamics ◽  
Differential Equation ◽  
Boundary Layer ◽  
Analytical Study ◽  
Stretching Sheet ◽  
Quadratic Function ◽  
Weakly Nonlinear ◽  
Governing Differential Equation ◽  
Layer Flow ◽  
Analytical Expressions

The paper discusses the boundary layer flow of Walters’ liquid B over a stretching sheet. The stretching is assumed to be a quadratic function of the coordinate along the direction of stretching. The study encompasses within its realm both Walters’ liquid B and second order liquid. The velocity distribution is obtained by solving the nonlinear governing differential equation. Analytical expressions are obtained for stream function and velocity components as functions of the viscoelastic and stretching related parameters. It is shown that the viscoelasticity goes hand in hand with quadratic stretching in enhancing the lifting of the liquid as we go along the sheet.

scholarly journals On Bending of Bernoulli-Euler Nanobeams for Nonlocal Composite Materials

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Luciano Feo ◽  
Rosa Penna
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Scale Parameter ◽  
Elastic Equilibrium ◽  
Functionally Graded ◽  
Constitutive Law ◽  
Bending Moments ◽  
Innovative Methodology ◽  
Governing Differential Equation ◽  
Nonlocality Effect

Evaluation of size effects in functionally graded elastic nanobeams is carried out by making recourse to the nonlocal continuum mechanics. The Bernoulli-Euler kinematic assumption and the Eringen nonlocal constitutive law are assumed in the formulation of the elastic equilibrium problem. An innovative methodology, characterized by a lowering in the order of governing differential equation, is adopted in the present manuscript in order to solve the boundary value problem of a nanobeam under flexure. Unlike standard treatments, a second-order differential equation of nonlocal equilibrium elastic is integrated in terms of transverse displacements and equilibrated bending moments. Benchmark examples are developed, thus providing the nonlocality effect in nanocantilever and clampled-simply supported nanobeams for selected values of the Eringen scale parameter.

scholarly journals Recurrence Formulae for the Functions which represent Solutions of the Differential Equation:

1914 ◽  
Vol 33 ◽  
pp. 107-117
Author(s):  
H. T. Flint
Keyword(s):  
Differential Equation ◽  
Positive Integer ◽  
In Series ◽  
Recurrence Formulae ◽  
Image Position

The contents of this paper were suggested by a discussion of the equation:in a paper by Glaisher which appears in the Philosophical Transactions, 1881, Part III.The solutions in series of (1) are:and in the paper referred to it is shewn that the coefficients of hp+1 in the expansions of and of satisfy equation (1) when p is a positive integer.

Rotating Disk Shape: Is the Equation Nonlinear?

1989 ◽  
Vol 111 (4) ◽  
pp. 456-458
Author(s):  
R. R. Jettappa
Keyword(s):  
Differential Equation ◽  
Linear Equation ◽  
Rotating Disk ◽  
Variable Coefficients ◽  
Thin Disk ◽  
Second Order ◽  
First Order ◽  
Governing Differential Equation ◽  
Centrifugal Loading

The determination of the shape of a rotating disk under centrifugal loading is considered. It is shown that the governing differential equation for the shape of a rotating thin disk is reducible to a linear equation of second order with variable coefficients. However, the form of this equation is such that it can be treated as an equation of first order thereby facilitating the integration by quadratures. All this is possible without any additional mathematical assumptions so that the results are exact within the limitations of the thin disk theory.

Dimensional Analysis in Design

1988 ◽  
Vol 110 (3) ◽  
pp. 401-407 ◽  
Author(s):  
J. R. Schnittger
Keyword(s):  
Differential Equation ◽  
Dimensional Analysis ◽  
Journal Bearing ◽  
Mechanical Design ◽  
Turbine Blades ◽  
Helical Spring ◽  
Disk Brake ◽  
Governing Differential Equation ◽  
Bearing Vibration ◽  
New Guidelines

New guidelines for dimensional analysis remove traditional road-blocks to its widespread use in mechanical design. Cases, with or without prior formula given, are exposed as well as those with a governing differential equation. The examples include bevel gear, helical spring, centrifugal pump, journal bearing, vibration of turbine blades, and a disk brake. A matrix method to determine nondimensional groups is reviewed.

scholarly journals Parametric Excitation and Evolutionary Dynamics

2013 ◽  
Vol 80 (5) ◽  
Author(s):  
Rocio E. Ruelas ◽  
David G. Rand ◽  
Richard H. Rand
Keyword(s):  
Differential Equation ◽  
Parametric Excitation ◽  
Variable Coefficient ◽  
Evolutionary Dynamics ◽  
Replicator Equation ◽  
Periodic Coefficients ◽  
Forcing Function ◽  
Governing Differential Equation ◽  
Social Applications ◽  
Dynamics Problems

Parametric excitation refers to dynamics problems in which the forcing function enters into the governing differential equation as a variable coefficient. Evolutionary dynamics refers to a mathematical model of natural selection (the “replicator” equation) which involves a combination of game theory and differential equations. In this paper we apply perturbation theory to investigate parametric resonance in a replicator equation having periodic coefficients. In particular, we study evolution in the Rock-Paper-Scissors game, which has biological and social applications. Here periodic coefficients could represent seasonal variation. We show that 2:1 subharmonic resonance can destabilize the usual “Rock-Paper-Scissors” equilibrium for parameters located in a resonant tongue in parameter space. However, we also show that the tongue may be absent or very small if the forcing parameters are chosen appropriately.

scholarly journals Approximation of Analytic Functions by Chebyshev Functions

2011 ◽  
Vol 2011 ◽  
pp. 1-10
Author(s):  
Soon-Mo Jung ◽  
Themistocles M. Rassias
Keyword(s):  
Differential Equation ◽  
Analytic Functions ◽  
Chebyshev Functions

We solve the inhomogeneous Chebyshev's differential equation and apply this result for approximating analytic functions by the Chebyshev functions.

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