Orthogonal Polynomial Series Research Articles

Sharp estimates for the convergence rate of “hyperbolic” partial sums of double fourier series in orthogonal polynomials

Two-variable functions f(x, y) from the class L 2 = L 2((a, b) × (c, d); p(x)q(y)) with the weight p(x)q(y) and the norm $$\left\| f \right\| = \sqrt {\int\limits_a^b {\int\limits_c^d {p(x)q(x)f^2 (x,y)dxdy} } }$$ are approximated by an orthonormal system of orthogonal P n (x)Q n (y), n, m = 0, 1, ..., with weights p(x) and q(y). Let $$E_N (f) = \mathop {\inf }\limits_{P_N } \left\| {f - P_N } \right\|$$ denote the best approximation of f ∈ L 2 by algebraic polynomials of the form $$\begin{array}{*{20}c} {P_N (x,y) = \sum\limits_{0 < n,m < N} {a_{m,n} x^n y^m ,} } \\ {P_1 (x,y) = const.} \\ \end{array}$$ . Consider a double Fourier series of f ∈ L 2 in the polynomials P n (x)Q m (y), n, m = 0, 1, ..., and its “hyperbolic” partial sums $$\begin{array}{*{20}c} {S_1 (f;x,y) = c_{0,0} (f)P_o (x)Q_o (y),} \\ {S_N (f;x,y) = \sum\limits_{0 < n,m < N} {c_{n,m} (f)P_n (x)Q_m (y), N = 2,3, \ldots .} } \\ \end{array}$$ A generalized shift operator Fh and a kth-order generalized modulus of continuity Ω k (A, h) of a function f ∈ L 2 are used to prove the following sharp estimate for the convergence rate of the approximation: $\begin{gathered} E_N (f) \leqslant (1 - (1 - h)^{2\sqrt N } )^{ - k} \Omega _k (f;h),h \in (0,1), \hfill \\ N = 4,5,...;k = 1,2,... \hfill \\ \end{gathered} $ . Moreover, for every fixed N = 4, 9, 16, ..., the constant on the right-hand side of this inequality is cannot be reduced.

A domain decomposition approach for vibration analysis of isotropic and composite cylindrical shells with arbitrary boundaries

An efficient domain decomposition method is proposed for solving the free, harmonic and transient vibrations of isotropic and composite cylindrical shells subjected to various combinations of classical and non-classical boundary conditions. Multi-segment partitioning strategy is adopted to accommodate the computing requirements of high-order vibration modes and responses. The continuity constraints on the segment interfaces are incorporated into the system potential functional by means of a modified variational principle and least-squares weighted residual method. An arbitrarily laminated version of Reissner–Naghdi’s shell theory is employed to formulate the theoretical model. Double mixed series, i.e., the Fourier series and orthogonal polynomials, are used as admissible displacement functions for each shell segment. The utility and robustness of the method for the application of various basis functions are evaluated with the following four sets of orthogonal polynomial series, i.e., the Chebyshev orthogonal polynomials of first and second kind, Legendre orthogonal polynomials of first kind and Hermite orthogonal polynomials. To test the convergence, efficiency and accuracy of the present method, free and forced vibrations (including the harmonic and transient vibrations) of isotropic and composite laminated cylindrical shells are examined under different combinations of free, shear-diaphragm, simply-supported, clamped and elastic supported boundaries. The theoretical results are compared with those previously published in literature, and the ones obtained by using the finite element program ANSYS. Very good agreement is observed.

On dispersion relations of waves in multilayered magneto-electro-elastic plates

The dispersion behavior of waves in a layered magneto-electro-elastic plate is studied. Formulations are given for open-circuit surface and short-circuit surface, respectively. Legendre orthogonal polynomial series approach is employed to solve the coupling magneto-electro-elastic controlling equations. The obtained dispersion curves for the open-circuit surface are compared with the published data to confirm the validity of the method. Through numerical examples, the convergence of the method is discussed. The differences of the two boundaries are illustrated. The influences of the piezoelectricity and piezomagnetism are illustrated and the influential factors of the piezoelectric and piezomagnetic effect are discussed. Finally, the influences of the volume fraction and stacking sequence on the piezoelectric effect and piezomagnetic effect are discussed.

Three Dimensional Vibration Analysis of a Class of Traction-Free Solid Elastic Bodies with an Eccentric Cavity

A three-dimensional elasticity-based continuum model is developed for describing the free vibrational characteristics of an important class of isotropic, homogeneous, and completely free structural bodies (i.e., finite cylinders, solid spheres, and rectangular parallelepipeds) containing an arbitrarily located simple inhomogeneity in form of a spherical or cylindrical defect. The solution method uses Ritz minimization procedure with triplicate series of orthogonal Chebyshev polynomials as the trial functions to approximate the displacement components in the associated elastic domains, and eventually arrive at the governing eigenvalue equations. An extensive review of the literature spanning over the past three decades is also given herein regarding the free vibration analysis of elastic structures using Ritz approach. Accuracy of the implemented approach is established through proper convergence studies, while the validity of results is demonstrated with the aid of a commercial FEM software, and whenever possible, by comparison with other published data. Numerical results are provided and discussed for the first few clusters of eigen-frequencies corresponding to various mode categories in a wide range of cavity eccentricities. Also, the corresponding 3D mode shapes are graphically illustrated for selected eccentricities. The numerical results disclose the vital influence of inner cavity eccentricity on the vibrational characteristics of the voided elastic structures. In particular, the activation of degenerate frequency splitting and incidence of internal/external mode crossings are confirmed and discussed. Most of the results reported herein are believed to be new to the existing literature and may serve as benchmark data for future developments in computational techniques.

Three dimensional vibration analysis of a class of traction-free solid elastic bodies with an eccentric cavity

A three-dimensional elasticity-based continuum model is developed for describing the free vibrational characteristics of an important class of isotropic, homogeneous, and completely free structural bodies (i.e., finite cylinders, solid spheres, and rectangular parallelepipeds) containing an arbitrarily located simple inhomogeneity in form of a spherical or cylindrical defect. The solution method uses Ritz minimization procedure with triplicate series of orthogonal Chebyshev polynomials as the trial functions to approximate the displacement components in the associated elastic domains, and eventually arrive at the governing eigenvalue equations. An extensive review of the literature spanning over the past three decades is also given herein regarding the free vibration analysis of elastic structures using Ritz approach. Accuracy of the implemented approach is established through proper convergence studies, while the validity of results is demonstrated with the aid of a commercial FEM software, and whenever possible, by comparison with other published data. Numerical results are provided and discussed for the first few clusters of eigen-frequencies corresponding to various mode categories in a wide range of cavity eccentricities. Also, the corresponding 3D mode shapes are graphically illustrated for selected eccentricities. The numerical results disclose the vital influence of inner cavity eccentricity on the vibrational characteristics of the voided elastic structures. In particular, the activation of degenerate frequency splitting and incidence of internal/external mode crossings are confirmed and discussed. Most of the results reported herein are believed to be new to the existing literature and may serve as benchmark data for future developments in computational techniques.

Three-dimensional vibration analysis of functionally graded thick, annular plates with variable thickness via polynomial-Ritz method

In this paper, free vibration analysis of functionally graded thick, annular plates with linear and nonlinear thickness variation along the radial direction is investigated by using the polynomial-Ritz method. The material properties of the functionally graded plates are assumed to be graded in the thickness direction according to the power-law distribution in terms of the volume fractions of the constituents. The solution procedure is based on the linear, small strain, three-dimensional elasticity theory. Potential (strain) and kinetic energies of the plates are formulated, and the polynomial-Ritz method is used to solve the eigenvalue problem. In this analysis method, a set of orthogonal polynomial series for three displacement components in a cylindrical polar coordinate are used to extract an eigenvalue equation yielding natural frequencies. Upper bound convergence of the non-dimensional frequencies to the exact values within at least four significant digits is demonstrated. Numerical results are presented and compared with the available literature. The vibration frequencies are given in several examples for various boundary conditions for the first time.

Circumferential SH Wave in the Inhomogeneous Magneto-Electro-Elastic Cylindrical Curved Plates

Based on linear magneto-electro-elasticity, the circumferential SH wave equation of transversely isotropy piezoelectricity-piezomagnetic functionally graded cylindrical curved plate is obtained. The Legendre orthogonal polynomial series expansion approach is employed to derive its asymptotic solution. The dispersion curves, electric potential and magnetic potential distributions are calculated. The influence of the piezoelectricity and piezomagnetism on the circumferential SH wave is investigated and the influential factors are analyzed. Finally, the influence of the gradient field and the ratio of the radius to thickness on the piezoelectric effect and piezomagnetic effect are illustrated.

Guided waves in functionally graded viscoelastic plates

In this paper, a dynamic solution for the propagating viscoelastic waves in functionally graded material (FGM) plates subjected to stress-free conditions is presented in the context of the Kelvin–Voigt viscoelastic theory. The FGM plate is composed of two orthotropic materials. The material properties are assumed to vary in the thickness direction according to a known variation law. The three obtained wave equations are divided into two groups, which control viscoelastic Lamb-like wave and viscoelastic SH wave, respectively. They are solved respectively by the Legendre orthogonal polynomial series approach. The validity of the method is confirmed through a comparison with the Lamb wave solution of a pure elastic FGM plate and a comparison with the SH wave solution of a viscoelastic homogeneous plate. The dispersion curves and attenuation curves for the graded and homogeneous viscoelastic plates are calculated to highlight their differences. The viscous effect on dispersion curves is shown. The influences of gradient variations are illustrated.

Guided thermoelastic wave propagation in layered plates without energy dissipation

In this paper, the propagation of guided thermoelastic waves in laminated orthotropic plates subjected to stress-free, isothermal boundary conditions is investigated in the context of the Green-Naghdi (GN) generalized thermoelastic theory (without energy dissipation). The coupled wave equations and heat conduction equation are solved by the Legendre orthogonal polynomial series expansion approach. The validity of the method is confirmed through a comparison. The dispersion curves of thermal modes and elastic modes are illustrated simultaneously. Dispersion curves of the corresponding pure elastic plate are also shown to analyze the influence of the thermoelasticity on elastic modes. The displacement and temperature distributions are shown to discuss the differences between the elastic modes and thermal modes.

On a criterion of rationality for a series in orthogonal polynomials

We present a criterion of rationality for a function determined by its expansion in a series in orthogonal polynomials. This criterion can be regarded as an analog of the well-known Kronecker criterion of rationality for functions given by power series.

Efficient algorithms for construction of recurrence relations for the expansion and connection coefficients in series of quantum classical orthogonal polynomials

Formulae expressing explicitly the q-difference derivatives and the moments of the polynomials P n ( x ; q) ∈ T ( T ={ P n ( x ; q) ∈ Askey–Wilson polynomials: Al-Salam-Carlitz I, Discrete q-Hermite I, Little (Big) q-Laguerre, Little (Big) q-Jacobi, q-Hahn, Alternative q-Charlier) of any degree and for any order in terms of P i ( x ; q) themselves are proved. We will also provide two other interesting formulae to expand the coefficients of general-order q-difference derivatives D q p f ( x ) , and for the moments x ℓ D q p f ( x ) , of an arbitrary function f( x) in terms of its original expansion coefficients. We used the underlying formulae to relate the coefficients of two different polynomial systems of basic hypergeometric orthogonal polynomials, belonging to the Askey–Wilson polynomials and P n ( x ; q) ∈ T. These formulae are useful in setting up the algebraic systems in the unknown coefficients, when applying the spectral methods for solving q-difference equations of any order.

Generalized thermoelastic waves in functionally graded plates without energy dissipation

Abstract In this paper, a dynamic solution of the propagating thermoelastic waves in functionally graded material (FGM) plate subjected to stress-free, isothermal boundary conditions is presented in the context of the Green–Naghdi (GN) generalized thermoelastic theory. The FGM plate is composed of two orthotropic materials. The materials properties are assumed to vary in the direction of the thickness according to a known variation law. The coupled wave equation and heat conduction equation are solved by the Legendre orthogonal polynomial series expansion approach. The convergency of the method is discussed through a numerical example. The dispersion curves of the inhomogeneous thermoelastic plate and the corresponding pure elastic plate are compared to show the characteristics of thermal modes and the influence of the thermoelasticity on elastic modes. The displacement, temperature and stress distributions of elastic modes and thermal modes are shown to discuss their differences. A plate with a different gradient variation is calculated to illustrate the influence of the gradient field on the wave characteristics.

Wave Characteristics in Magneto-electro-elastic Functionally Graded Spherical Curved Plates

Piezoelectric-piezomagnetic functionally graded materials (FGM), with a gradual change of the mechanical, electric and magnetic properties, have great promise for different applications. Based on the Legendre orthogonal polynomial series expansion approach, a dynamic solution is presented for the propagation of harmonic waves in piezoelectric-piezomagnetic FGM spherical curved plates. The material properties are assumed to vary in the direction of the thickness according to a known variation law. The dispersion curves of the piezoelectric-piezomagnetic FGM spherical curved plate and the corresponding non-piezoelectric, non-piezomagnetic plates are calculated to show the influences of the piezoelectricity and piezomagnetism. Electric potential and magnetic potential distributions are also obtained to illustrate the different influences of the piezoelectricity and piezomagnetism. Finally, a spherical curved plate at different ratio of radius to thickness is calculated to show the influence f the ratio on the piezoelectric effect and piezomagnetic effect.

Generalized thermoelastic waves in spherical curved plates without energy dissipation

In this article, the propagation of thermoelastic waves in orthotropic spherical curved plates subjected to stress-free, isothermal boundary conditions is investigated in the context of the Green–Naghdi (GN) generalized thermoelastic theory (without energy dissipation). The theoretical formulation is based on the linear GN thermoelastic theory. The coupled wave equation and heat conduction equation expressed by the displacement and temperature are obtained. By the Legendre orthogonal polynomial series expansion approach, the coupled controlling equations are solved. The convergence of the method is demonstrated through a numerical example. The dispersion curves of thermal modes and elastic modes are illustrated simultaneously. Dispersion curves of the corresponding purely elastic spherical plate are also shown to analyze the influence of thermoelasticity on elastic modes. The displacement, temperature and stress distributions of both elastic modes and thermal modes are calculated to show their differences. A thermoelastic spherical plate with a different ratio of radius to thickness is considered to show the influence of the ratio on the characteristics of thermoelastic waves.

Circumferential thermoelastic waves in orthotropic cylindrical curved plates without energy dissipation

In this article, the propagation of guided thermoelastic waves in the circumferential direction of orthotropic cylindrical curved plates subjected to stress-free, isothermal boundary conditions is investigated in the context of the Green–Naghdi (GN) generalized thermoelastic theory (without energy dissipation). The coupled wave equations and heat conduction equation are solved by the Legendre orthogonal polynomial series expansion approach. The convergency of the method is discussed through a numerical example. The dispersion curves of thermal modes and elastic modes are illustrated simultaneously. Dispersion curves of the corresponding pure elastic cylindrical plate are also shown to analyze the influence of the thermoelasticity on elastic modes. The displacement, temperature and stress distributions are shown to discuss the differences between the elastic modes and thermal modes. A thermoelastic cylindrical plate with a different ratio of radius to thickness is considered to discuss the influence of the ratio on the characteristics of circumferential thermoelastic waves.

Fractional radiative transfer equation within Chebyshev spectral approach

In this work we report the convergence of the Chebyshev polynomials combined with the S N method for the steady state transport equation using the fractional derivative. The procedure is based on the expansion of the angular flux in a truncated series of orthogonal polynomials that results in the transformation of the multidimensional problem into a system of fractional differential equations. The convergence of this approach is studied in the context of the multidimensional discrete-ordinates equations.

Method of summation of some slowly convergent series

A new method of summation of slowly convergent series is proposed. It may be successfully applied to the summation of generalized and basic hypergeometric series, as well as some classical orthogonal polynomial series expansions. In some special cases, our algorithm is equivalent to Wynn’s epsilon algorithm, Weniger transformation [E.J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Computer Physics Reports 10 (1989) 189–371] or the technique recently introduced by Čížek et al. [J. Čížek, J. Zamastil, L. Skála, New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field, Journal of Mathematical Physics 44 (3) (2003) 962–968]. In the case of trigonometric series, our method is very similar to the Homeier’s H transformation, while in the case of orthogonal series — to the K transformation. Two iterated methods related to the proposed method are considered. Some theoretical results and several illustrative numerical examples are given.

Recurrence relation approach for expansion and connection coefficients in series of classical discrete orthogonal polynomials

The two formulae expressing explicitly the difference derivatives and the moments of discrete orthogonal polynomials {P n (x): Meixner, Kravchuk and Charlier} of any degree and for any order in terms of P n (x) themselves are proved. Two other formulae for the expansion coefficients of general-order difference derivatives Δ q f(x), and for the moments x ℓΔ q f(x), of an arbitrary function f(x) of a discrete variable in terms of its original expansion coefficients are also obtained. Application of these formulae for solving ordinary difference equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica) in order to build and solve recursively for the connection coefficients between Hahn–Charlier, Hahn–Meixner and Hahn–Kravchuk are described.

Circumferential Waves in Piezoelectric Cylindrical Curved Plate

Based on linear three-dimensional piezoelasticity, an orthogonal polynomial series expansion approach is used to determine the characteristics of guided circumferential waves in homogeneous anisotropic piezoelectric cylindrical curved plates. The displacement components and electrical potential, expanded in a series of Legendre polynomials, are introduced into the governing equations along with position-dependent material constants, so that the solution of the wave equation is reduced to an eigenvalue problem. The dispersion curves for PZT-4 and Ba 2 NaNb 5 O 15 cylindrical curved plates are calculated. The effect of piezoelectricity is also shown. Mechanical displacement distribution and electric potential distribution are illustrated and the influence of the radius to thickness ratio on the circumferential wave characteristics is discussed. Finally, the influence of the polarizing direction on the piezoelectric effect is shown.

Wave characteristics in functionally graded piezoelectric hollow cylinders

Based on linear three-dimensional piezoelasticity, the Legendre orthogonal polynomial series expansion approach is used for determining the wave characteristics in hollow cylinders composed of the functionally graded piezoelectric materials (FGPM) with open circuit. The displacement and electric potential components, expanded in a series of Legendre polynomials, are introduced into the governing equations along with position-dependent material constants so that the solution of the wave equation is reduced to an eigenvalue problem. Dispersion curves for FGPM and the corresponding non-piezoelectric hollow cylinders are calculated to show the piezoelectric effect. The influence of the ratio of radius to thickness is discussed. Electric potential and displacement distributions are used to show the piezoelectric effect on the flexural torsional mode. The influence of the polarizing direction on the piezoelectric effect is illustrated. For the radial and axial polarization, the piezoelectric effect reacts mostly on the longitudinal mode. For circumferential polarization, the piezoelectric effect reacts mostly on the torsional mode. In the FGPM hollow cylinder, piezoelectricity can weaken the guided wave dispersion.