Kind Chebyshev Polynomials Research Articles

Fracture Analysis of Double-Side Adhesively Bonded Composite Repairs to Cracked Aluminium Plate Using Line Spring Model

A line spring model is developed for analyzing the fracture problem of cracked metallic plate repaired with the double-sided adhesively bonded composite patch. The restraining action of the bonded patch is modeled as continuous distributed linear springs bridging the crack faces provided that the cracked plate is subjected to extensional load. The effective spring constant is determined from 1-D bonded joint theory. The hyper-singular integral equation (HSIE), which can be solved using the second kind Chebyshev polynomial expansion method, is applied to determine the crack opening displacements (COD) and the crack tip stress intensity factors (SIF) of the repaired cracked plate. The numerical result of SIF for the crack-tip correlates very well with the finite element (FE) computations based on the virtual crack closure technique (VCCT). The present analysis approaches and mathematical techniques are critical to the successful design, analysis and implementation of crack patching.

Matrix method based on the second kind Chebyshev polynomials for solving time fractional diffusion-wave equations

In this paper, the second kind Chebyshev polynomials (SKCPs) basis is used to solve time-fractional diffusion-wave equations with damping. We present some notations and definitions of the fractional calculus and introduce some basic properties of the SKCPs. Bivariate shifted SKCPs are defined and the operational matrix of fractional integration and some other needed operational matrices are constructed. Our approach uses the properties of bivariate shifted SKCPs to transform the considered problem to a matrix equation without using any collocation points. The main characteristic of this technique is that only a small number of the basic functions is needed to obtain a satisfactory result. An estimation of the error is given in the sense of Sobolev norms. Numerical examples are given to demonstrate the efficiency and accuracy of the proposed method.

Operational Matrix of Fractional Integration Based on the Shifted Second Kind Chebyshev Polynomials for Solving Fractional Differential Equations

This paper is devoted to studying a computational method for solving multi-term differential equations based on new operational matrix of shifted second kind Chebyshev polynomials. The properties of the operational matrix of fractional integration are exploited to reduce the main problem to an algebraic equation. We present an upper bound for the error in our estimation that leads to achieve the convergence rate of O(M −κ ). Numerical experiments are reported to demonstrate the applicability and efficiency of the proposed method.

Pouzet–Runge–Kutta–Chebyshev method for Volterra integral equations of the second kind

We construct a stabilized Runge–Kutta method of Pouzet type (Pouzet–Runge–Kutta–Chebyshev method, PRKC) for nonlinear stiff Volterra integral equations of the second kind. The PRKC method is an explicit second-order scheme which possesses extended stability region due to the first kind Chebyshev polynomial used to establish the stability function.

Application of Chebyshev polynomials to classes of analytic functions

Our objective in this paper is to consider some basic properties of the familiar Chebyshev polynomials in the theory of analytic functions. We investigate some basic useful characteristics for a class H(t), t∈(1/2,1], of functions f, with f(0)=0, f′(0)=1, analytic in the open unit disc U={z:|z|<1} satisfying the condition that1+zf″(z)f′(z)≺H(z,t)=11−2tz+z2(z∈U), where H(z,t) is the generating function of the second kind of Chebyshev polynomials. The Fekete–Szegö problem in the class is also solved.

Some Identities Involving the Derivative of the First Kind Chebyshev Polynomials

We use the combinatorial method and algebraic manipulations to obtain several interesting identities involving the power sums of the derivative of the first kind Chebyshev polynomials. This solved an open problem proposed by Li (2015).

Integral representations and new generating functions of Chebyshev polynomials

In this paper we use the two-variable Hermite polynomials and their operational rules to derive integral representations of Chebyshev polynomials. The concepts and the formalism of the Hermite polynomials Hn(x;y) are a powerful tool to obtain most of the properties of the Chebyshev polynomials. By using these results, we also show how it is possible to introduce relevant generalizations of these classes of polynomials and we derive for them new identities and integral representations. In particular we state new generating functions for the first and second kind Chebyshev polynomials. 2000 AMS Classification: 33C45, 33D45.

Numerical approach for the solution of hypersingular integro-differential equations

The main purpose of this article is to present an approximation method of hypersingular integro-differential equations in the most general form under the mixed conditions in terms of the second kind Chebyshev polynomials. This method transforms mixed hypersingular integro-differential equations and the given conditions into matrix equation which corresponding to a system of linear algebraic equation. The error analysis and convergence for the proposed method is also introduced. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system Maple10.

Дискретные преобразования со свойством прилипания на основе системы {sin x sin kx} и системы полиномов Чебышёва второго рода

Дискретные преобразования со свойством прилипания на основе системы {sin x sin kx} и системы полиномов Чебышёва второго рода

Calculation of the critical thickness for one-speed neutrons in a reflected slab with backward and forward scattering using modified T N method

Abstract The reflected critical slab problem is investigated in one-speed neutron transport theory for reflecting boundary conditions. The neutron angular flux is expanded in a series of first kind Chebyshev polynomials and isotropic, backward and forward scattering kernel is used as the scattering function. The critical sizes for one-speed neutrons in a uniform finite slab surrounded by identical reflectors from both sides are calculated for various values of the collision parameter, backward and forward scattering and reflection coefficient. Numerical values calculated for the critical slab are presented in the tables and are found to be in good agreement with literature values.

Factorizations of negatively subscripted balancing and Lucas-balancing numbers

In this paper, we find some tridigonal matrices whose determinant and permanent are equal to the negatively subscripted balancing and Lucas- balancing numbers. Also using the First and second kind of Chebyshev polynomials, we obtain the factorization of these numbers.

Spectral collocation and a two-level continuation scheme for dipolar Bose–Einstein condensates

We exploit the high accuracy of spectral collocation methods in the context of a two-level continuation scheme for computing ground state solutions of dipolar Bose–Einstein condensates (BEC), where the first kind Chebyshev polynomials and Fourier sine functions are used as the basis functions for the trial function space. The governing Gross–Pitaevskii equation (or Schrödinger equation) can be reformulated as a Schrödinger–Poisson (SP) type system [13]. The two-level continuation scheme is developed for tracing the first solution curves of the SP system, which in turn provide an appropriate initial guess for the Newton method to compute ground state solutions for 3D dipolar BEC. Extensive numerical experiments on 3D dipolar BEC and dipolar BEC in optical lattices are reported.

A two-parameter continuation method for computing numerical solutions of spin-1 Bose–Einstein condensates

We describe a novel two-parameter continuation method combined with a spectral-collocation method (SCM) for computing the ground state and excited-state solutions of spin-1 Bose–Einstein condensates (BEC), where the second kind Chebyshev polynomials are used as the basis functions for the trial function space. To compute the ground state solution of spin-1 BEC, we implement the single parameter continuation algorithm with the chemical potential μ as the continuation parameter, and trace the first solution branch of the Gross–Pitaevskii equations (GPEs). When the curve-tracing is close enough to the target point, where the normalization condition of the wave function is going to be satisfied, we add the magnetic potential λ as the second continuation parameter with the magnetization M as the additional constraint condition. Then we implement the two-parameter continuation algorithm until the target point is reached, and the ground state solution of the GPEs is obtained. The excited state solutions of the GPEs can be treated in a similar way. Some numerical experiments on Na23 and Rb87 are reported. The numerical results on the spin-1 BEC are the same as those reported in [10]. Further numerical experiments on excited-state solutions of spin-1 BEC suffice to show the robustness and efficiency of the proposed two-parameter continuation algorithm.

Determinant Representations of Polynomial Sequences of Riordan Type

In this paper, using the production matrix of a Riordan array, we obtain a recurrence relation for polynomial sequence associated with the Riordan array, and we also show that the general term for the sequence can be expressed as the characteristic polynomial of the principal submatrix of the production matrix. As applications, a unified determinant expression for the four kinds of Chebyshev polynomials is given.

A Two-Parameter Continuation Method for Rotating Two-Component Bose-Einstein Condensates in Optical Lattices

AbstractWe study efficient spectral-collocation and continuation methods (SCCM) for rotating two-component Bose-Einstein condensates (BECs) and rotating two-component BECs in optical lattices, where the second kind Chebyshev polynomials are used as the basis functions for the trial function space. A novel two-parameter continuation algorithm is proposed for computing the ground state and first excited state solutions of the governing Gross-Pitaevskii equations (GPEs), where the classical tangent vector is split into two constraint conditions for the bordered linear systems. Numerical results on rotating two-component BECs and rotating two-component BECs in optical lattices are reported. The results on the former are consistent with the published numerical results.

Approximating an Infinitely-Lived Homogeneous Agents Model by Chebyshev Collocation

Paper approximates long run utility implications of revenue neutral tax schedules, i.e. using first kind Chebyshev polynomials. Utility approximations are made on steady state utility equations of Hall's infinitely-lived-representative-agent model. I endogenized budget neutral tax schedules to exogenous levels of steady state capital. Utilities of such tax plans are then fixed to appropriate Chebyshev nodes.

A two-parameter continuation algorithm for vortex pinning in rotating Bose–Einstein condensates

We describe an efficient two-parameter continuation algorithm combined with spectral collocation methods for computing the ground state and central vortex state solutions of rotating Bose–Einstein condensates in optical lattices, where the first kind and second kind Chebyshev polynomials are used as the basis functions for the trial function space. By treating the chemical potential and angular velocity as the continuation parameters simultaneously under the additional constraint of normalization condition, the proposed algorithm can effectively compute numerical solutions for a rich variety of physical phenomena observed in physical experiments with very little cost. Comparisons with various numerical methods on some sample test problems are reported.

Array synthesis and wide band system

Acoustic arrays play important role as the key devices in wide band system designs. The Laguerre polynomials are successfully applied to the array synthesis first time. Conventional uniform array is the simplest design, because each element is excited at the same weight, which leads to high side lobe levels, artifacts and noise to advanced systems. Dolph utilized the first kind Chebyshev polynomials to synthesize the array beam pattern for side lobe control. However, it comes with the equal levels of the side lobes, due to the mathematical nature of the first kind Chebyshev polynomials. Taylor introduced a modified version Dolph-Chebyshev synthesis technique, which displayed tapered down side lobe levels in the region away from the main lobe. The key characteristics from the sample numerically simulated arrays by the Laguerre polynomials synthesis, i. e. radiation patterns, half-power beam width, directivity and the beam efficiency are compared with those from the synthesis of the Dolph-Chebyshev of the first kind or second kind, Taylor shading, Legendre and Hermite polynomials techniques. Work supported by the U.S. Navy.

Orthogonal Functions Based on Chebyshev Polynomials

It is known that Chebyshev polynomials are an orthogonal set associated with a certain weight function. In this paper, we present an approach for the contruction of a special wavelet function as well as a special scaling function. Main tool of the special wavelet is a first kind Chebyshev polynomial. Based on Chebyshev polynomials and their zero, we define our scaling function and wavelets, and by using Christoffel-Darboux formula for Chebyshev polynomials, we prove that these functions are orthogonal. Finally, we provide several examples of scaling function and wavelets for illustration. Keywords: Chebyshev polynomial; Christoffel-Darboux formula; Wavelets; and Scaling function. 2010 Mathematics Subject Classification 76D04.

Modified UN method for the reflected critical slab problem with forward and backward scattering

Abstract The critical slab problem is investigated in one-speed neutron transport theory for reflecting boundary conditions using second kind Chebyshev polynomials approximation. The forward-backward-isotropic scattering kernel is used in a uniform homogeneous slab. The critical slab thicknesses for various values of the collision parameter, forward and backward scattering and reflection coefficient are given in the tables together with the ones available in literature and they are in good agreement.