Terms Of Chebyshev Polynomials Research Articles

A computational method for a thermoacoustic boundary layer in a gas-filled tube.

This paper demonstrates a computational method to solve equations for a thermoacoustic boundary layer in a gas-filled tube subjected to a temperature gradient axially. With applications to numerical simulations of thermoacoustic systems, the purpose is to establish a method to evaluate numerically a velocity vb at the edge of a boundary layer directed into a core region in the outside of the layer. The computational method exploits a series expansion in terms of Chebyshev polynomials. Solved is a benchmark example for which analytical solutions are available. Assuming that a quiescent gas is started by an impulse and an acoustic field in the core region is given, an axial velocity and a temperature disturbance in the boundary layer are sought, from which vb is derived. Comparing the numerical solutions obtained against the analytical ones, it is found that the acoustic field and ultimately vb are well obtained with good accuracy. It is thus concluded that the present computational method will apply to numerical simulations based on the boundary-layer theory.

Scattering of water waves by multiple segmented concentric semi-closed breakwaters and an interior cylinder: An analytical approach

This study aims to provide a theoretical analytical model for the design of breakwaters like Palm Jumeirah breakwater in Dubai by investigating the hydrodynamic characteristics of segmental semi-enclosed breakwaters simplified from it. An analytical solution to the interaction of water waves and a cylinder and multiple concentric segmented arc-shaped breakwaters has been developed based on linear potential theory. The eigenfunction matching method and separation of variables have been applied to expand the unknown function in terms of Chebyshev polynomials. By considering the boundary conditions, the integral equation has been simplified into a set of algebraic equations, which are then solved to determine the unknown function. The accuracy of the model has been confirmed by comparing the results with those in previous articles. The numerical analysis has revealed the relationship between variations in the hydrodynamic loads, wave elevation of the cylinder, and factors including the gap width between the arcs, opening angle, wave incidence angle, and porous-effect coefficient. The results indicate that segmented breakwaters offer better protection to the internal structure than unsegmented breakwaters. Moreover, segmented breakwaters with smaller gaps are less sensitive to diagonal waves and the gap can serve as a navigational channel, which is an extra benefit.

Passage of an electromagnetic wave through a rectangular waveguide with a section of a semiconductor film

Background. In microwave technology, of great interest is the creation of amplifiers and generators based on thin-layered longitudinal inhomogeneities in rectangular waveguides, which are an integral part of many functional devices in the microwave and EHF ranges. Aim. Therefore, it is of practical interest to calculate one of these inhomogeneities in a rectangular waveguide. Methods. One of the methods for calculating such structures is the diffraction of an electromagnetic wave by a segment of a semiconductor film with negative differential conductivity on a narrow wall of a rectangular waveguide. Based on the boundary conditions, a singular integral equation for the electric component of the electromagnetic field was obtained. By the method of inversion of the integral equation and by means of expansion in terms of Chebyshev polynomials, the integral equation is given in a system of two algebraic equations. The coefficients of the system are calculated according to the theory of residues. Then an equation was obtained for calculating the transmission coefficient, which was calculated in the Matcad environment. Results. Based on the integral equation, the values for calculating the transmission coefficient for the structure under consideration are obtained. Conclusion. From the graphs obtained for the calculated transmission coefficient, it can be concluded that there are frequency zones where an electromagnetic wave is amplified in the structure. So, on the basis of the structure under consideration, it is possible to create amplifiers and generators of the microwave and EHF ranges.

A High-Order-Accurate 3D Surface Integral Equation Solver for Uniaxial Anisotropic Media

This paper introduces a high-order accurate surface integral equation method for solving 3D electromagnetic scattering for dielectric objects with uniaxially anisotropic permittivity tensors. The N-Müller formulation is leveraged resulting in a second-kind integral formulation, and a finite-difference-based approach is used to deal with the strongly singular terms resulting from the dyadic Green’s functions for uniaxially anisotropic media while maintaining the high-order accuracy of the discretization strategy. The integral operators are discretized via a Nyström-collocation approach, which represents the unknown surface densities in terms of Chebyshev polynomials on curvilinear quadrilateral surface patches. The convergence is investigated for various geometries, including a sphere, cube, a complicated NURBS geometry imported from a 3D CAD modeler software, and a nanophotonic silicon waveguide, and results are compared against a commercial finite element solver. To the best of our knowledge, this is the first demonstration of high-order accuracy for objects with uniaxially anisotropic materials using surface integral equations.

A Galerkin formulation for the nonlinear analysis of a flexible beam in channel flow

In this paper we propose a Galerkin formulation for the 1-D model describing the nonlinear flow-structure interaction of a flexible beam in confined flow. In broad terms, the system of PDEs is converted into a set of time-dependent equations (ODEs and algebraic) by developing all variables in terms of series of space-dependent orthogonal functions. The beam motion is developed in terms of its mode shapes while the flow pressure and velocity fields in each channel are developed in terms of Chebyshev polynomials. Additionally, a tau-variant of the Galerkin approach enables the enforcement of the nonlinear time-dependent boundary conditions in a well-posed manner. Ultimately, the resulting system is a set of nonlinear differential–algebraic equations that can be truncated at any suitable numbers of terms, leading to exploitable reduced formulations. Compared to CFD methods, this type of formulations is not only more computationally efficient, but also provide an easy discernment of the relevant parameters and often a more intuitive interpretation of results. Convergence studies, in terms of both the truncation of Chebyshev polynomials and beam modes, are performed to access to what extent reduced formulations are viable in different scenarios, including linear stability analysis as well as the calculation of limit-cycle oscillations with or without intermittent impacts between the beam and the side-walls. The presented framework can serve as a basis for a comprehensive analysis of the nonlinear dynamics of flexible beams in confined flow. Namely, it is well adapted to the use of bifurcation analysis tools for the continuation of periodic solutions, which will contribute to a richer understanding of the underlying physics occurring in this type of FSI systems. Moreover, the generic methodology presented here can also be adapted to different systems in the field of fluid–structure interaction, providing compact time-dependent formulations for nonlinear analysis.

GW space-time method: Energy band gap of solid hydrogen

We implement the $GW$ space-time method at finite temperatures, in which the Green's function $G$ and the screened Coulomb interaction $W$ are represented in the real space on a suitable mesh and in imaginary time in terms of Chebyshev polynomials, paying particular attention to controlling systematic errors of the representation. Having validated the technique by the canonical application to silicon and germanium, we apply it to the calculation of band gaps in hexagonal solid hydrogen with the bare Green's function obtained from density functional approximation and the interaction screened within the random phase approximation. The results, obtained from the asymptotic decay of the full Green's function without resorting to analytic continuation, suggest that the solid hydrogen above 150 GPa cannot adopt an orientationally ordered hexagonal-closed-pack structure due to its metallic behavior. The demonstrated ability of the method to store the full $G$ and $W$ functions in memory with sufficient accuracy is crucial for its subsequent extensions to include higher orders of the diagrammatic series by means of diagrammatic Monte Carlo algorithms.

ChiliPDF: Chebyshev interpolation for parton distributions

Parton distribution functions (PDFs) are an essential ingredient for theoretical predictions at colliders. Since their exact form is unknown, their handling and delivery for practical applications relies on approximate numerical methods. We discuss the implementation of PDFs based on a global interpolation in terms of Chebyshev polynomials. We demonstrate that this allows for significantly higher numerical accuracy at lower computational cost compared with local interpolation methods such as splines. Whilst the numerical inaccuracy of currently used local methods can become a nontrivial limitation in high-precision applications, in our approach it is negligible for practical purposes. This holds in particular for differentiation and for Mellin convolution with kernels that have end point singularities. We illustrate our approach for these and other important numerical operations, including DGLAP evolution, and find that they are performed accurately and fast. Our results are implemented in the C++ library ChiliPDF.

Time fractional advection-dispersion model to study transportation of particles with time-memory for unsteady nonequilibrium suspension in open-channel turbulent flows

In this study, the effects of time-memory on the mixing and nonequilibrium transportation of particles in an unsteady turbulent flow are investigated. The memory effect of particles is captured through a time-fractional advection-dispersion equation rather than a traditional advection-dispersion equation. The time-fractional derivative is considered in Caputo sense which includes a power-law memory kernel that captures the power-law jumps of particles. The time-fractional model is solved using the Chebyshev collocation method. To make the solution procedure more robust three different kinds of Chebyshev polynomials are considered. The time-fractional derivative is approximated using the finite difference method at small time intervals and numerical solutions are obtained in terms of Chebyshev polynomials. The model solutions are compared with existing experimental data of traditional conditions and satisfactory results are obtained. Apart from this, the effects of time-memory are analyzed for bottom concentration and transient concentration distribution of particles. The results show that for uniform initial conditions, bottom concentration increases with time as the order of fractional derivative decreases. In the case of transient concentration, the value of concentration initially decreases when T < 1 and thereafter increases throughout the flow depth. The effects of time-memory are also analyzed under steady flow conditions. Results show that under steady conditions, transient concentration is more sensitive for linear, parabolic, and parabolic-constant models of sediment diffusivity rather than the constant model.

Novel Expressions for the Derivatives of Sixth Kind Chebyshev Polynomials: Spectral Solution of the Non-Linear One-Dimensional Burgers’ Equation

This paper is concerned with establishing novel expressions that express the derivative of any order of the orthogonal polynomials, namely, Chebyshev polynomials of the sixth kind in terms of Chebyshev polynomials themselves. We will prove that these expressions involve certain terminating hypergeometric functions of the type 4F3(1) that can be reduced in some specific cases. The derived expressions along with the linearization formula of Chebyshev polynomials of the sixth kind serve in obtaining a numerical solution of the non-linear one-dimensional Burgers’ equation based on the application of the spectral tau method. Convergence analysis of the proposed double shifted Chebyshev expansion of the sixth kind is investigated. Numerical results are displayed aiming to show the efficiency and applicability of the proposed algorithm.

Quantifying alignment in carbon nanotube yarns and similar two‐dimensional anisotropic systems

AbstractThe uniaxial orientational order in a macromolecular system is usually specified using the Hermans factor which is equivalent to the second moment of the system's orientation distribution function (ODF) expanded in terms of Legendre polynomials. In this work, we show that for aligned materials that are two‐dimensional (2D) or have a measurable 2D intensity distribution, such as carbon nanotube (CNT) textiles, the Hermans factor is not appropriate. The ODF must be expanded in terms of Chebyshev polynomials and therefore, its second moment is a better measure of orientation in 2D. We also demonstrate that both orientation parameters (Hermans in three dimensional (3D) and Chebyshev in 2D) depend not only on the respective full‐width‐at‐half‐maximum of the peaks in the ODF but also on the shape of the fitted functions. Most importantly, we demonstrate a method to rapidly estimate the Chebyshev orientation parameter from a sample's 2D Fourier power spectrum, using an analysis program written in Python which is available for open access. As validation examples, we use digital photographs of dry spaghetti as well as scanning electron microscopy images of direct‐spun carbon nanotube fibers, proving the technique's applicability to a wide variety of fibers and images.

Hidden duality and accidental degeneracy in cycloacene and Möbius cycloacene

The accidental degeneracy appearing in cycloacenes as triplets and quadruplets is explained with the concept of segmentation, introduced here with the aim of describing the effective disconnection of π orbitals on these organic compounds. For periodic systems with time reversal symmetry, the emergent nodal domains are shown to divide the atomic chains into simpler carbon structures analog to benzene rings, diallyl chains, anthracene (triacene) chains, and tetramethyl-naphthalene skeletal forms. The common electronic levels of these segments are identified as members of degenerate multiplets of the global system. The peculiar degeneracy of Möbius cycloacene is also explained by segmentation. In the last part, it is shown that the multiplicity of energies for cycloacene can be foreseen by studying the continuous limit of the tight-binding model; the degeneracy conditions are put in terms of Chebyshev polynomials. The results obtained in this work have important consequences on the physics of electronic transport in organic wires, together with their artificial realizations.

Nonlinear flexure mechanics of beams: stress gradient and nonlocal integral theory

In order to study the intrinsic size-effects, the stress gradient theory is implemented to a nano-scale beam model in nonlinear flexure. The nonlocal integral elasticity model is considered as a suitable counterpart to examine the softening behavior of nano-beams. Reissner variational principle is extended consistent with the stress gradient theory and applied to establish the differential, constitutive and boundary conditions of a nano-sized beam in nonlinear flexure. The nonlinear integro-differential and boundary conditions of inflected beams in the framework of the nonlocal integral elasticity are determined utilizing the total elastic strain energy formulation. A practical series solution approach in terms of Chebyshev polynomials is introduced to appropriately estimate the kinematic and kinetic field variables. A softening structural behavior is observed in the flexure of the stress gradient and the nonlocal beam in terms of the characteristic parameter and the smaller-is-softer phenomenon is, therefore, confirmed. The flexural response associated with the stress gradient theory is demonstrated to be in excellent agreement with the counterpart results of the nonlocal elasticity model equipped with the Helmholtz kernel function. The nonlocal elasticity theory endowed with the Error kernel function is illustrated to underestimate the flexural results of the stress gradient beam model. Detected numerical benchmark can be efficiently exploited for structural design and optimization of pioneering nano-engineering devices broadly utilized in advanced nano-electro-mechanical systems.

Waveguide band-gap N -qubit array with a tunable transparency resonance

We study a single photon transmission through 1D N- qubit chain. The qubits are supposed to be identical with equal distance between neighbors. We express the transfer matrix of N- qubit chain in terms of Chebyshev polynomials, which allows us to obtain simple expressions for the transmission and reflection amplitudes for arbitrarily large N. If the distance between neighbor qubits is equal to half wavelength, the transmission spectrum exhibits a flat bandgap structure with very steep walls. We show that for odd N the tuning of the excitation frequency of a central qubit gives rise to the appearance within a bandgap of a narrow resonance with a full transmission. The position of the resonance and its width can be controlled by the frequency of a central qubit. We show that the formation of the bandgap and of the transmission resonance is conditioned by the overlapping the widths of individual qubits which results from the strong coupling between qubits and waveguide photons.

Bending-torsion coupling bursting oscillation of a sandwich conical panel under parametric excitation

Regarding the dynamic behavior of the functionally graded materials (FGM) conical panels under the static and harmonic in-plane loading, most studies focus on parametric resonances and dynamic instability which determine the stability regions. Studies on the bursting oscillation of the conical panels under the lower frequency excitation are quite rare. In this paper, we present a novel nonautonomous system of the cantilever functionally graded materials sandwich conical panels with a slowly varying parametric excitation and study the bending-torsion coupling bursting oscillations. The face sheets and core layer are the metal ceramic composite layer and homogeneous metal layer, respectively. In two surfaces layers, the component materials are gradually changed in the transverse direction and the temperature dependent properties show the form of the power law distribution. The excitations in the meridional direction include the static preload and lower frequency harmonic excitation. The mode functions of the first-order bending mode and the first-order torsion mode are obtained in terms of Chebyshev polynomials with the help of energy principle. The bending-torsion coupling nonautonomous nonlinear oscillation system is derived by Lagrange equations. Under the slowly varying in-plane load, the equilibrium solutions of the system are obtained by treating them as a generalized autonomous system and they will be no longer stable through the pitchfork bifurcation with the symmetry breaking. The bursting oscillations of the novel dynamic system are analyzed by numerical methods. It is interesting to find that the mechanisms of bursting oscillation are associated with the pitchfork bifurcation with the symmetry breaking.

Formal weight enumerators and Chebyshev polynomials

A formal weight enumerator is a homogeneous polynomial in two variables which behaves like the Hamming weight enumerator of a self-dual linear code except that the coefficients are not necessarily nonnegative integers. The notion of formal weight enumerator was first introduced by Ozeki in connection with modular forms, and a systematic investigation of formal weight enumerators has been conducted by Chinen in connection with zeta functions and Riemann hypothesis for linear codes. In this paper, we establish a relation between formal weight enumerators and Chebyshev polynomials. Specifically, the condition for the existence of formal weight enumerators with prescribed parameters $$(n,\varepsilon ,q)$$ is given in terms of Chebyshev polynomials. According to the parity of n and the sign $$\varepsilon$$ , the four kinds of Chebyshev polynomials appear in the statement of the result. Further, we obtain explicit expressions of formal weight enumerators in the case where n is odd or $$\varepsilon =-1$$ using Dickson polynomials, which generalize Chebyshev polynomials. We also state a conjecture from a viewpoint of binomial moments, and see that the results in this paper partially support the conjecture.

Nonlinear flexure of Timoshenko–Ehrenfest nano-beams via nonlocal integral elasticity

Nonlinear flexural behavior of elastic nano-beams under static loads is investigated in the framework of the Eringen’s nonlocal integral elasticity. The integro-differential and boundary conditions of equilibrium for inflected Timoshenko–Ehrenfest elastic beams with von Karman nonlinear strains are established by using the minimum total potential energy principle. Eringen’s nonlocal differential law, consequent (but not equivalent) to Eringen’s nonlocal integral convolution, is well established to be inconsistent when applied to bounded domains of nano-mechanics interest. Accordingly, nonlocal integral elasticity formulation is utilized to capture scale effects in the nonlinear flexure of nano-beams. An analytical series solution in terms of Chebyshev polynomials is proposed to suitably approximate the displacement field of inflected nano-beams. Selected case studies of applicative interest are investigated, and the approximated nonlinear solutions of nonlocal nano-beams are detected. The demonstrated benchmark examples can be effectively employed in the assessment of beam-type elements of nano-electromechanical systems.

Chebyshev spectral methods for multi-order fractional neutral pantograph equations

This paper is concerned with the application of the spectral tau and collocation methods to delay multi-order fractional differential equations with vanishing delay $$rx\ (0<r<1)$$ . The fractional derivatives are described in the Caputo sense. The model solution is expanded in terms of Chebyshev polynomials. The convergence of the proposed approaches is investigated in the weighted $$L^2$$ -norm. Numerical examples are provided to highlight the convergence rate and the flexibility of this approach. Our results confirm that nonlocal numerical methods are best suited to discretize fractional differential equations as they naturally take the global behavior of the solution into account.

Stability Analysis of MHD Fluid Flow over a Moving Plate with Pressure Gradient Using the Chebyshev Spectral Method

The purpose of this paper is to investigate the magnetohydrodynamic stability external flow of a viscous, incompressible and electrically conducting fluid over a moving flat plate using temporal linear stability analysis. Using a similarity variables based on the Skan-Falkner transformation, the governing differential equations of mean flow are transformed into a nonlinear ordinary differential equation, which is then solved numerically by the Runge-Kutta method. The MHD stability equation is solved numerically by using the Chebyshev spectral collocation method, which is based on the eigenfunction expansion in terms of Chebyshev polynomials, collocation points, and the subsequent solution of the resulting generalized eigenvalue problem with the QZ algorithm. The influence of the pressure gradient, magnetic field and wall velocity on the dimensionless mean velocity are presented graphically and discussed. For the disturbance flow in the presence of magnetic field and moving wall, the critical Reynolds number increases with increasing the magnetic field parameter and wall velocity, indicating that these parameters stabilize the flow.

Complexity of the circulant foliation over a graph

In the present paper, we investigate the complexity of infinite family of graphs \(H_n=H_n(G_1,\,G_2,\ldots ,G_m)\) obtained as a circulant foliation over a graph H on m vertices with fibers \(G_1,\,G_2,\ldots ,G_m.\) Each fiber \(G_i=C_n(s_{i,1},\,s_{i,2},\ldots ,s_{i,k_i})\) of this foliation is the circulant graph on n vertices with jumps \(s_{i,1},\,s_{i,2},\ldots ,s_{i,k_i}.\) This family includes the family of generalized Petersen graphs, I-graphs, sandwiches of circulant graphs, discrete torus graphs and others. We obtain a closed formula for the number \(\tau (n)\) of spanning trees in \(H_n\) in terms of Chebyshev polynomials, investigate some arithmetical properties of this function and find its asymptotics as \(n\rightarrow \infty .\)

Relations between single and repeated qubit gates: coherent error amplification for high-fidelity quantum-gate tomography

In quantum computation the target fidelity of the qubit gates is very high, with the admissible error being in the range from 10−3 to 10−4 and even less, depending on the protocol. The direct experimental determination of such a tiny error is very challenging by standard quantum-process tomography. Instead, the method of randomized benchmarking, which uses a random sequence of Clifford gates, has become a standard tool for determination of the average gate error as the decay constant in the exponentially decaying fidelity. In this paper, the task for determining a tiny error is addressed by sequentially repeating the same gate multiple times, which leads to the coherent amplification of the error, until it reaches large enough values to be measured reliably. If the transition probability is p = 1 − ϵ with ϵ ≪ 1 in the single process, then classical intuition dictates that the probability after N passes should be . However, this classical expectation is misleading because it neglects interference effects. This paper presents a rigorous theoretical analysis based on the SU(2) symmetry of the qubit propagator, resulting in explicit analytic relations that link the N-pass propagator to the single-pass one in terms of Chebyshev polynomials. In particular, the relations suggest that in some special cases the N-pass transition probability degrades as , i.e. dramatically faster than the classical probability estimate. In the general case, however, the relation between the single-pass and N-pass propagators is much more involved. Recipes are proposed for unambiguous determination of the gate errors in the general case for both Clifford and non-Clifford gates.