Modified Chebyshev Polynomials Research Articles

Spectral Solutions of Specific Singular Differential Equations Using A Unified Spectral Galerkin-Collocation Algorithm

This paper presents a novel numerical approach to addressing three types of high-order singular boundary value problems. We introduce and consider three modified Chebyshev polynomials (CPs) of the third kind as proposed basis functions for these problems. We develop new derivative operational matrices for the three modified CPs of the third kind by deriving formulas for their first derivatives. Our approach follows a unified method for numerically handling singular differential equations (DEs). To transform these equations into algebraic systems suitable for numerical treatment, we employ the collocation method in combination with the introduced operational matrices of derivatives of the modified CPs of the third kind. We address the convergence examination for the three expansions in a unified manner. We present numerous numerical examples to demonstrate the accuracy and efficiency of our unified numerical approach.

Spectral Treatment of High-Order Emden–Fowler Equations Based on Modified Chebyshev Polynomials

This paper is devoted to proposing numerical algorithms based on the use of the tau and collocation procedures, two widely used spectral approaches for the numerical treatment of the initial high-order linear and non-linear equations of the singular type, especially those of the high-order Emden–Fowler type. The class of modified Chebyshev polynomials of the third-kind is constructed. This class of polynomials generalizes the class of the third-kind Chebyshev polynomials. A new formula that expresses the first-order derivative of the modified Chebyshev polynomials in terms of their original modified polynomials is established. The establishment of this essential formula is based on reducing a certain terminating hypergeometric function of the type 5F4(1). The development of our suggested numerical algorithms begins with the extraction of a new operational derivative matrix from this derivative formula. Expansion’s convergence study is performed in detail. Some illustrative examples of linear and non-linear Emden–Flower-type equations of different orders are displayed. Our proposed algorithms are compared with some other methods in the literature. This confirms the accuracy and high efficiency of our presented algorithms.

An improved design method for even-degree optimum allpole filters with equiripple passband responses

The maximum power transfer between source and load is extremely important in analog signal processing. The main disadvantage of signal processing using the even-degree Chebyshev filter is the impossibility of LC ladder network realization between two equal resistors. Therefore, either the utilization of the ideal transformer or the Chebyshev filter of odd-degree is required to achieve the maximum power transfer. To overcome this limitation, the class of even-degree polynomials with the equiripple behavior, referred to as scaled modified Chebyshev polynomials, can be used as the filter’s characteristic function. Passband ripples of obtained filters can be controlled through an embedded parameter, while normalized half-power frequency is kept equal to unity. Furthermore, these filters can be optimized for the minimum amount of reflected power at the input terminals. In this paper, the optimum modified Chebyshev filters are first compared to the classic Chebyshev filters to show that both filters exhibit similar performances. Second, proposed filters are compared to the Least Square Monotonic (LSM) filters that are optimized by minimizing the area under the characteristic function in the passband. Results of comparison reveal that the proposed even-degree scaled modified Chebyshev filters outperform the LSM filters, as is already proved for the odd-degree filters.

A comprehensive study on the coupled multi-mode vibrations of cylindrical shells

As is discussed in the previous work (Dong et al. 2021), the functions of beam-like modes are not suitable for simulating the cylindrical shells with the clamped and free ends. Also, ignoring the influences of the in-plane inertias can lead to large relative differences of the solutions of vibration characteristics. This work aims to accurately calculate the vibration characteristics of the cylindrical shells under various classical boundary conditions comprised of movable simply-supported, immovable simply-supported, sliding, clamped and free ends. Meanwhile, this work deals with the nonlinear coupled multi-mode vibrations of the cylindrical shell under different boundary conditions with the in-plane inertias taken into consideration. Firstly, several modified Chebyshev polynomials in terms of the axial coordinate and new harmonic functions in terms of the circumferential coordinate are employed to establish the equations of natural frequency in the frame of Lagrange equations for all the single-modes, where the polynomials are independent of geometries and material properties of the shells. And then, the circumferential-wave-dependent mode functions of the cylindrical shells are proposed, based on these mode functions coming from the linear vibration, a unified nonlinear differential equation of motion of the cylindrical shells under different boundary conditions is established for the first endeavor, in which the coupled multi-mode vibrations containing various vibration waves are considered. Finally, the high dimensional differential equation with the square and cubic nonlinearities is investigated by employing the incremental harmonic balance method. Results shown that polynomials accurately satisfy various boundary conditions, and the contribution of the single-mode to the coupled multi-mode vibration is related to the loading patterns.

On the analytical and meshless numerical approaches to mixture stress gradient functionally graded nano-bar in tension

The mixture stress gradient theory of elasticity is conceived via consistent unification of the classical elasticity theory and the stress gradient theory within a stationary variational framework. The boundary-value problem associated with a functionally graded nano-bar is rigorously formulated. The constitutive law of the axial force field is determined and equipped with proper non-standard boundary conditions. Evidences of well-posedness of the mixture stress gradient problems, defined on finite structural domains, are demonstrated by analytical analysis of the axial displacement field of structural schemes of practical interest in nano-mechanics. An effective meshless numerical approach is, moreover, introduced based on the proposed stationary variational principle while employing autonomous series solution of the kinematic and kinetic field variables. Suitable mathematical forms of the coordinate functions are set forth in terms of the modified Chebyshev polynomials, satisfying the required classical and non-standard boundary conditions. An excellent agreement between the numerical results of the axial displacement field of the functionally graded nano-bar and the analytical solution counterpart is confirmed on the entire span of the nano-sized bar, in terms of the mixture parameter and the stress gradient characteristic parameter. The effectiveness of the established meshless numerical approach, demonstrating a fast convergence rate and an admissible convergence region, is hence ensured. The established mixture stress gradient theory can effectively characterize the peculiar size-dependent response of functionally graded structural elements of advanced ultra-small systems.

New operational matrices approach for optimal control based on modified Chebyshev polynomials


 
 
 
 The purpose of this paper is to introduce interesting modified Chebyshev orthogonal polynomial. Then, their new operational matrices of derivative and integration or modified Chebyshev polynomials of the first kind are introduced with explicit formulas. A direct computational method for solving a special class of optimal control problem, named, the quadratic optimal control problem is proposed using the obtained operational matrices. More precisely, this method is based on a state parameterization scheme, which gives an accurate approximation of the exact solution by utilizing a small number of unknown coefficients with the aid of modified Chebyshev polynomials. In addition, the constraint is reduced to some algebraic equations and the original optimal control problem reduces to optimization technique, which can be solved easily, and the approximate value of the performance index is calculated. Moreover, special attention is presented to discuss the convergence analysis and an upper bound of the error for the presented approximate solution is derived. Finally, some important illustrative examples of obtained results are shown and proved that powerful method in a simple way to get an optimal control of the considered.
 
 
 

Using Chebyshev Polynomials in Solving Diffusion Equations

In this work, we modified Chebyshev polynomials of the first kind to match the characteristics of a second order differential equation that is a result of a separation of the variable technique used to solve a partial differential diffusion equation. The resultant polynomials solution in one spatial dimension is found and the corresponding changing parameters for the first order differential equation in time are extracted accordingly. The method is tested for applicability, stability, and convergence.

A robust spectral treatment of a class of initial value problems using modified Chebyshev polynomials

New modified shifted Chebyshev polynomials (MSCPs) have been constructed over the interval [α, β]. These polynomials are utilized as basis functions with the application of the spectral collocation method. The operational matrix of integer order derivatives of these polynomials is introduced. The elements of this matrix are explicitly given. The introduced operational matrix along with the collocation method is used to find a direct solver of linear/nonlinear class of IVPs. Furthermore, the convergence and error analysis of the modified Chebyshev expansion are discussed. Some specific numerical examples are given to ascertain the wide applicability and the good efficiency of the suggested algorithm. The obtained results from the tested numerical examples are convincing. Also, the introduced approximate solutions are very close to the analytical ones.

The application of modified Chebyshev polynomials in asymmetric cryptography

Based on Chebyshev polynomials, you can create an asymmetric cryptosystem that allows secure communication. Such a cryptosystem uses the fact that these polynomials form a semi-group due to the composition operation. This article presents new cryptosystems that use other than semi-group property dependencies. Based on these dependencies as well as modifications of Chebyshev's polynomials, two cryptosystems have been proposed. The presented analysis shows that their security is the same as in the case of algorithms associated with the problem of discrete logarithms. The article also shows methods that allow faster calculation of Chebyshev polynomials.

Model reduction of aerothermodynamic for hypersonic aerothermoelasticity based on POD and Chebyshev method

Accurate modeling of aerothermodynamics with low computational cost takes on a crucial role for the optimization and control of hypersonic vehicles. This study examines three reduced-order models (ROMs) to provide a reliable and efficient alternative approach for obtaining the aerothermodynamics of a hypersonic control surface. Coupled computational fluid dynamics (CFD) and computational thermostructural dynamics (CTSD) approaches are used to generate the snapshots for ROMs considering the interactions between aerothermodynamics, structural dynamics and heat transfer. One ROM adopts a surrogate approach named Kriging. The second ROM is constructed by the combination of Proper Orthogonal Decomposition (POD) and Kriging, namely, POD-Kriging. The accuracy of Kriging-based ROM is higher than that of POD-Kriging-based ROM, but the efficiency is lower. Therefore, to address the shortcomings of the above two approaches, a new ROM is developed that is composed of POD and modified Chebyshev polynomials, namely, POD-Chebyshev. The ROM based on POD-Chebyshev has the best precision and efficiency among the three ROMs and generally has less than 2% average maximum error for the studied problem.

Multi-variable regression methods using modified Chebyshev polynomials of class 2

So far, many regression works have been implemented by using linear regression methods. Although more accurate predictions results could be obtained, polynomial regression is not used as much as compared to linear regression in real applications due to occurrence of coefficient explosion. To overcome this problem, two regression algorithms using Chebyshev polynomials of class 2 based on cascade regression and feature selection are proposed in this paper. In the experimental part, three separate experiments including function interpolation and real-case regression were conducted on three datasets to test the proposed algorithms. As shown by the experimental results, the proposed algorithms performed better than other regression methods in terms of both accuracy and processing time.

Integro-differential nonlocal theory of elasticity

The second-order integro-differential nonlocal theory of elasticity is established as an extension of the Eringen nonlocal integral model. The present research introduces an appropriate thermodynamically consistent model allowing for the higher-order strain gradient effects within the nonlocal theory of elasticity. The thermodynamic framework for third-grade nonlocal elastic materials is developed and employed to establish the Helmholtz free energy and the associated constitutive equations. Establishing the minimum total potential energy principle, the integro-differential conditions of dynamic equilibrium along with the associated classical and higher-order boundary conditions are derived and comprehensively discussed. A rigorous formulation of the third-grade nonlocal elastic Bernoulli–Euler nano-beam is also presented. A novel series solution based on the modified Chebyshev polynomials is introduced to examine the flexural response of the size-dependent beam. The proposed size-dependent beam model is demonstrated to reveal the stiffening or softening flexural behaviors, depending on the competitions of the characteristic length-scale parameters. The higher-order gradients of strain fields are illustrated to have more dominant effects on the beam stiffening.

Synthesis of Highly Directive Endfire Arrays Using Modified Chebyshev Polynomials

In this paper an additional parameter in the Chebyshev polynomial is introduced so as to design an endfire array by controlling two factors of the radiation characteristics- namely the Half Power Beam Width(HPBW) and the side lobe level(SLL) simultaneously. It has been shown that for a given SLL improved HPBW and First Null Beam Width (FNBW) can be achieved. Individual elements in the array are assumed to be isotropic. Simulations were carried out to find bounds on the choice of HPBW and SLL. Simulation results also shows that the directivity can be improved over the Dolph-Chebyshev array. Improvement in the roll off characteristics are also shown. Implementation of the array using the coefficients obtained by this synthesized procedure can result in a highly directive array.

Automated knowledge based filter synthesis using modified Chebyshev polynomials of the first kind

The paper presents the automated design of active RC and programmable filters. The approximating function is derived using Chebyshev orthogonal polynomials of the first kind. Optimization is performed using symbolic manipulation of expressions inputted into computer algebra system. The code is in the form of template notebook, and the user should specify the minimal number of numeric values, such as the number of second-order filter sections, the pass-band edge frequency, the reflection coefficient, and the preferred values of component values.

Dimensional Reductions for the Computation of Time–Dependent Quantum Expectations

We consider dimension reduction techniques for the Liouville–von Neumann equation for the evaluation of the expectation values in a mixed quantum system. We describe several existing methods that have appeared in the literature, showing the failure of them when the system is scaled up. We introduce a new method termed direct expectation values via Chebyshev based on evaluation of a trace formula combined with a direct expansion in modified Chebyshev polynomials. This reduction is highly efficient and does not destroy any information. We demonstrate the practical application of the scheme for a nuclear spin system and compare with popular alternatives. In nuclear spin dynamics the main goal for simulations is being able to simulate a system with as many spins as possible; for this reason it is very important to have an efficient method that scales the least with respect to the number of particles. This method may be applied to autonomous quantum problems where the desired outcome of quantum simulation, rather than being a full description of the system, is only the expectation value of some observables.

Some identities for the moments of random variables having a linear rescaling the Student's t distribution

Taking advantage of Nadarajah's representation of modified Chebyshev polynomials by means of the random variable that has the Student's t distribution [S. Nadarajah, On modified Chebyshev polynomials, J. Math. Anal. Appl. 334 (2007) 1492–1494] the scope of this paper is the generation of the identities for the moments of linear rescaling random variables that have the Student's t distribution.

On modified Chebyshev polynomials

Two elegant representations are derived for the modified Chebyshev polynomials discussed by Witula and Slota [R. Witula, D. Slota, On modified Chebyshev polynomials, J. Math. Anal. Appl. 324 (2006) 321–343].

Linear stability of flow in an internally heated rectangular duct

The linear stability of flow in a vertical rectangular duct subject to homogeneous internal heating, constant-temperature no-slip walls and a driving pressure gradient is investigated numerically in the framework of the Boussinesq hypothesis. A Galerkin method based upon modified Chebyshev polynomials is used to discretize the linearized Navier-Stokes equations in their most compact form, i.e. eliminating the pressure and the streamwise velocity and making use of the intrinsic symmetry properties. A classification of the basic flow in Grashof and Reynolds space in terms of inflectional properties is proposed. It is found that the flow loses stability at all aspect ratios for a combination of finite thermal buoyancy and pressure forces with opposed signs. In the square duct, the unstable region lies inside the range where the basic velocity profile exhibits additional inflection lines. Unstable eigenfunctions are obtained for all basic symmetry modes, consisting of rapidly propagating large-scale structures located in the vicinity of the inflection lines near the centre of the duct.

On modified Chebyshev polynomials

In this paper some new properties and applications of modified Chebyshev polynomials and Morgan–Voyce polynomials will be presented. The aim of the paper is to complete the knowledge about all of these types of polynomials.

Acoustical wave propagator with modified Chebyshev expansion

This paper presents an improved expansion scheme for the acoustical wave propagator (AWP). This expansion employs a set of modified Chebyshev polynomials with the Bessel functions of the first kind as its expansion coefficients. As a result, the previous requirement of operator splitting in the time domain is removed, and the convergence property of the AWP scheme is significantly improved.